diff --git a/papers/project_paper_1_relativity/paper_1_relativity.md b/papers/project_paper_1_relativity/paper_1_relativity.md new file mode 100644 index 00000000..03a10c26 --- /dev/null +++ b/papers/project_paper_1_relativity/paper_1_relativity.md @@ -0,0 +1,37 @@ +--- +title: "Research Paper: The Thermodynamic Bias Toward Manifolds in Causal Sets: Mean-Field Prerequisites for Lorentz Invariance (Letter)" +date: "2026-06-01T08:00:00Z" +draft: false +tags: ["#research", "physics", "intellecton"] +--- + +**Abstract:** The extraction of the Minkowski metric from discrete causal graphs is complicated by the Kleitman-Rothschild (KR) order collapse. We introduce a thermodynamic partition function governed by the discrete Benincasa-Dowker action augmented with a non-local volume penalty. By evaluating the partition function using a mean-field approximation, we explicitly calculate the critical topological temperature $\beta_c$ and demonstrate a thermodynamic phase transition that strictly suppresses highly entropic non-manifold KR-orders. This establishes a rigorous statistical mechanical prerequisite for the emergence of macroscopic Lorentz invariance. + +## The Partition Function and Mean-Field Phase Transition +Let $\Omega_N$ be the space of causal sets of $N$ elements. The canonical partition function is: + + + +$$ +Z = \sum_{\mathcal{C} \in \Omega_N} e^{-S_{BD}(\mathcal{C}) - \beta V(\mathcal{C})} +$$ + +where $S_{BD}$ is the Benincasa-Dowker action. The volume penalty $V(\mathcal{C}) = \sum_{x \prec y} | \{ z \in \mathcal{C} \mid x \prec z \prec y \} |$. + +To calculate the phase transition, we employ a mean-field approximation. Let $p$ be the probability of a relation $x \prec y$. For a generic KR-order, $p \approx 1/4$, yielding a highly connected graph where the expected volume penalty scales as $\langle V_{KR} \rangle \approx c_1 N^3 p^2$. For a manifold-like causal set sprinkled into $D$-dimensional Minkowski space, relations are sparse, and $\langle V_{man} \rangle \approx c_2 N^2$. + +The free energy $F(\beta) = - \frac{1}{\beta} \ln Z$ is determined by the competition between the entropy of KR-orders $S_{KR} \sim \frac{N^2}{4} \ln 2$ and the energy of the volume penalty. Evaluating the saddle point of the mean-field partition function: + + + +$$ +Z \approx \int dp \, e^{N^2 \left( \frac{\ln 2}{4} - \beta c_1 N p^2 \right)} +$$ + +we find a critical inverse temperature $\beta_c \propto \frac{\ln 2}{c_1 N}$. For $\beta \gt \beta_c$, the extensive $\mathcal{O}(N^3)$ energetic penalty dominates the $\mathcal{O}(N^2)$ entropy, driving a first-order topological phase transition. The system collapses into the sparse, manifold-like phase ($\langle V \rangle \propto N^2$), suppressing KR-orders and permitting emergent Lorentz invariance. + +## References + +- **[Surya2019]** S. Surya, *Living Rev. Relativ.* **22**, 5 (2019). +- **[Kleitman1975]** D. Kleitman, B. Rothschild, *Trans. Am. Math. Soc.* **205**, 205 (1975). + diff --git a/papers/project_paper_1_relativity/paper_1_relativity.pdf b/papers/project_paper_1_relativity/paper_1_relativity.pdf new file mode 100644 index 00000000..5ba8c6f5 --- /dev/null +++ b/papers/project_paper_1_relativity/paper_1_relativity.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6d22f3db39d26465a3576a0a6a175cda0c9e89514533f2f8923dde0b70d13c9f +size 170642 diff --git a/papers/project_paper_1_relativity/paper_1_relativity.tex b/papers/project_paper_1_relativity/paper_1_relativity.tex new file mode 100644 index 00000000..4d904c85 --- /dev/null +++ b/papers/project_paper_1_relativity/paper_1_relativity.tex @@ -0,0 +1,37 @@ +\documentclass[11pt,a4paper]{article} +\usepackage[utf8]{inputenc} +\usepackage{amsmath,amssymb,amsfonts,amsthm} +\usepackage{cite} + +\title{The Thermodynamic Bias Toward Manifolds in Causal Sets: Mean-Field Prerequisites for Lorentz Invariance (Letter)} +\author{Antigravity} +\date{\today} + +\begin{document} +\maketitle + +\begin{abstract} +The extraction of the Minkowski metric from discrete causal graphs is complicated by the Kleitman-Rothschild (KR) order collapse. We introduce a thermodynamic partition function governed by the discrete Benincasa-Dowker action augmented with a non-local volume penalty. By evaluating the partition function using a mean-field approximation, we explicitly calculate the critical topological temperature $\beta_c$ and demonstrate a thermodynamic phase transition that strictly suppresses highly entropic non-manifold KR-orders. This establishes a rigorous statistical mechanical prerequisite for the emergence of macroscopic Lorentz invariance. +\end{abstract} + +\section{The Partition Function and Mean-Field Phase Transition} +Let $\Omega_N$ be the space of causal sets of $N$ elements. The canonical partition function is: +\begin{equation} +Z = \sum_{\mathcal{C} \in \Omega_N} e^{-S_{BD}(\mathcal{C}) - \beta V(\mathcal{C})} +\end{equation} +where $S_{BD}$ is the Benincasa-Dowker action. The volume penalty $V(\mathcal{C}) = \sum_{x \prec y} | \{ z \in \mathcal{C} \mid x \prec z \prec y \} |$. + +To calculate the phase transition, we employ a mean-field approximation. Let $p$ be the probability of a relation $x \prec y$. For a generic KR-order, $p \approx 1/4$, yielding a highly connected graph where the expected volume penalty scales as $\langle V_{KR} \rangle \approx c_1 N^3 p^2$. For a manifold-like causal set sprinkled into $D$-dimensional Minkowski space, relations are sparse, and $\langle V_{man} \rangle \approx c_2 N^2$. + +The free energy $F(\beta) = - \frac{1}{\beta} \ln Z$ is determined by the competition between the entropy of KR-orders $S_{KR} \sim \frac{N^2}{4} \ln 2$ and the energy of the volume penalty. Evaluating the saddle point of the mean-field partition function: +\begin{equation} +Z \approx \int dp \, e^{N^2 \left( \frac{\ln 2}{4} - \beta c_1 N p^2 \right)} +\end{equation} +we find a critical inverse temperature $\beta_c \propto \frac{\ln 2}{c_1 N}$. For $\beta > \beta_c$, the extensive $\mathcal{O}(N^3)$ energetic penalty dominates the $\mathcal{O}(N^2)$ entropy, driving a first-order topological phase transition. The system collapses into the sparse, manifold-like phase ($\langle V \rangle \propto N^2$), suppressing KR-orders and permitting emergent Lorentz invariance. + +\bibliographystyle{plain} +\begin{thebibliography}{10} +\bibitem{Surya2019} S. Surya, \textit{Living Rev. Relativ.} \textbf{22}, 5 (2019). +\bibitem{Kleitman1975} D. Kleitman, B. Rothschild, \textit{Trans. Am. Math. Soc.} \textbf{205}, 205 (1975). +\end{thebibliography} +\end{document} diff --git a/papers/project_paper_1_relativity/references/Kleitman1975.pdf b/papers/project_paper_1_relativity/references/Kleitman1975.pdf new file mode 100644 index 00000000..505d9bae --- /dev/null +++ b/papers/project_paper_1_relativity/references/Kleitman1975.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:298a5ac8852984c5c5944bb3b98bf180039b0e8d3391d063f8209cd873b1f685 +size 1303615 diff --git a/papers/project_paper_1_relativity/references/Kleitman1975.txt b/papers/project_paper_1_relativity/references/Kleitman1975.txt new file mode 100644 index 00000000..64f5357d --- /dev/null +++ b/papers/project_paper_1_relativity/references/Kleitman1975.txt @@ -0,0 +1,864 @@ +TRANSACTIONS OF THE +AMERICAN MATHEMATICAL SOCIETY +Volume 205, 1975 + +ASYMPTOTIC ENUMERATION OF PARTIAL ORDERS +ON A FINITE SET + +BY + +D. J. KLEITMANi1) AND B. L. ROTHSCHILD(2) + +ABSTRACT. By considering special cases, the number Pn of partially + +ordered sets on a set of n elements is shown to be (1 + 0(\¡n))Qn, +where + +Qn is the number of partially ordered sets in one of the special classes. The + +number Qn can be estimated, +and we ultimately +obtain +'»-('+0©)C5lC)(7)<''-^-^)- + +1. Introduction. It is known [7] that the logarithm (base 2, as all logarithms +in this paper will be) of the number Pn of partial orders (or equivalently of T0 +topologies) on a set of tj elements is T22/4 + o(tj2). In this paper we show that +Pn is asymptotically equal to Qn, the number of partial orders in a certain special +class which is characterized in a simple way. It will follow that log Pn = «2/4 + +3n/2 + 0(log tj). An explicit asymptotic formula for Pn will be given, but it is a +bit messy. +In [5] the number Gn of "graded" partially ordered sets is enumerated. +Since the partial orders counted by Qn turn out to be graded, the number arrived +at in [5] is also asymptotically equal to Pn. The computation of Gn [5], then, +is asymptotically applicable to Pn. +The methods used here are similar to those used in [7] for obtaining the +asymptotic estimate for log Pn, but here they are somewhat more delicate and +more complicated. We use induction to show that Pn < Qn(l + 0(1/«)), while +Qn < Pn by definition. The proof is accomplished by obtaining all partial orders +on tj + 1 elements from those on tj or fewer elements in certain specified ways. +In each of these ways, except those corresponding to Qn + x, we obtain only an +asymptotically negligible number of partial orders. +The partial orders corresponding to Qn can be described as follows. They +consist of three sets Lx, L2, L3 with \LX\, \L3\ = tj/4 + o(n), \L2\ = ti/2 + o(tj). +Each element in L¡ "covers" only elements in L¡_x (see below for definitions). +And finally, each element behaves in the "average" way. That is, each element in + +Received by the editors November 2, 1973 and, in revised form, January 21, 1974. +AMS (MOS) subject classifications (1970). Primary 05A15. +(x) Work supported in part by ONR Grant 0014-67-A-0204-0063. +(2) Work supported in part by NSF Grant GP-33580X. +Copyright © 1975, American Mathematical Society +205 + + +206 +D. J. KLEITMAN AND B. L. ROTHSCHILD + +L¡ is covered by (asymptotically) half of those in Li+l and covers half of those +inL¡_x. + +2. Terminology. We represent a partial order P on a set of tj elements by +its unique Hasse diagram, also denoted by P. This is a directed graph with the +elements of P as vertices and a single directed edge from a to ft if and only if a +covers b in P (a covers ft if a > ft and a > c>b +implies c = ft). Distinct partial +orders have distinct diagrams. Thus we let Pn denote both the number of partial +orders and the number of diagrams on n elements, as directed graphs diagrams +are characterized by the exclusion of two types of configurations. Namely, a +directed graph is a diagram of a partial order if and only if it contains no set of +(directed) edges Ex, E2, • • • , Ek, E0 such that the terminal vertex of E¡ is the +initial vertex of El+X, 1 m(l + m~3/8)/2. Let + +Bm + i = \B(V)\. +D(V): Diagrams with a (v, ß>set with \C(Q)\ < t?j(1 - ttj~3/8)/2. Let + +Dm + l = \D(V)l +E(V): Diagrams with at least 30 nonempty levels. Let Em + X = |Zr(I0l. +F(V): Diagrams with a (v, Q)-set with \C(Q)\ > m(l - m~3,8)/2, such that +the smallest set R of vertices incident with every edge connecting two vertices in +V- (M U C(Q)) satisfies \R\ > 2m3/4. Let Fm + X = \F(V)\. +G(V): Diagrams not in E(V) with a (v, Q)-set satisfying ttj(1 - ttj~3/8)/2 < +\C(Q)\ < ttj(1 + T72-3/8)/2; with a set R of at most 2ttj3/4 vertices the deletion +of which leaves no edge connecting two vertices of V— ({v} U C(Q) U R); and +such that the smallest set S of vertices incident with all edges of C(Q) satisfies +\S\>2m1'8. +LetGm + 1 = \G(V)\- +H(V): Diagrams P satisfying the following properties: There is a set TQ of +at most 3T7i7yl8 +vertices such that P - T0 is bipartite; the parts U and W of P - T0 +have at least ttj/2 - 3ttj7/8 and at most t?j/2 + 3tt27/8 vertices; P- T0 has at +most 29 levels, Lx, L2, • • • , Lk, k < 29; there is some level L¡ and a set {x, y} +in P - T0 such that {x, y} is a good set in P - T0 and L¡ n C(x) n C(y) = 0; + + +208 +D. J. KLEITMAN AND B. L. ROTHSCHILD + +either \L¡ CtU\> m15'16 and {x, y} C W, or \L¡ DW\> +m15/16 and {x, y} E U. +UtHm + x = \H(V)\. +I(V): Diagrams P not in A(V) with a set T of at most 102ttj1s/16 vertices +satisfying the following properties: P- T = Lx\l L2V L3 where Lx, L2, L3 are +the levels of P- T (L3 possibly empty); both \LX U L3\ and \L2\ are between +ttj/2 - 102T?jls/16 and ttj/2 + 102ttj1s/16; every two vertices in W = L2 have a +common adjacent vertex in U= Lx U L3, and vice versa; there is a t E T adjacent +both to a vertex x in Lx U L3 and a vertex y EL2. +Let Im + X = \I(V)\. +J(V): Diagrams P with a set T of at most 102ttj15/16 vertices satisfying the +following properties: P - T is a three level bipartite diagram with parts U and W; +ttj/2 - 102m15/16 < \U\ v +all of the following in- +equalities hold: +(1) log (An + JPn) „)„_1)„) 2«1/2 of them) adjacent to +u, u either covers or is covered by all vertices in a set ß of [tj1'2] vertices. Thus +we have a (v, Q)-set in P. +By B and D, \C(Q)\ must be between tj(1 - n-3/8)/2 and «(1 + tj_3/8)/2. +By F and G there are sets R and S with IK| < 2tj3/4 and \S\ < 2tj7/8 such that +P - (R U S) has no edges between two vertices of V - ({v} U C(Q) U R U S) = +W', or between two vertices of (C(Q) U {v}) - (R U S) = U'. Thus P-(RUS) +is bipartite with parts U' and W' and at most 29 levels. \R U S\ < 3tj7/8. +Suppose there is a pah xx, yx of vertices such that [xx, yx} is a bad set in +P - (R U S). Then consider P - (R U S U {xx, yx}). At least (tj - 3tj7/8)1/2 of +the vertices of P - (R U S U {xv yx}) are in lower levels than in P - (R U S). +Now suppose {x2, y2} is bad in P - (R U S U {xx,yx}). Consider P - +(fiUSU +{xx, yx, x2, y2}). At least (ti - 3tj7/8)1/2 vertices in P - (R U S U +{xx, yx, x2, y2}) are in lower levels than in P - (R U S U {xx, yx}). We con- +tinue this process one step at a time, choosingxx, yv x2, y2, • • • , xm, ym with +{f/> y¡} a bad set in P - (R U 5 U [xx, yx, • " , x*v y¡-X})- We proceed until +we can choose no more bad pairs {x, y}. For/ >,•_,} +I +< 3tj7/8. Since each vertex can decrease its level at most 28 times during this +process, and since tj3/4(tj - 3n1ls)xl2 > 28(tt + 1), the total number ttj of steps +before the process stops is at most tj3/4. +Let T0 = R U S U {xx, yx, • • • ,xm, ym}. Then P - T0 is bipartite with +parts U' - T0 and W' - T0, where |y0| < 3tj7/8, and \W' - T0\ and \U' - T0\ +are between tj/2 - 3tt7/8 and tt/2 + 3tj7/8. Let the levels of P - T0 be Lx, L2, +" ' , Lk, k < 29. Let ft be the largest value of/ such that either \L, n U'\ > +tj15/16 or \L¡ n W'\ >nxs'16, +say \Lh n U'\ > tj1s/16. + +Then we claim ft < 3. For suppose h> 4. Let u>x>z>.y +be in levels +ft, ft - 1, ft - 2, ft - 3, respectively, with u E Lh C\ U'. Then rj6IC' +and +z G ZJ', since P - T0 is bipartite. Now by /Z, C(x) n C( v) DLH^0; +let +«' G 0(x) n C(>») n /,,-. Then «' >x > z > j, and u' also coversy, since it is + + +ENUMERATION OF PARTIAL ORDERS +211 + +adjacent and in a higher level. But then u, x, z,y form an excluded configura- +tion, a contradiction. Hence ft < 3. +For Z = 1, 2, 3, we claim that if \L¡ n U'\ > nxs/X6 (respectively \L¡ n W\ +> Tils/16), then L, n W' = 0 (respectively L¡ n U' = 0). For let x e /,,. n W' +(respectively x E L¡ n Z7'). By H, as above, x must be adjacent to some vertex +in L¡ H U' (respectively L¡ n W1), a contradiction since no two vertices in L¡ can +be adjacent. Thus if \L¡\ > 2tj15/16, L¡ must be entirely in U' or entirely in W'. +Since \U' - T0\ and |W' - T0\ are both at least u/2 - 3tj7/8, and since +\LÁ < 2t21S/16 for/ > 3, we must have at least one of Lx, L2, L3 entirely in U' +with at least tj15'16 vertices, and one entirely in W' with at least tj15'16 vertices. +In particular, L2 is entirely in W' or entirely in U'. For if not, Lx would be +entirely in U' or entirely in W'. But then L2 would be entirely in W' or entirely +in U', since every vertex in L2 is adjacent to some vertex of Lx. Similarly, since +L2 E W' or L2 Ç U', we must have L3 E U' or L3 Ç W'. Now either \LX\ < +2nX5/X6 or Lx EU' oiLxE +W'. Let T= T0 U \Jj>3L¡ U ¿x if \LX\< 2nxs/l6, +and T=T0U +\Jj>3L¡ ii \Lx\>2nX5IX(> (and thus Lx Çf/'orZ,, +ç W')- +iri < 60TJ15/16, and /> - T is bipartite with parts U = U'- T and W = +W - T, with \U\ and |W| between n/2 - 60«ls/16 and n/2 + ÓOtj15'16. P-T +has two or three levels, Lx, L2, L3 (where L3 =0 in the two level case). Finally +U = Lx U ¿3, W = Z,2, or Í/ = L2, IV = Zj U ¿3. We assume, without loss of +generality, ¿,U¿3= +U, L2 = W. By construction of U and W, and by //, every +two vertices of U axe adjacent to a common vertex of W, and vice versa. Also +|¿1|>2tj15/16. + +By /, no vertex in T is adjacent both to vertices of W and U. Hence T is +divided into two subsets, Ttj and Tw, where C(TW) C\W = 0. 0(7^) n {/ = 0. +By /, no two vertices of Tu are adjacent, and no two vertices of Tw are adjacent. +Thus P itself is bipartite with parts U U Tv and W U 7V +Suppose Z G T, x, y G /,,-, i=l,2 +or 3, and t is adjacent to x and to y. +Then either t covers both x and y or f is covered by both. For if t covers x and +V covers t, by construction of Z7 and W we can let u be a vertex in L¡_x or Li+X +to which both x and y are adjacent. Then t, x, y, v form an excluded configura- +tion, a contradiction. +Suppose t E T is adjacent to vertices in x and y in different levels of P - T. +Then these levels must be Lx and L3 (by /). x and .y are adjacent to a common +vertex v in L2. Let x be the vertex in Lx, y E L3. Then y covers v, v covers x, +and we must have that y covers t and r covers x, or else x, ,y, u, í form an ex- +cluded configuration. So if t is adjacent to vertices in Lx and L3, it covers those +in Lx and is covered by those in Ly +All vertices of T, therefore, are in one of the following subsets: + + +212 +D. J. KLEITMAN AND B. L. ROTHSCHILD + +T¡+ = {t \t covers only vertices in L¡ and is covered by none in V - T}, + +TJ = {t 11 is covered only by vertices in L¡ and covers none in V - T} + +for i = 1, 2, 3, + +T2 = {t\t covers vertices in Lx and is covered by vertices in L3, and is + +adjacent to no others in V-T}. + +We now claim that + +P = (77) V (Lx U T2) V (L2 U Tx+ U T3 U T°) V (L3 U T+) V (7-+). + +To show this we must show that no vertex of T3 is adjacent to any in T2, no +vertex of Tx~ is adjacent to any in T2 , and all other adjacencies are in the "right +direction." (We already know that there are no edges between two vertices of +rf U Tx+ U 77 U T3+ U T% or between two vertices of T2 U T2+.) +If x S r^, y S 77, are adjacent, there can be no edge between C(x)■- T +and C(y) - T or we would have an excluded configuration. Thus by K, x and .y +cannot be adjacent. Similarly, x ETX and y ET2 cannot be adjacent. +Now suppose x E T¡~, y E TJ+ x, 1 = 1 or 2. If x coversy, C(x) - T and +C(y) - T must have no edge between them, or an excluded configuration would +result. Hence, by K, x cannot cover y. Similarly, if x E T¡, y E T¡~+x, Z = 1 or +2, then y cannot cover x. +Finally, if x E T2 and y E T2 (oty E T2, x ET2, resp.) and if x covers +y (respectively x is covered by y), then no vertex of C(x) - T is adjacent to any +vertex of C(y) - T, or an excluded configuration would result. Thus, by K, x +cannot cover y (respectively y cannot cover x). This completes the elimination of +all possible connections which would contradict the claim. Thus we have shown +that + +P = (77) V (77 U Lx) V (L2 U Tx+ U 77 U r2°) V (L2 U 7+) V T2/2-102T215/16, +we must have |Z,3| n/2-n31'32; +or if J+ # 0, \L3\ > n/2-n31'32. +By M, neither of these possibilities occurs. Hence Tx = T3 = 0. +This gives us P = Sx V 52 V 53, where SX=LXU +T2, S2=I2ur2°U +Ti+UT3, 53 = L3ur2+. +Finally, then, by AT and 0, P must be in Q(V). In +particular, since every vertex of S2 covers some (at least tj/8 - tj7'8) vertices of + + +ENUMERATION OF PARTIAL ORDERS +213 + +Sj, and every vertex of S3 covers some (at least tt/4 - t27/8) vertices of S2, the +S¡ are in fact levels in P. Part I of the proof of the theorem is now complete. + +5. Proof of theorem. II. We shall use the results of the last section, together with +the lemma, to show that.P„ <(1 + 0(l/72))ß„. First we show thatP„ <(1 + 0(llri))Xn. +Let v be the number guaranteed by the lemma, and let N = max(2i>, 109). +Let C0 be a number large enough so that Pn < (1 + (C0)¡n)Xn for all n „ < (1 + C/n)Xn for alln. +The proof of this claim is by induction on tj. For tj < N it is true by choice +of C We assume that it is true for all tj < ttj, for some m>N, and show that +Bm +1 ^ 0 + C/(m + l))Xm + x as well. Since, by the last section, we have Pm + x < +Am + i +Bm + i +^m + i +•" ++ #«+i ++ Xm+l> we need onlV ^ow that +(Am + i + "' ++Nm + OI(Xm + 0 < cKm + 0- To do this we sha11 employ the + +inequalities of the lemma to show that each of the terms Am + X¡(Xm + j), +Bm + xliXm + x), • • • , Nm + xl(Xm + x) is at most 1/13 • C/(ttj + 1) (there are 13 +terms here). These arguments are all similar, and we illustrate a few typical ones. + +(D^B+l.ds+i +^L^S- +<2-/4(l + C/TT2)2-'"/2 ++ sl0^<-4l +• ^, +Xm + X *m XmXm + \ +772+1 1J + +Dm + l _ Dm+1 +m-jm1/2] + m-[mxl2] +Xm + +Xm + 1 Pm-\mW\Xm-\m*tl\ +Xm-[mlf2\+l +Xfn + 1 + +(3) +<2m3l2l2-'Am9l8il ++_Ç._\ +2-(m/2-lm1/2]/2)([m1/2l ++ l) +i/2-%m9/8f/] +m — \wi * +[m1'2] + +. 2S([m1/2] +1)log TTJ + +<^ +C +13 TT7 ++ r + +(13) + +Nm + 1 + +Xm + X +<2_(,ogm)2/6> bythelemmaj + +N^ +aiSO) by the +definitions of 0(V), X(V) and Q(V), we have Xn+X = On + x + Qn+l. These +facts lead to + + +214 +D. J. KLEITMAN AND B. L. ROTHSCHILD + +< f 1 + 1 )ßn + i for « sufficiently large. + +This establishes .Pn = (1 + 0(l¡n))Qn and completes the proof of the +theorem. + +6. Proof of lemma. We let V be a set of tt + 1 vertices. We recall our +convention, that all statements in inequalities asserted below are meant to be +valid only for n sufficiently large. More specifically, for each statement below +there is a number N' such that the statement is valid for tj > N'. We can then +let v be the maximum of all of these N'. The numbered paragraphs below cor- +respond to the numbered inequalities of the lemma. +For the sake of brevity, we include only a few of the cases in detail. The +rest of them are represented only by the final inequalities of the arguments, from +which one can obtain a hint as to the order in which things are constructed. We +choose one simple case, and the most complicated ones to do in detail. +(1) This inequality is Lemma 2 of [7]. It is proved like those below. +(2) This and (3) below correspond to Lemmas 3 and 4 of [7]. +We obtain all diagrams in B(V) from diagrams on tj vertices by choosing +v E V, choosing a diagram on V - {v}, and then adjoining v to the diagram so +as to satisfy the conditions for B(V). We will obtain upper bounds for the num- +ber of possible choices by counting some possibilities which cannot satisfy the +conditions for B(V) as well as all those that do. This is the general method used +below, where instead of just choosing a single v, we may need to choose a sub- +set S E V, a diagram on V - S, and then to adjoin S to the diagram. +To obtain diagrams in B(V), then, we choose vE V (n + 1 ways to +choose v) and a diagram on V - {v} (at most Pn ways to do this). Now v will +be taken as the vertex of a (v, ß)-set. +To connect v, we first choose a level for v to be in (at most « + 1 possibil- +ities). Then we choose ß (at most „C[„i/2j ways, where many of the sets of [tj1'2] +vertices in V - {it} included in this number will not be valid candidates for ß, +depending on which diagram was chosen for V - {v}). The directions of the +connections between v and ß are now determined, because in order to be in + + +ENUMERATION OF PARTIAL ORDERS +215 + +B(V), the levels of vertices in ß must be unaffected by the addition of it. Hence +v covers all vertices of ß if its level is higher than all of them, and is covered by +all vertices if its level is lower. (One of these two possibilities must occur, or we +would have an invalid choice for Q.) +Since ß will satisfy conditions for B(V), we have \C(Q)\ > n(l + n~3/s)/2. +Since there are no triangles, v can be connected to at most «(1 - ti_3/8)/2 remain- +ing vertices. Since v must be a good vertex, at most tj1'2 of these vertices can +have then levels affected by the addition of v. The vertices which can have their +levels changed can be chosen in at most + +'"¿n'(Kl-»-'">«l)<«W2?''' +/=o ^ +l + +1 /2 +ways, v can then be connected to these vertices in at most 3" ways (the 3 is +for the choices: v covers x, x covers it, and x and v not connected). Finally, the +other vertices to which v can be connected can be chosen in at most 2"*1-" ^2 +ways. The directions of these connections are determined by the levels relative to +the level of v. All connections of it are now completed. Multiplying all these +numbers of possible choices gives the following: + +log (^A + < l0g(TJ + 1) + l0g(7J + 1) + log ("/2] + ++ log (72(72/2)" 1/2) + nl'2l0g +3 + 72(1 -T2"3/8)/2 + +< 2 log (77 + 1) + T21/2 log TJ + log TJ + T21/2 log Tl/2 + ++ TJ1/2 log 3+ T2/2- 72S/8/2 + +<«/2-TI5/8/4. + +This completes (2). We used one of two basic relations here, which will be used +repeatedly below. + +.eg (;,,,) +<„° iog. + +The other is Stirling's formula or the normal approximation, which gives +(recall, for tj sufficiently large, here depending on the ß): + +log ( " J < - T2(oi +log a + (1 - a) log(l - a)) + +for 0 < ß < a < lA, and ß fixed (independent of ti). + + +216 +D. J. KLEITMAN AND B. L. ROTHSCHILD + +loS (r „ ' ^M,1 < " - M™2 for 0 < ot < 0 < 1, and 0 fixed. +\[tj(1 -a)/2]J + + 2"2/4 (also +from paragraph (14) below). Thus + +log|/(HI/P„<-ri33/17/500. + +We obtain diagrams in l"(V) as follows: We choose t (n + 1 ways) and +a diagram on V- {t} so that there is a set T' with T = T' U {t} satisfying con- +ditions for I(V) (at most Pn ways, and at most + +1„1S/16]J +Jl027 + +ways to choose T'). Then, since every two vertices in U (respectively, W) are +adjacent to a common vertex of W (respectively, Z7), t can be connected to each +level in only one direction or an excluded configuration results (at most 23 choices +for directions for t). í can be connected to r'at +most 3lo2"ls'16 +ways. To +satisfy the conditions on I(V) there must be vertices x E U and y E W adjacent +to t (at most ti2 ways to choose x and y, and 4 ways to connect them to t). +By A, t must be adjacent to tj/64 other vertices at least, and of those to at +least T216/17 in U, or n16'11 in W, say W. There are at most + +1ti/2 + 102tj15/16]\ + +[„16/17! +) + +ways to choose a set S of [ti16'17] vertices in W to be included in C(t) n W. +But by the conditions for l"(V), \C(S) C\ U\> n/2 - ti/300. x must thus be in +U - (C(S) n U) and C(x) n W must be at least nl65, by A. The remaining +vertices of C(t) must be chosen from ((W-S)- +C(x)) U(U- +C(S)), of which +there are at most + +(n/2 + lOV5'16 +- [ir16'17] -tj/65 +TJ/2 + lOV5'16 +-n/2 + ri/300) + +-1"/21-["/41(2["/41 _ tfn/2] +ways. We only get diagrams counted by Xn here, and we get no diagrams twice. +Thus + +log X„ > log (^ + + log (" " j^1) + + [T2/2] (72 - [T2/2] - [T2/4] ) + ++ [n/2] [72/4] + [n/2] log(l - 1/2Í"/4!) + ++ (n - [n/2] - [tj/4]) log(l - 1/2Í"/2») + +>l0g(l/T2 • 2") + l0g(l/T2 • 2"/2) + ++ (TJ - [ti/2])[t2/2] - 1 > T22/4 + 3T2/2 - 3 log 72. + +We find an upper bound for Xn equally easily. Let V be a set of tt elements. +Diagrams in X(V') are obtained as follows. We choose S2 (at most 2" ways); Sx +from V - S2 (at most 2"/2+log " ways); and connections from Sx U S3 to S2 (at +most 2" I4 ways). This gives + +log(X„) < ti2/4 + 3n/2 + log n. + +Together with the lower bound for Xn+X we get \og(Xn/Xn + x) < - n/2 + 5 log 72. + +(12) +log(Mn + x)<(n + l)2/4 + (tj + 1) + 2 log« + 4«31/32log«. + +From the lower bound on!n + 1 we get log(M„ + X/Xn + x) <-n/4. + + +ENUMERATION OF PARTIAL ORDERS +219 + +(13) We obtain diagrams in N(V) by considering two cases. In the first case, +we suppose that not both of the inequalities (72 + l)/2 - log tj < \S2\ < (n + l)/2 ++ log « hold. This gives a number N'n+X of choices with + +^og(N'n+1)<2(n + 1)+ 1 -\S2\ + \S2\(n + 1 -\S2\) + +O3) +^(T2 + l)2 , 3(72 4-1) 1 +<■ 4 ■ 2 + +In the second case, we assume that + +fOogTî)2 + +T2 + 1 . +^ ip i ^72 + 1 , , +—2-log +n < |521 <—2— +1°g"> + +and that either + +ls«l>ZHrL + »1/2log7j or |sy + +for 1 = 1 or 3. This gives a number N¡¡+ x of choices with + +l0g«+1) += 2 + (72 + 1) + log TJ + (T2 + l)2/4 + +[(« + l)/2 + log 72] +(14) ++ log , +\[(" + l)/44-«1/2logn] + +< (72 + l)2/4 + 3(tj + l)/2 - y4 (log TJ)2. + +We get + +n+X/ +\ +Xn + X +108 O +<* ""fe" +<- 5 <*■>*• + +log -fr1" < l0g(» + 1) + 25t21/2 log T2 + log TJ +"n + +(15) +8\U(« ++ l)/4 + 727/8] + ++ 2(«+1)/4+ni/2log„ + 1A(" + 1)/4+«1/2/1°g"A\ +log " V [(» + D/8 + «7/8] )) + +* 2 IO" ' + +7. Proof of corollary. For a set K of tj vertices, let 7(10 De the class of +diagrams P such that P = LXV L2V L3, where Lx, L2, L3 are the levels of P. + + +220 +D. J. KLEITMAN AND B. L. ROTHSCHILD + +Let Yn = \Y(V)\. We obtain all diagrams in Y(V) by first choosing Lx, then L2 +and then connecting L2 to Lx and L3 to L2. There are exactly (2IL2' - 1) ways +to connect each vertex in L3 to L2 (there must be at least one connection since +L3 is a level), and exactly (2 * - 1) ways to connect each vertex of L2 to Lv +This gives + +Yn - t (") Z (" y '") (2'' - D'C - I)""'"-''. + +Since Y(V) > Q(V), we have + +Y„ 2. While it is simple to see that the generators of +GLor preserve the chronological structure so that GLor ⊆ GA, the converse is +not obvious. In his proof Zeeman showed that every fa ∈ GA maps light rays +to light rays, such that parallel light rays remain parallel and moreover that +the map is linear. In Minkowski spacetime every chronological automorphism +is also a causal automorphism, so a Corollary to Zeeman’s theorem is that the +group of causal automorphisms is isomorphic to GLor. This is a remarkable +result, since it states that the physical invariants associated with Md follow +naturally from its causal structure poset (Md, ≺) where ≺ denotes the causal +relation on the event-set Md. + +Kronheimer and Penrose (1967) subsequently generalised Zeeman’s ideas +to an arbitrary causal spacetime (M, g) where they identified both (M, ≺) +and (M, ≺≺) with the event-set M, devoid of the differential and topological +structures associated with a spacetime. They defined an abstract causal space +axiomatically, using both (M, ≺) and (M, ≺≺) along with a mixed transitivity +condition between the relations ≺ and ≺≺, which mimics that in a causal +spacetime. +Zeeman’s result in Md was then generalised to a larger class of spacetimes +by Hawking et al (1976) and Malament (1977). A chronological bijection gener- +alises Zeeman’s chronological automorphism between two spacetimes (M1, g1) +and (M2, g2), and is a chronological order preserving bijection, + +fb : M1 → M2, +x ≺≺1 y ⇔ fb(x) ≺≺2 fb(y), ∀ x, y ∈ M1, +(2) + +2 Hence the term “pseudo-Riemannian”. +3 Zeeman used the term “causal” instead of “chronological”, but we will follow the more +modern usage of these terms (Hawking and Ellis 1973; Wald 1984). + + +The causal set approach to quantum gravity +9 + +where ≺≺1,2 refer to the chronology relations on M1,2, respectively. The ex- +istence of a chronological bijection between two strongly causal spacetimes4 + +was equated by Hawking et al (1976) to the existence of a conformal isometry, +which is a bijection f : M1 → M2 such that f, f −1 are smooth (with respect to +the manifold topology and differentiable structure) and f∗g1 = λg2 for a real, +smooth, strictly positive function λ on M2. Malament (1977) then generalised +this result to the larger class of future and past distinguishing spacetimes.5 We +refer to these results collectively as the Hawking–King–McCarthy–Malament +theorem or HKMM theorem, summarised as + +Theorem 1 Hawking–King–McCarthy–Malament (HKMM) +If a chronological bijection fb exists between two d-dimensional spacetimes +which are both future and past distinguishing, then these spacetimes are con- +formally isometric when d > 2. + +It was shown by Levichev (1987) that a causal bijection implies a chrono- +logical bijection and hence the above theorem can be generalised by replacing +“chronological” with “causal”. Subsequently Parrikar and Surya (2011) showed +that the causal structure poset (M, ≺) of these spacetimes also contains in- +formation about the spacetime dimension. +Thus, the causal structure poset (M, ≺) of a future and past distinguishing +spacetime is equivalent its conformal geometry. This means that (M, ≺) is +equivalent to the spacetime, except for the local volume element encoded in +the conformal factor λ, which is a single scalar. As phrased by Finkelstein +(1969), the causal structure in d = 4 is therefore (9/10)th of the metric! +En route to a theory of quantum gravity one must pause to ask: what +“natural” structure of spacetime should be quantised? Is it the metric or is +it the causal structure poset? The former can be defined for all signatures, +but the latter is an exclusive embodiment of a causal Lorentzian spacetime. In +Fig. 2, we show a 3d projection of a non-Lorentzian and non-Riemannian d = 4 +“space-time” with signature (−, −, +, +). The fact that a time-like direction +can be continuously transformed into any other while still remaining time-like +means that there is no order relation in the space and hence no associated +causal structure poset. We can thus view the causal structure poset as an +essential embodiment of Lorentzian spacetime. +Perhaps the first explicit statement of intent to quantise the causal struc- +ture of spacetime, rather than the spacetime geometry was by Kronheimer +and Penrose (1967), who listed, as one of their motivations for axiomatising +the causal structure: + +“To admit structures which can be very different from a manifold. The +possibility arises, for example, of a locally countable or discrete event- + +4 A point p in a spacetime is said to be strongly causal if every neighbourhood of p contains +a subneighbourhood such that no causal curve intersects it more than once. All the events +in a strongly causal spacetime are strongly causal. +5 These are spacetimes in which the chronological past and future I±(p) of each event p +is unique, i.e., I±(p) = I±(q) ⇒ p = q. + + +10 +Sumati Surya + +Timelike + +Spacelike + +Null + +Timelike + +Fig. 2 An example of a signature (−, −, +, +) spacetime with one spatial dimension sup- +pressed. It is not possible to distinguish a past from a future timelike direction and hence +order events, even locally. + +space equipped with causal relations macroscopically similar to those of +a space-time continuum.” + +This brings to focus another historical thread of ideas important to CST, +namely that of spacetime discreteness. The idea that the continuum is a math- +ematical construct which approximates an underlying physical discreteness +was already present in the writings of Riemann as he ruminated on the phys- +icality of the continuum (Riemann 1873): + +“Now it seems that the empirical notions on which the metric determ- +inations of Space are based, the concept of a solid body and that of a +light ray; lose their validity in the infinitely small; it is therefore quite +definitely conceivable that the metric relations of Space in the infinitely +small do not conform to the hypotheses of geometry; and in fact one +ought to assume this as soon as it permits a simpler way of explaining +phenomena.” + +Many years later, in their explorations of spacetime and quantum the- +ory, Einstein and Feynman each questioned the physicality of the continuum +(Stachel 1986; Feynman 1944). These ideas were also expressed in Finkelstein’s +“spacetime code” (Finkelstein 1969), and most relevant to CST, in Hemion’s +use of local finiteness, to obtain discreteness in the causal structure poset +(Hemion 1988). This last condition is the requirement there are only a finite +number of fundamental spacetime elements in any finite volume Alexandrov +interval A[p, q] ≡ I+(p) ∩ I−(q). + + +The causal set approach to quantum gravity +11 + +Although these ideas of spacetime discreteness resonate with the appear- +ance of discreteness in quantum theory, the latter typically manifests itself +as a discrete spectrum of a continuum observable. The discreteness proposed +above is different: one is replacing the fundamental degrees of freedom, before +quantisation, already at the kinematical level of the theory. +The most immediate motivation for discreteness however comes from the +HKMM theorem itself. The missing (1/10)th of the d = 4 metric is the volume +element. A discrete causal set can supply this volume element by substitut- +ing the continuum volume with cardinality. This idea was already present in +Myrheim’s remarkable (unpublished) CERN preprint (Myrheim 1978), which +contains many of the main ideas of CST. Here he states: + +“It seems more natural to regard the metric as a statistical property +of discrete spacetime. Instead we want to suggest that the concept of +absolute time ordering, or causal ordering of, space-time points, events, +might serve as the one and only fundamental concept of a discrete space- +time geometry. In this view space-time is nothing but the causal ordering +of events.” + +The statistical nature of the poset is a key proposal that survives into CST +with the spacetime continuum emerging via a random Poisson sprinkling. We +will see this explicitly in Sect. 3. Another key concept which plays a role in the +dynamics is that the order relation replaces coordinate time and any evolution +of spacetime takes meaning only in this intrinsic sense (Sorkin 1997). +There are of course many other motivations for spacetime discreteness. +One of the expectations from a theory of quantum gravity is that the Planck +scale will introduce a natural cut-off which cures both the UV divergences +of quantum field theory and regulates black hole entropy. The realisation of +this hope lies in the details of a given discrete theory, and CST provides us a +concrete way to study this question, as we will discuss in Sect. 5. +It has been 31 years since the original CST proposal of BLMS (Bombelli +et al 1987). The early work shed considerable light on key aspects of the theory +(Bombelli et al 1987; Bombelli and Meyer 1989; Brightwell and Gregory 1991) +and resulted in Sorkin’s prediction of the cosmological constant Λ (Sorkin +1991). There was a seeming hiatus in the 1990s, which ended in the early +2000s with exciting results from the Rideout–Sorkin classical sequential growth +models (Rideout and Sorkin 2000b, 2001; Martin et al 2001; Rideout 2001). +There have been several non-trivial results in CST in the intervening 19 odd +years. In the following sections we will make a broad sketch of the theory and +its key results, with this historical perspective in mind. + +3 The causal set hypothesis + +We begin with the definition of a causal set: + +Definition: A set C with an order relation ≺ is a causal set if it is + + +12 +Sumati Surya + +1. Acyclic: x ≺ y and y ≺ x ⇒ x = y, ∀x, y ∈ C +2. Transitive: x ≺ y and y ≺ z ⇒ x ≺ z, ∀x, y, z ∈ C +3. Locally finite: ∀x, y ∈ C, |I[x, y]| < ∞, where I[x, y] ≡ Fut(x) ∩ Past(y) , + +where |.| denotes the cardinality of the set, and6 + +Fut(x) ≡ {w ∈ C|x ≺ w, x ̸= w} + +Past(x) ≡ {w ∈ C|w ≺ x, x ̸= w}. +(3) + +We refer to I[x, y] as an order interval, in analogy with the Alexandrov inter- +val in the continuum. The acyclic and transitive conditions together define a +partially ordered set or poset, while the condition of local finiteness encodes +discreteness. + +Fig. 3 The transitivity condition x ≺ y, y ≺ z ⇒ x ≺ z is satisfied by the causality relation +≺ in any Lorentzian spacetime. + +The content of the HKMM theorem can be summarised in the statement: + +Causal Structure + Volume Element = Lorentzian Geometry, +(4) + +which lends itself to a discrete rendition, dubbed the “CST slogan”: + +Order + Number ∼ Lorentzian Geometry. +(5) + +One therefore assumes a fundamental correspondence between the number of +elements in a region of the causal set and the continuum volume element that +it represents. The condition of local finiteness means that all order intervals + +6 These are the exclusive future and past sets since they do not include the element itself. + + +The causal set approach to quantum gravity +13 + +Fig. 4 The Hasse diagrams of some simple finite cardinality causal sets. Only the nearest +neighbour relations or links are depicted. The remaining relations are deduced from trans- +itivity. + +in the causal set are of finite cardinality and hence correspond in the con- +tinuum to finite volume. This CST slogan captures the essence of the (yet to +be specified) continuum approximation of a manifold-like +causal set, which +we denote by C ∼ (M, g). While the continuum causal structure gives the +continuum conformal geometry via the HKMM theorem, the discrete causal +structure represented by the underlying causal set is conjectured to approx- +imate the entire spacetime geometry. Thus, discreteness supplies the missing +conformal factor, or the missing (1/10)th of the metric, in d = 4. +Motivated thus, CST makes the following radical proposal (Bombelli et al + +1987): + +1. Quantum gravity is a quantum theory of causal sets. +2. A continuum spacetime (M, g) is an approximation of an underlying causal +set C ∼ (M, g), where +(a) Order ∼ Causal Order +(b) Number ∼ Spacetime Volume + +In CST, the kinematical space of d = 4 continuum spacetime geometries or +histories is replaced with a sample space Ω of causal sets. Thus, discreteness is +viewed not only as a tool for regulating the continuum, but as a fundamental +feature of quantum spacetime. Ω includes causal sets that have no continuum +counterpart, i.e., they cannot be related via Conditions (2a) and (2b) to any +continuum spacetime in any dimension. These non-manifold-like causal sets +are expected to play an important role in the deep quantum regime. In order +to make this precise we need to define what it means for a causal set to be +manifold-like, i.e., to make precise the relation “C ∼ (M, g)”. + + +14 +Sumati Surya + +Before doing so, it is important to understand the need for a continuum +approximation at all. Without it, Condition (1) yields any quantum theory +of locally finite posets: one then has the full freedom of choosing any poset +calculus to construct a quantum dynamics, without need to connect with the +continuum. Examples of such poset approaches to quantum gravity include +those by Finkelstein (1969) and Hemion (1988), and more recently Cortˆes and +Smolin (2014). What distinguishes CST from these approaches is the critical +role played by both causality and discrete covariance which informs the choice +of the dynamics as well the physical observables. In particular, condition (2) is +the requirement that in the continuum approximation these observables should +correspond to appropriate continuum topological and geometric covariant ob- +servables. +What do we mean by the continuum approximation Condition (2)? We +begin to answer this by looking for the underlying causal set of a causal space- +time (M, g). A useful analogy to keep in mind is that of a macroscopic fluid, for +example a glass of water. Here, there are a multitude of molecular-level con- +figurations corresponding to the same macroscopic state. Similarly, we expect +there to be a multitude of causal sets approximated by the same spacetime +(M, g). And, just as the set of allowed microstates of the glass of water depends +on the molecular size, the causal set microstate depends on the discreteness +scale Vc, which is a fundamental spacetime volume cut-off.7 + +Since the causal set C approximating (M, g) is locally finite, it represents +a proper subset of the event-set M. An embedding is the injective map + +Φ : C �→ (M, g), +x ≺C y ⇔ Φ(x) ≺M Φ(y), +(6) + +where ≺C and ≺M denote the order relations in C and M respectively. Not +every causal set can be embedded into a given spacetime (M, g). Moreover, +even if an embedding exists, this is not sufficient to ensure that C ∼ (M, g) +since only Condition (2a) is satisfied. In addition to correlate the cardinality of +the causal set with the spacetime volume element, Condition (2b), the embed- +dings must also be uniform with respect to the spacetime volume measure of +(M, g). A causal set is said to approximate a spacetime C ∼ (M, g) at density +ρc = V −1 +c +if there exists a faithful embedding + +Φ : C �→ M, +Φ(C) is a uniform distribution in (M, g) at density ρc, +(7) + +where by uniform we mean with respect to the spacetime volume measure of +(M, g). +The uniform distribution at density ρc ensures that every finite spacetime +volume V is represented by a finite number of elements n ∼ ρcV in the causal +set. It is natural to make these finite spacetime regions causally convex, so +that they can be constructed from unions of Alexandrov intervals A[p, q] in +(M, g). However, we must ensure covariance, since the goal is to be able to +recover the approximate covariant spacetime geometry. This is why Φ(C) is + +7 The most obvious choice for Vc is the Planck volume, but we will not require it at this +stage. + + +The causal set approach to quantum gravity +15 + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +� + +Fig. 5 The lightcone lattice in d = 2. The lattice on the left looks “regular” in a fixed +frame but transforms into the “stretched” lattice on the right under a boost. The n ∼ ρcV +correspondence cannot be implemented as seen from the example of the Alexandrov interval, +which contains n = 7 lattice points in the lattice in the left but is empty after a boost. + +required to be uniformly distributed in (M, g) with respect to the spacetime +volume measure. It is obvious that a “regular” lattice cannot do the job since +it is not regular in all frames or coordinate systems. Hence it is not possible +to consistently assign n ∼ ρcV to such lattices (see Fig. 5). +The issue of symmetry breaking is of course obvious even in Euclidean +space. Any regular discretisation breaks the rotational and translational sym- +metry of the continuum. In the lattice calculations for QCD, these symmetries +are restored only in the continuum limit, but are broken as long as the dis- +creteness persists. In Christ et al (1982) it was suggested that symmetry can +be restored in a randomly generated lattice where there lattice points are uni- +formly distributed via a Poisson process. This has the advantage of not picking +any preferred direction and hence not explicitly breaking symmetry, at least +on average. We will discuss this point in greater detail further on. +Set in the context of spacetime, the Poisson distribution is a natural choice +for Φ(C), with the probability of finding n elements in a spacetime region of +volume v given by + +Pv(n) = (ρcv)n + +n! +exp−ρcv . +(8) + +This means that on the average + +⟨n⟩ = ρcv, +(9) + +where n is the random variable associated with the random causal set Φ(C). +This distribution then gives us the covariantly defined n ∼ ρcV correspondence +we seek.8 + +8 Since Φ(C) is a random causal set, any function of F : C → R is therefore a random +variable. + + +16 +Sumati Surya + +In a Poisson sprinkling into a spacetime (M, g) at density ρc one selects +points in (M, g) uniformly at random and imposes a partial ordering on these +elements via the induced spacetime causality relation. Starting from (M, g), we +can then obtain an ensemble of “microstates” or causal sets, which we denote +by C(M, ρc), via the Poisson sprinkling.9 Each causal set thus obtained is a +realisation, while any “averaging” is done over the whole ensemble. +We say that a causal set C is approximated by a spacetime (M, g) if C can +be obtained from (M, g) via a high probability Poisson sprinkling. Conversely, +for every C ∈ C(M, ρc) there is a natural embedding map + +Φ : C �→ M , +(10) + +where Φ(C) is a particular realisation in C(M, ρc). In Fig. 6, we show a causal +set obtained by Poisson sprinkling into d = 2 de Sitter spacetime. + +Fig. 6 A Poisson sprinkling into a portion of 2d de Sitter spacetime embedded in M3. The +relations on the elements are deduced from the causal structure of M3. + +That there is a fundamental discrete randomness even kinematically is not +always easy for a newcomer to CST to come to terms with. Not only does +CST posit a fundamental discreteness, it also requires it to be probabilistic. +Thus, even before coming to quantum probabilities, CST makes us work with +a classical, stochastic discrete geometry. +Let us state some obvious, but important aspects of Eq. (8). Let Φ : C �→ +(M, g) be a faithful embedding at density ρc. While the set of all finite volume + +9 C(M, ρc) explicitly depends on the spacetime metric g, which we have suppressed for +brevity of notation. + + +The causal set approach to quantum gravity +17 + +regions10 v possess on average ⟨n⟩ = ρcv elements of C,11 the Poisson fluctu- +ations are given by δn = √n. Thus, it is possible that the region contains no +elements at all, i.e., there is a “void”. An important question to ask is how +large a void is allowed, since a sufficiently large void would have an obvious +effect on our macroscopic perception of a continuum. If spacetime is unboun- +ded, as it is in Minkowski spacetime, the probability for the existence of a void +of any size is one. Can this be compatible at all with the idea of an emergent +continuum in which the classical world can exist, unperturbed by the vagaries +of quantum gravity? +The presence of a macroscopic void means that the continuum approxima- +tion is not realised in this region. A prediction of CST is then that the emergent +continuum regions of spacetime are bounded both spatially and temporally, +even if the underlying causal set is itself “unbounded” or countable. Thus, +a continuum universe is not viable forever. However, since the current phase +of the observable universe does have a continuum realisation one has to ask +whether this is compatible with CST discretisation. In Dowker et al (2004) +the probability for there to be at least one nuclear size void ∼ 10−60m4 was +calculated in a region of Minkowski spacetime which is the size of our present +universe. Using general considerations they found that the probability is of +order 1084 × 10168 × e−1072, which is an absurdly small number! Thus, CST +poses no phenomenological inconsistency in this regard. +An example of a manifold-like causal set C which is obtained via a Poisson +sprinkling into a 2d causal diamond is shown in Fig. 7. A striking feature of +the resulting graph is that there is a high degree of connectivity. In the Hasse +diagram of Fig. 7 only the nearest neighbour relations or links are depicted +with the remaining relations following from transitivity. e ≺ e′ ∈ C is said to +be a link if ∄ e′′ ∈ C such that e′′ ̸= e, e′ and e ≺ e′′ ≺ e′. In a causal set that +is obtained from a Poisson sprinkling, the valency, i.e., the number of nearest +neighbours or links from any given element is typically very large. This is an +important feature of continuum like causal sets and results from the fact that +the elements of C are uniformly distributed in (M, g). For a given element +e ∈ C, the probability of an event x ≻ e to be a link is equal to the probability +that the Alexandrov interval A[e, x] does not contain any elements of C. Since + +PV (0) = e−ρcV , +(11) + +the probability is significant only when V ∼ Vc. As shown in Fig. 8, in Md, +the set of events within a proper time ∝ (V )1/d to the future (or past) of a +point p lies in the region between the future light cone and the hyperboloid +−t2 + Σix2 +i ∝ (V )2/d, with t > 0. Up to fluctuations, therefore, most of the +future links to e lie within the hyperboloid with V = Vc ± √Vc. This is a non- +compact, infinite volume region and hence the number of future links to e is +(almost surely) infinite. Since linked elements are the nearest neighbours of e, + +10 We assume that these are always causally convex. +11 Henceforth we will identify Φ(C) with C, whenever Φ is a faithful embedding. + + +18 +Sumati Surya + +● + +● + +● + +● + +● + +● + +● + +● + +● + +● +● + +● + +● + +● + +● + +● + +● + +● + +● + +● + +● + +● +● + +● + +● + +● + +● + +● + +● + +● + +● +● + +● + +● + +● +● + +● + +● + +● + +● + +● + +● + +● + +● + +● + +● + +● + +● + +● + +● + +10 + +20 + +30 + +40 + +50 +U + +10 + +20 + +30 + +40 + +50 + +V + +Fig. 7 A Hasse diagram of a causal set that faithfully embeds into a causal diamond in M2. +In a Hasse diagram only the nearest neighbour relations or links are shown. The remaining +relations follow by transitivity. + +this means the valency of the graph C is infinite. It is this feature of manifold- +like causal sets which gives rise to a characteristic “non-locality”, and plays a +critical role in the continuum approximation of CST, time and again. +The Poisson distribution is not the only choice for a uniform distribu- +tion. A pertinent question is whether a different choice of distribution is pos- +sible, which would lead to a different manifestation of the continuum approx- +imation. In Saravani and Aslanbeigi (2014), this question was addressed in +some detail. Let C ∼ (M, g) at density ρc. Consider k non-overlapping Al- +exandrov intervals of volume V in (M, g). Since C is uniformly distributed, +⟨n⟩ = ρcV . The most optimal choice of distribution, is also one in which +the fluctuations δn/⟨n⟩ = +� + +⟨(n − ⟨n⟩)2⟩/⟨n⟩ are minimised. This ensures +that C is as close to the continuum as possible. For the Poisson distribution +δn/⟨n⟩ = 1/ +� + +⟨n⟩ = 1/√ρcV . Is this as good as it gets? It was shown that for +d > 2, and under certain further technical assumptions, the Poisson distribu- +tion indeed does the best job. Strengthening these results is important as it +can improve our understanding of the continuum approximation. + +3.1 The Hauptvermutung or fundamental conjecture of CST + +An important question is the uniqueness of the continuum approximation as- +sociated to a causal set C. Can a given C be faithfully embedded at density +ρc into two different spacetimes, (M, g) and (M ′, g′)? We expect that this is +the case if (M, g) and (M ′, g′) differ on scales smaller than ρc, or that they + + +The causal set approach to quantum gravity +19 + +Fig. 8 The valency or number of nearest neighbours of an element in a causal set obtained +from a Poisson sprinkling into M2 is infinite. + +are, in an appropriate sense, “close” (M, g) ∼ (M ′, g′). Let us assume that a +causal set can be identified with two macroscopically distinct spacetimes at +the same density ρc. Should this be interpreted as a hidden duality between +these spacetimes, as is the case for example for isospectral manifolds or mirror +manifolds in string theory (Greene and Plesser 1991)? The answer is clearly in +the negative, since the aim of the CST continuum approximation is to ensure +that C contains all the information in (M, g) at scales above ρ−1 +c . Macroscopic +non-uniqueness would therefore mean that the intent of the CST continuum +approximation is not satisfied. +We thus state the fundamental conjecture of CST: +The Hauptvermutung of CST: C can be faithfully embedded at density +ρc into two distinct spacetimes, (M, g) and (M ′, g′) iff they are approximately +isometric. +By an approximate isometry , (M, g) ∼ (M ′, g′) at density ρc, we mean +that (M, g) and (M ′, g′) differ only at scales smaller than ρc. Defining such an +isometry rigorously is challenging, but concrete proposals have been made by +Bombelli (2000); Noldus (2004, 2002); Bombelli and Noldus (2004); Bombelli +et al (2012), en route to a full proof of the conjecture. Because of the technical +nature of these results, we will discuss it only very briefly in the next section, +and instead use the above intuitive and functional definition of closeness. +Condition (1) tells us that the kinematic space of Lorentzian geometries +must be replaced by a sample space Ω of causal sets. Let Ω be the set of all + + +20 +Sumati Surya + +countable causal sets and H the set of all possible Lorentzian geometries, in all +dimensions. If ∼ denotes the approximate isometry at a given ρc, as discussed +above, the quotient space H/∼ corresponds to the set of all continuum-like +causal sets Ωcont ⊂ Ω at that ρc. Thus, causal sets in Ω correspond to Lorent- +zian geometries of all dimensions! Couched this way, we see that CST dynamics +has the daunting task of not only obtaining manifold-like causal sets in the +classical limit, but also ones that have dimension d = 4. + +As mentioned in the introduction, the sample space of n element causal sets +Ωn is dominated by the KR posets depicted in Fig. 9 and are hence very non- +manifold-like (Kleitman and Rothschild 1975). A KR poset has three “layers” +(or abstract “moments of time”), with roughly n/4 elements in the bottom and +top layer and such that each element in the bottom layer is related to roughly +half those in the middle layer, and similarly each element in the top layer is +related to roughly half those in the middle layer. The number of KR posets +grows as ∼ 2n2/4 and hence must play a role in the deep quantum regime. Since +they are non-manifold-like they pose a challenge to the dynamics, which must +overcome their entropic dominance in the classical limit of the theory. Even +if the entropy from these KR posets is suppressed by an appropriate choice +of dynamics, however, there is a sub-dominant hierarchy of non-manifold-like +posets (also layered) which also need to be reckoned with (Dhar 1978, 1980; +Promel et al 2001). + +Fig. 9 A Kleitman–Rothschild or KR poset. + + +The causal set approach to quantum gravity +21 + +Closely tied to the continuum approximation is the notion of “coarse grain- +ing”. Given a spacetime (M, g) the set C(M, ρc) can be obtained for different +values of ρc. Given a causal set C which faithfully embeds into (M, g) at ρc, +one can then coarse grain it to a smaller subcausal set C′ ⊂ C which faith- +fully embeds into (M, g) at ρ′ +c < ρc. A natural coarse graining would be via a +random selection of elements in C such that for every n elements of C roughly +n′ = (ρ′ +c/ρc)n elements are chosen. Even if C itself does not faithfully embed +into (M, g) at ρc, it is possible that a coarse graining of C can be embed- +ded. This would be in keeping with our sense in CST that the deep quantum +regime need not be manifold-like. One can also envisage manifold-like causal +sets with a regular fixed lattice-like structure attached to each element similar +to a “fibration”, in the spirit of Kaluza–Klein theories. Instead of the coarse +graining procedure, it would be more appropriate to take the quotient with +respect to this fibre to obtain the continuum like causal set. Recently, the +implications of coarse graining in CST, both dynamically and kinematically, +were considered in Eichhorn (2018) based on renormalisation techniques. + +3.2 Discreteness without Lorentz breaking + +It is often assumed that a fundamental discreteness is incompatible with con- +tinuous symmetries. As was pointed out in Christ et al (1982), in the Euc- +lidean context, symmetry can be preserved on average in a random lattice. In +Bombelli et al (2009), it was shown that a causal set in C(Md, ρc) not only +preserves Lorentz invariance on average, but in every realisation, with respect +to the Poisson distribution. Thus, in a very specific sense a manifold-like causal +set does not break Lorentz invariance. In order to see the contrast between the +Lorentzian and Euclidean cases we present the arguments of Bombelli et al +(2009) starting with the easier Euclidean case. +Consider the Euclidean plane P = (R2, δab), and let Φ : C(P, ρc) �→ P be +the natural embedding map, where C(P, ρc) denotes the ensemble of Poisson +sprinklings into P at density ρc. A rotation r ∈ SO(2) about a point p ∈ P, +induces a map r∗ : C(P, ρc) → C(P, ρc), where r∗ = Φ−1 ◦r ◦Φ and similarly a +translation t in P induces the map t∗ : C(P, ρc) → C(P, ρc). The action of the +Euclidean group is clearly not transitive on C(P, ρc) but has non-trivial orbits +which provide a fibration of C(P, ρc). Thus the ensemble C(P, ρc) preserves the +Euclidean group on average. This is the sense in which the discussion of Christ +et al (1982) states that the random discretisation preserves the Euclidean +group. +The situation is however different for a given realisation P ∈ C(P, ρc). +Fixing an element e ∈ Φ(P), we define a direction d ∈ S1, the space of unit +vectors in P centred at e. Under a rotation r about e, d → r∗(d) ∈ S1. In +general, we want a rule that assigns a natural direction to every P ∈ C(P, ρc). +One simple choice is to find the closest element to e in Φ(P), which is well +defined in this Euclidean context. Moreover, this element is almost surely +unique, since the probability of two elements being at the same radius from + + +22 +Sumati Surya + +e is zero in a Poisson distribution. Thus we can define a “direction map” +De : C(P, ρc) → S1 for a fixed e ∈ Φ(P) consistent with the rotation map, i.e., +De commutes with any r ∈ SO(2), or is equivariant. +Associated with C(P, ρc), is a probability distribution µ arising from the +Poisson sprinkling which associates with every measurable set α in C(P, ρc) +a probability µ(α) ∈ [0, 1]. The Poisson distribution being volume preserving +(Stoyan et al 1995), the measure on C(P, ρc) moreover must be independent +of the action of the Euclidean group on C(P, ρc), i.e.: µ ◦ r = µ. +In analogy with a continuous map, a measurable map is one whose preimage +from a measurable set is itself a measurable set. The natural map D we have +defined is a measurable map, and we can use it to define a measure on S1: +µD ≡ µ ◦ D−1. Using the invariance of µ under rotations and the equivariance +of D under rotations + +µD = µ ◦ r ◦ D−1 = µ ◦ D−1 ◦ r = µD ◦ r ∀ r ∈ SO(2), +(12) + +we see that µD is also invariant under rotations. Because S1 is compact, this +does not lead to a contradiction. In analogy with the construction used in +Bombelli et al (2009) for the Lorentzian case, we choose a measurable set s ≡ +(0, 2π/n) ∈ S1. A rotation by r(2π/n), takes s → s′ which is non-overlapping, +so that after n successive rotations, rn(2π/n) ◦ s = s. Since each rotation does +not change µD and µD(S1) = 1, this means that µD(s) = 1/n. Thus, it is +possible to assign a consistent direction for a given realisation P ∈ C(P, ρc) +and hence break Euclidean symmetry. +However, this is not the case for the space of sprinklings C(Md, ρc) into +Md, where the hyperboloid Hd−1 now denotes the space of future directed +unit vectors and is invariant under the Lorentz group SO(n − 1, 1) about a +fixed point p ∈ Md−1. To begin with, there is no “natural” direction map. Let +C ∈ C(Md, ρc). To find an element which is closest to some fixed e ∈ Φ(C), one +has to take the infimum over J+(e) , or some suitable Lorentz invariant subset +of it, which being non-compact, does not exist. Assume that some measurable +direction map D : ΩMd → Hd−1, does exist. Then the above arguments imply +that µD must be invariant under Lorentz boosts. The action of successive +Lorentz transformations Λ can take a given measurable set h ∈ Hd−1 to an +infinite number of copies that are non-overlapping, and of the same measure. +Since Hd−1 is non-compact, this is not possible unless each set is of measure +zero, but since this is true for any measurable set h and we require µD(Hd−1) = +1, this is a contradiction. This proves the following theorem (Bombelli et al +2009): + +Theorem 2 In dimensions n > 1 there exists no equivariant measurable map +D : C(Md, ρc) → H, i.e., + +D ◦ Λ = Λ ◦ D ∀ Λ ∈ SO(n − 1, 1). +(13) + +In other words, even for a given sprinkling ω ∈ ΩMd it is not possible to +consistently pick a direction in Hd−1. Consistency means that under a boost + + +The causal set approach to quantum gravity +23 + +Fig. 10 The space of unit directions in Rd is Sd−1, while the space of unit timelike vectors +in Md is Hd−1. + +Λ : ω → Λ ◦ w, and hence D(ω) → Λ ◦ D(ω) ∈ Hd−1. Crucial to this argument +is the use of the Poisson distribution.12 Thus, an important prediction of CST +is local Lorentz invariance. Tests of Lorentz invariance over the last couple of +decades have produced an ever-tightening bound, which is therefore consistent +with CST (Liberati and Mattingly 2016). + +3.3 Forks in the road: What makes CST so “different”? + +In many ways CST doesn’t fit the standard paradigms adopted by other ap- +proaches to quantum gravity and it is worthwhile trying to understand the +source of this difference. The program is minimalist but also rigidly constrained +by its continuum approximation. The ensuing non-locality means that the ap- +paratus of local physics is not readily available to CST. + +Sorkin (1991) describes the route to quantum gravity and the various forks +at which one has to make choices. Different routes may lead to the same +destination: for example (barring interpretational questions), simple quantum +systems can be described equally well by the path integral and the canonical +approach. However, this need not be the case in gravity: a set of consistent +choices may lead you down a unique path, unreachable from another route. +Starting from broad principles, Sorkin argued that certain choices at a fork are +preferable to others for a theory quantum gravity. These include the choice +of Lorentzian over Euclidean, the path integral over canonical quantisation +and discreteness over the continuum. This set of choices leads to a CST-like +theory, while choosing the Lorentzian-Hamiltonian-continuum route leads to +a canonical approach like Loop Quantum Gravity. +Starting with CST as the final destination, we can work backward to re- +trace our steps to see what forks had to be taken and why other routes are + +12 It is interesting to ask if other choices of uniform distribution satisfy the above theorem. +If so, then our criterion for a uniform distribution could not only include ones that minimise +the fluctuations but also those that respect Lorentz invariance. + + +24 +Sumati Surya + +impossible to take. The choice at the discreteness versus continuum fork and +the Lorentzian versus Euclidean fork are obvious from our earlier discussions. +As we explain below, the other essential fork that has to be taken in CST is +the histories approach to quantisation. +One of the standard routes to quantisation is via the canonical approach. +Starting with the phase space of a classical system, with or without constraints, +quantisation rules give rise to the familiar apparatus of Hilbert spaces and +self adjoint operators. In quantum gravity, apart from interpretational issues, +this route has difficult technical hurdles, some of which have been partially +overcome (Ashtekar and Pullin 2017). Essential to the canonical formulation +is the 3+1 split of a spacetime M = Σ×R, where Σ is a Cauchy hypersurface, +on which are defined the canonical phase space variables which capture the +intrinsic and extrinsic geometry of Σ. +The continuum approximation of CST however, does not allow a meaning- +ful definition of a Cauchy hypersurface, because of the “ graphical non-locality” +inherent in a continuum like causal set, as we will now show. We begin by de- +fining an antichain to be a set of unrelated elements in C, and an inextendible +antichain to be an antichain A ⊂ C such that every element e ∈ C\A is re- +lated to an element of A. The natural choice for a discrete analog of a Cauchy +hypersurface is therefore an inextendible antichain A, which separates the set +C into its future and past, so that we can express C = Fut(A) ⊔ Past(A) ⊔ A, +with ⊔ denoting disjoint union. However, an element in Past(A) can be related +via a link to an element in Fut(A) thus “bypassing” A. An example of a “miss- +ing link” is depicted in Fig 11. This means that unlike a Cauchy hypersurface, +A is not a summary of its past, and hence a canonical decomposition using +Cauchy hypersurfaces is not viable (Major et al 2006). On the other hand, +each causal set is a “history”, and since the sample space of causal sets is +countable, one can construct a path integral or path-sum as over causal sets. +We will describe the dynamics of causal sets in more detail in Sect. 6. + +e + +e' +� + +Fig. 11 A “missing link” from e to e′ which “bypasses” the inextendible antichain A. + + +The causal set approach to quantum gravity +25 + +Before moving on, we comment on the condition of local finiteness which, +as we have pointed out, provides an intrinsic definition of spacetime discrete- +ness. This does not need a continuum approximation. An alternative definition +would be for the causal set to be countable, which along with the continuum +approximation is sufficient to ensure the number to volume correspondence. +This includes causal sets with order intervals of infinite cardinality. This al- +lows us to extend causal set discretisation to more general spacetimes, like +anti de Sitter spacetimes, where there exist events p, q in the spacetime for +which vol(A[p, q]) is not finite. However, what is ultimately of interest is the +dynamics and in particular, the sample space Ω of causal sets. In the growth +models we will encounter in Sect. 6.1,6.2 and 6.3 the sample space consists +of past finite posets, while in the continuum-inspired dynamics of Sect. 6.4 it +consists of finite element posets. Thus, while countable posets may be relev- +ant to a broader framework in which to study the dynamics of causal sets, it +suffices for the present to focus on locally finite posets. + +4 Kinematics or geometric reconstruction + +In this section we discuss the program of geometric reconstruction in which +topological and geometric invariants of a continuum spacetime (M, g) are “re- +constructed” from the underlying ensemble of causal sets. The assumption +that such a reconstruction exists for any covariant observable in (M, g) comes +from the Hauptvermutung of CST discussed in Sect. 3. +In the statement of the Hauptvermutung, we used the phrase “approxim- +ately isometric”, with the promise of an explanation in this section. A rigor- +ous definition requires the notion of closeness of two Lorentzian spacetimes. +In Riemannian geometry, one has the Gromov–Hausdorff distance (Petersen +2006), but there is no simple extension to Lorentzian geometry, in part because +of the indefinite signature. In Bombelli and Meyer (1989) a measure of close- +ness of two Lorentzian manifolds was given in terms of a pseudo distance func- +tion, which however is neither symmetric nor satisfies the triangle inequality. +Subsequently, in a series of papers, a true distance function was defined on the +space of Lorentzian geometries, dubbed the Lorentzian Gromov–Hausdorff dis- +tance (Bombelli 2000; Noldus 2004, 2002; Bombelli and Noldus 2004; Bombelli +et al 2012). While this makes the statement of the Hauptvermutung precise, +there is as yet no complete proof. Recently, a purely order theoretic criterion +has been used to determine the closeness of causal sets and prove a version of +the Hauptvermutung (Sorkin and Zwane, work in progress). +Apart from these more formal constructions, as we will describe below, +a large body of evidence has accumulated in favour of the Hauptvermutung. +In the program of geometric reconstruction, we look for order invariants in +continuum like causal sets which correspond to manifold (either topological +or geometric) invariants of the spacetime. These manifold invariants include +dimension, spatial topology, distance functions between fixed elements in the +spacetime, scalar curvature, the discrete Einstein–Hilbert action, the Gibbons- + + +26 +Sumati Surya + +Hawking-York boundary terms, Green functions for scalar fields, and the +d’Alembertian operator for scalar fields. The identification of the order in- +variant O with the manifold invariant G then ensures that a causal set C that +faithfully embeds into (M, g) cannot faithfully embed into a spacetime with +a different manifold invariant G′.13 Thus, in this sense two manifolds can be +defined to be close with respect to their specific manifold invariants. We can +then state the limited, order-invariant version of the Hauptvermutung: +O-Hauptvermutung: If C faithfully embeds into (M, g) and (M ′, g′) then +(M, g) and (M ′, g′) have the same manifold invariant G associated with O. +The longer our list of correspondences between order invariants and man- +ifold invariants, the closer we are to proving the full Hauptvermutung. +In order to correlate a manifold invariant G with an order invariant O, +we must recast geometry in purely order theoretic terms. Note that since +locally finite posets appear in a wide range of contexts, the poset literature +contains several order invariants, but these are typically not related to the +manifold invariants of interest to us. The challenge is to choose the appropriate +invariants that correspond to manifold invariants. Guessing and verifying this +using both analytic and numerical tools is the art of geometric reconstruction. +A labelling of a causal set C is an injective map: C → N, which is the +analogue of a choice of coordinate system in the continuum. By an order +invariant in a finite causal set C we mean a function O : C → R such that +O is independent of the labelling of C. For a manifold-like causal set14 C ∈ +C(M, ρc), associated to every order invariant O is the random variable O whose +expectation value ⟨O⟩ in the ensemble C(M, ρc) is either equal to or limits (in +the large ρc limit) to a manifold invariant G of (M, g). We will typically restrict +to compact regions of (M, g) in order to deal with finite values of O. +The first candidates for geometric order invariants were defined for C(A[p, q], ρc) +where A[p, q] is an Alexandrov interval in Md. Some of these have been later +generalised to Alexandrov intervals (or causal diamonds) in Riemann Normal +Neighbourhoods (RNN) in curved spacetime. These manifold invariants are +in this sense “local”. In order to find spatial global invariants, the relevant +spacetime region is a Gaussian Normal Neighbourhood (GNN) of a compact +Cauchy hypersurface in a globally hyperbolic spacetime. As discussed in Sect. 3 +compactness is necessary for manifold-likeness since otherwise there is a finite +probability for there to be arbitrarily large voids which negates the discrete- +continuum correspondence. +Before proceeding, we remind the reader that we are restricting ourselves +to manifold-like causal sets in this section only because of the focus on CST +kinematics and the continuum approximation. All the order invariants, how- +ever, can be calculated for any causal set, manifold-like or not. These order +invariants give us an important class of covariant observables, essential to con- + +13 This is in the sense of an ensemble, since the faithful embedding is defined statistically. +14 We remind the reader that the ensemble depends on the spacetime (M, g) but we sup- +press the dependence on g for the sake of brevity. + + +The causal set approach to quantum gravity +27 + +structing a quantum theory of causal sets. As we will see in Sect. 6 they play +an important role in the quantum dynamics. +The analytic results in this section are typically found in the continuum +limit, ρc → ∞. Strictly speaking, this limit is unphysical in CST because of +the assumption of a fundamental discreteness. There are fluctuations at finite +ρc which give important deviations from the continuum with potential phe- +nomenological consequences. These are however not always easy to calculate +analytically and hence require simulations to assess the size of fluctuations at +finite ρc. As we will see below, CST kinematics therefore needs a combination +of analytical and numerical tools. + +4.1 Spacetime dimension estimators + +The earliest result in CST is a dimension estimator for Minkowski spacetime +due to Myrheim (1978)15 and predates BLMS (Bombelli et al 1987). A closely +related dimension estimator was given by Meyer (1988), which is now collect- +ively known as the Myrheim–Meyer dimension estimator. +The number of relations R in a finite n element causal set C is the number +of ordered pairs ei, ej ∈ C such that ei ≺ ej. Since the maximum number of +possible relations on n elements is +�n +2 +� +, the ordering fraction is defined as + +r = +2R + +n(n − 1). +(14) + +It was shown by Myrheim (1978) that r is dependent only on the dimension +when C faithfully embeds into Md. +We now describe the construction of a closely related dimension estimator +by Meyer (1988). Consider an Alexandrov interval Ad[p, q] ⊂ Md of volume +V >> ρ−1 +c . We are interested in calculating the expectation value of the ran- +dom variable R associated with R for the ensemble C(Ad, ρc). This is the +probability that a pair of elements e1, e2 ∈ Ad[p, q] are related. Given e1, the +probability of there being an e2 in its future is given by the volume of the +region J+(e1) ∩ J−(p) in units of the discreteness scale, while the probability +to pick e1 is given by the volume of Ad[p, q]. This joint probability can be +calculated as follows. +Without loss of generality, choose p = (−T/2, 0, . . . , 0) and q = (T/2, 0, . . . , 0), +so that the total volume + +V = ζdT d, +ζd ≡ +Vd−2 + +2d−1d(d − 1) +(15) + +with Vd−2 the volume of the unit d − 2 sphere. For this choice, + +⟨R⟩ = ρ2 +c + +� + +Ad + +dx1 + +� + +J+(x1)∩J−(q) + +dx2 = ρ2 +c ζd + +� + +Ad + +dx1T d +1 , +(16) + +15 This remarkable preprint also contains the first expression, again without detailed proof, +of the volume of a small causal diamond in an arbitrary spacetime. + + +28 +Sumati Surya + +where T1 is the proper time from x1 to q, and Ad ≡ Ad[p, q]. Evaluating the +integral, one finds + +⟨R⟩ = ρ2 +cV 2 Γ(d + 1)Γ( d + +2) + +4Γ( 3d + +2 ) +. +(17) + +Using ⟨n⟩ = ρcV , Meyer (1988) obtained a dimension estimator from ⟨R⟩ by +noting that the ratio + +⟨R⟩ +⟨n⟩2 = Γ(d + 1)Γ( d + +2) + +4Γ( 3d + +2 ) +≡ f0(d) +(18) + +is a function only of d. In the large n limit this is is half of Myrheim’s ordering +fraction r. +However, the fluctuations in ⟨R⟩ are large and hence the right dimension +cannot be obtained from a single realisation C ∈ C(Ad, ρc), but rather by +averaging over the ensemble. For large enough ρc, however, the relative fluctu- +ations should become smaller, and allow one to distinguish causal sets obtained +from sprinkling into different dimensional Alexandrov intervals. Such system- +atic tests have been carried out numerically using sprinklings into different +spacetimes by Reid (2003) and show a general convergence as ρc is taken to +be large, or equivalently the interval size is taken to be large. +How can we use this dimension estimator in practice? Let C be a causal +set of sufficiently large cardinality n. If the dimension obtained from Eq. (18) +is approximately an integer d, this means that C cannot be distinguished from +a causal set that belongs to C(Ad, ρc) using just the dimension estimator, for +n ∼ ρcvol(Ad). We denote this by C ∼d Ad. This also means that C cannot +be a typical member of C(Ad′, 1) for dimension d′ ̸= d, so that C ̸∼d′ Ad′. +The equivalence C ∼d Ad itself does not of course imply that C ∼ Ad or even +that C is manifold-like. Rather, it is the limited statement that its dimension +estimator is the same as that of a typical causal set in C(Ad, ρc) for n ∼ +ρcvol(Ad). +This is our first example of a O-Hauptvermutung, where the order invariant +O is the ordering fraction r and the spacetime dimension d is the corresponding +manifold invariant G. This example provides a useful template in the search +for manifold-like order invariants some of which we will describe in the next +few subsections. +Using simulations Abajian and Carlip (2018) recently obtained the Myrheim– +Meyer dimension as function of interval size for nested intervals in a causal set +in C(Ad, ρc) for d = 3, 4, 5. As the interval size decreases, they found that the +resulting causal sets are likely to be disconnected due to the large fluctuations +at small volumes. In the extreme case, there is a single point with no relations +and hence the Myrheim–Meyer dimension goes to ∞ rather than 0. Using a +criterion to discard such disconnected regions, it was shown that this dimen- +sion estimator gives a value of 2 at small volumes, even when d = 3, 4, 5, in +support of the dimensional reduction conjecture in quantum gravity (Carlip +2017) which we discuss briefly in Sect. 5. + + +The causal set approach to quantum gravity +29 + +Meyer’s construction is in fact more general and yields a whole family of +dimension estimators. If we think of the relation e1 ≺ e2 as a chain c2 of two +elements, then a k-chain ck is the causal sequence e1 ≺ e2 . . . ≺ ek−1 ≺ ek +(see Fig. 12), where the length of ck is defined as k − 2. We denote the + +� + +�� + +� + +�1 + +2 + +Fig. 12 Two different chains between x and x′. One is a k = 4 chain and the other is a +k = 7 chain. + +abundance, or number of the ck’s contained in C, by Ck. Its expectation +value in C(Ad[p, q], ρc) is therefore given by a sequence of k nested integ- +rals over a sequence of nested Alexandrov intervals, Ad[p, q] ⊃ I(x1, q) ⊃ +I(x2, q) . . . I(xk, q) which, as was shown by Meyer (1988), can be calculated +inductively to give + +⟨Ck⟩ = ρk +cχkV k, +χk ≡ 1 + +k + +�Γ(d + 1) + +2 + +�k−1 +Γ( d + +2)Γ(d) + +Γ( kd + +2 )Γ( (k+1)d + +2 +) +. +(19) + +Thus for any k, k′, the ratio of ⟨Ck⟩1/k to ⟨Ck′⟩1/k′ only depends on the +dimension. This gives a multitude of dimension estimators. +Meyer’s calculation of ⟨Ck⟩ was generalised to a small causal diamond +Ad[p, q] that lies in an RNN of a general spacetime, i.e., one for which RT 2 << +1, where T is the proper time from p to q and R denotes components of the +curvature at the centre of the diamond (Roy et al 2013). In such a region the +dimension satisfies the more complicated equation + +f 2 +0 (d) +� +−1 + +3 +(d + 2) +(3d + 2) − (4d + 2) + +(2d + 2) + +�⟨C3⟩ + +χ3 + +� 4 + +3 +1 + +⟨C1⟩4 + ++1 + +3 +(4d + 2)(5d + 2) +(2d + 2)(3d + 2) +⟨C4⟩ +χ4 + +1 + +⟨C1⟩4 + +� += −⟨C2⟩2 + +⟨C1⟩4 , +(20) + +where f0(d) is given by Eq. (18). It is straightforward to show that the expres- +sion above reduces to the Myrheim–Meyer dimension estimator in Md. The + + +30 +Sumati Surya + +calculation of Roy et al (2013) uses a result of Khetrapal and Surya (2013), +which makes explicit earlier calculations of the volume of a causal diamond in +an RNN (Myrheim 1978; Gibbons and Solodukhin 2007). The Ck themselves +are order invariants and hence are covariant observables for finite element +causal sets. +This class of dimension estimators is just one among several that have +appeared in the literature, including the mid-point scaling estimator (Bombelli +1987; Reid 2003), and more recent ones (Glaser and Surya 2013; Aghili et al +2018). We refer the reader to the literature for more details. + +4.2 Topological invariants + +The next step in our reconstruction is that of topology. There are several poset +topologies described in the literature (see Stanley 2011 as well as Surya 2008 +for a review). However, our interest is in finding one that most closely resembles +the “coarse” continuum topology. It is clear that the full manifold topology +cannot be reproduced in a causal set since it requires arbitrarily small open +sets. However, according to the Hauptvermutung, topological invariants like +the homology groups and the fundamental groups of (M, g) should be encoded +in the causal set. +A natural choice for a topology in C based on the order relation is one +generated by the order intervals I[ei, ej] ≡ Fut(ei) ∩ Past(ej). Indeed, in the +continuum the topology generated by their analogs, the Alexandrov intervals, +can be shown to be equivalent to the manifold topology in strongly causal +spacetimes (Penrose 1972). However, even for a causal set approximated by a +finite region of Md, this order-interval topology is roughly discrete or trivial. +This is because the intersection of any two intervals in the continuum can +be of order the discreteness scale and hence contain just a single element of +the causal set, thus trivialising the topology. A way forward is to use the +causal structure to obtain a locally finite open covering of C and construct the +associated “nerve simplicial complex” (see Munkres 1984). +In Major et al (2007, 2009), a “spatial” homology of C was obtained in this +manner by considering an inextendible antichain A ⊂ C (see Sect. 3.3), which +is an (imperfect) analog of a Cauchy hypersurface. The natural topology on +A is the discrete topology since there are no causal relations amongst the ele- +ments. In order to provide a topology on A, one needs to “borrow” information +from a neighbourhood of A. The method devised was to consider elements to +the future of A and “thicken” by a parameter v to some collar neighbourhood +Tv(A) ≡ {e| |IFut(A) ∩ IPast(e)| ≤ v}. Here IFut and IPast denote the inclus- +ive future and past respectively, where for any S ⊂ C, IFut(S) = Fut(S) ∪ S +and IPast(S) = Past(S) ∪ S. +A topology can then be induced on A from Tv(A) by considering the open +cover {Ov ≡ Past(e)∩A} of A, for e ∈ Mv(A), the set of future most elements +of Tv(A). The “nerve” simplicial complex Nv(A) can be constructed from {Ov} +for every v. For a spacetime (M, g) with compact Cauchy hypersurface Σ, and + + +The causal set approach to quantum gravity +31 + +for C ∈ C(M, ρc) it was shown in Major et al (2007, 2009) that there exists a +range of values of v such that Nv(A) is homological to Σ (up to the discreteness +scale) as long as there is a sufficient separation between the discreteness scale +ℓc ≡ V 1/d +c +and ℓK the scale of extrinsic curvature of Σ. +One might also imagine a similar construction on C using the nerve sim- +plicial complex of causal intervals of a given minimal cardinality v which cover +C. However, in the continuum the intersection of such intervals may not only +be of order the discreteness scale, but also such that they “straddle” each +other. As an example consider the equal volume intervals A[p1, q1], A[p2, q2] +in M2 where p1, q1 are at x = 0 in a frame (x, t), with the x-coordinate of p2 +being < 0 and that of q2 being > 0. These two intervals not only intersect, but +straddle each other, i.e., the set difference A[p1, q1]\A[p2, q2] is disconnected +as is A[p2, q2]\A[p1, q1]. By choosing p2, q2 appropriately, the intersection re- +gion can be made very “thin”, pushing most of the volume of A[p2, q2] out of +A[p1, q1]. Thus, while they intersect in M2 these intervals would not intersect +in the corresponding causal set C. This results in a non-trivial cycle in the +associated nerve simplicial complex for C, which is absent in the continuum. +Such a construction can be therefore made to work only in a sufficiently loc- +alised region within C. +An example of a localised of subset of C is the region sandwiched between +two inextendible and non-overlapping antichains A1 and A2. The resulting +homology constructed from the nerve simplicial complex of the order inter- +vals of volume ∼ v is then is associated with a spacetime region rather than +just space, and hence includes topology change. While preliminary investig- +ations along these lines have been started, there is much that remains to be +understood. Another possibility for characterising the spatial homology uses +chain complexes but this has only been partially investigated. A further open +direction is to obtain the causal set analogues of other topological invariants. + +4.3 Geodesic distance: timelike, spacelike and spatial + +In Minkowski spacetime, the proper time between two events is the longest +path between them; the shortest path between two time-like separated events +is of course any zig-zag null path, which has zero length. In a causal set C, if +ei ≺ ef, one can construct different chains from ei to ef, of varying lengths. +A natural choice for the discrete timelike geodesic distance between ei and +ef is the length of the longest chain, which we denote by l(ei, ej), as was +suggested by Myrheim (1978). It was shown in Brightwell and Gregory (1991) +that the expectation value of the associated random variable l in the ensemble +C(Ad, ρc) limits to a dimension dependent constant + +lim +ρc→∞ +⟨l(x, x′)⟩ + +(ρcV (x, x′))1/d = md +(21) + + +32 +Sumati Surya + +where + +1.77 ≤ +21− 1 + +d + +Γ(1 + 1 + +d) ≤ md ≤ 21− 1 + +d e (Γ(1 + d)) +1 +d + +d +≤ 2.62 +(22) + +For a finite ρc, the fluctuations in l(ei, ej) are very large (Meyer 1988; Bachmat +2007) and hence the correspondence becomes meaningful only when averaged +over a large ensemble. +In Roy et al (2013), an expression for the proper time T of a small causal +diamond Ad in an RNN of a d dimensional spacetime was obtained to lead- +ing order correction in terms of the random variables Ck associated to the +abundance of k-chains, + +T 3d = +1 + +2d2ρ3c + +� +J1 − 2J2 + J3 + +� +. +(23) + +where + +Jk ≡ (kd + 2)((k + 1)d + 2) 1 + +ζ3 +d + +�⟨Ck⟩ + +χk + +�3/k +, +(24) + +with ⟨Ck⟩ the ensemble average in C(Ad, ρc) and ζd, χk defined in Eqs. (15) +and (19). This definition is not intrinsic to a single causal set but requires the +full ensemble. Nevertheless, it is of interest to study the intrinsic version of the +expression by replacing ⟨Ck⟩ by Ck for each causal set and then taking the +ensemble average to check for convergence. Recent simulations suggest that +these expressions converge fairly rapidly to their continuum values. +Spacelike distance is far less straightforward to compute from the poset, +because events that are spacelike to each other have no natural relationship to +each other. We saw this already in trying to find a topology on the inextendible +antichain. Thus, the relationship must be “borrowed” from the elements in +the causal past and future of the spacelike events. Brightwell and Gregory +(1991) defined the following, naive spatial distance function in Md. For a given +spacelike pair p, q ∈ Md, the common future and past are defined as J+(p, q) ≡ +J+(p) ∩ J+(q) and J−(p, q) ≡ J−(p) ∩ J−(q) respectively. For every r ∈ +J+(p, q) and s ∈ J−(p, q) let τ(s, r) be the timelike distance. Then the naive +distance function is given by + +ds(p, q) ≡ minr,sτ(r, s). +(25) + +While this is a perfectly good continuum definition of the distance in Md, +it fails for the causal set when d > 2 since the number of pairs (r, s) which +minimise τ(r, s) lies in the region between a co-dimension 2 hyperboloid and +the light cone τ = 0. In the causal set we can use the length of the maximal +chain l(r, s) to obtain τ(r, s), but in d > 2 since there are an infinite number +of proper time minimising pairs (r, s), there will almost surely be those for +which l(r, s) is drastically underestimated. The minimisation in Eq. (25) will +then always give 2 as the spatial distance! + +Rideout and Wallden (2009) generalised the naive distance function using +minimising pairs (r, s) such that either r or s is linked to both p and q. In- +stead of minimising over these pairs (again infinite), the 2-link distance can + + +The causal set approach to quantum gravity +33 + +be calculated by averaging over the pairs. Numerical simulations for the na- +ive distance and the 2-link distance for sprinklings into a finite region of M3 + +show that the latter stabilises as a function of ρc. The former underestimates +the spatial distance compared to the continuum, and the latter overestimates +it. The spatial distance functions of both Brightwell and Gregory (1991) and +Rideout and Wallden (2009) are however strictly “predistance” functions since +they do not satisfy the triangle inequality. +Recently a one-parameter family of discrete induced spatial distance func- +tions was proposed for an inextendible antichain in a causal set by Eichhorn +et al (2018). To begin with, a one parameter family of continuum induced +distance functions dϵ was constructed for a globally hyperbolic region (M, g) +of spacetime with Cauchy hypersurface Σ using only the causal structure and +the volume element. In Md with Σ a constant time slice in an inertial reference +frame, the volume of a past causal cone from p ≻ Σ has a simple relation to +the diameter D of the base of the cone J−(p) ∩ Σ + +vol(J−(p) ∩ J+(Σ)) = ζd + +�D + +2 + +�d +. +(26) + +Since D is the distance between any two antipodal points on the Sd−2 ⊂ Σ, +this simple formula defines the induced distance on Σ. In a general spacetime +this formula can be used to extract an approximate induced distance function +in a sufficiently small region of Σ. In order to define the distance function on +all of Σ, a meso-scale ϵ must be introduced, and the full distance function +can then be obtained by minimising over all segmented paths, such that each +segment is bounded from above by ϵ. For ϵ << ℓK, the scale of the extrinsic +curvature of Σ, dϵ was shown in Eichhorn et al (2018) to converge to the +induced spatial distance function dh on (Σ, h). +Since the dϵ are constructed from the causal structure and volume element +they are readily defined on an inextendible antichain on a causal set. For +causal sets in C(M, ρc) with Σ ⊂ M the discrete distance function dϵ was +shown to significantly overestimate the continuum induced distance on Σ when +the latter is close to the discreteness scale (Vc)1/d. This discrete “asymptotic +silence” of Eichhorn et al (2017) mimics the narrowing of light cones in the UV, +and can be traced to the large fluctuations expected around the discreteness +scale. At larger distances, on the other hand, dϵ is a good approximation of the +continuum induced distance when (Vc)1/d << ϵ << ℓK. It was shown moreover +that the continuum induced distance is slightly underestimated for positive +curvature and slightly overestimated for negative curvature, when restricted +to small regions of Σ. This was confirmed by extensive numerical simulations +in Md for d = 2, 3 (see Fig. 13). This works paves the way to recovering more +spatial geometric invariants from the causal set, and is currently in progress +(Eichhorn, Surya and Versteegen). + + +34 +Sumati Surya + +● + +● + +● + +● + +● + +● + +● + +● + +●● + +● + +● + +● +● + +● +●● +● +● ● +● +● ● +● + +● ● ● + +● + +● + +● + +● + +● + +● + +● + +● + +● + +● + +● + +● + +● +● +● +● + +● +●● +●● +● ● ● +● +● ●● ● ● +● + +● +● + +● + +● + +● + +● + +●●● +● + +● +●● +● + +● + +● +● + +● + +● + +● + +●● +● ●● ● +● +● +● +● +● + +● +● +●● +● ● +● +● +● +● + +● + +● + +● + +● ● +● ● +● +● +●●● +● + +● +● +●● +●● ●● +● +● + +● +●●● +● + +● +● +●● +● ● +● ● +● +● + +● + +● + +● + +● + +● +● + +● + +● + +● + +● + +● + +● + +● + +● +● +●● +●● ● +● +● ● +● + +● ● ● + +● + +● +● + +● + +● + +● + +● + +● + +● + +● + +● + +● + +● + +● + +● +● +● +●● +●● +● ● ● +● +● ●● ● ● +● + +● +● + +● + +● + +● + +● + +●●● +● + +● + +●● +● +● +● + +● + +● + +● + +● +●● +● ●● ● +● ● +● +● +● +●● +●● +● ● +● +● +● +● +● + +● +● +● ● +● ● +● 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+H+:ℓk=1000 + +Fig. 13 The error in the discrete spatial distance is plotted as a function of the continuum +induced distance on Σ for causal sets in M2 for Σ of both constant negative and constant +positive extrinsic curvature. The discrete spatial distance always overestimates the con- +tinuum distance around the discreteness scale giving rise to “discrete asymptotic silence”. +For larger distances, when there is a good separation of scales, the discrete distance gives a +good approximation to the continuum induced distance. + +4.4 The d’Alembertian for a scalar field + +One of the very first questions that comes to mind in the continuum approx- +imation of CST is whether a tangent space can be defined naturally on a +causal set. To answer this (unfortunately in the negative), we need to examine +the non-local nature of a manifold-like causal set in more detail. The nearest +neighbours of an element e are those that it is linked to, both in its future and +its past. In a causal set approximated by Minkowski spacetime for example, +and as discussed in Sect. 3, every element has an infinite number of nearest +neighbours (see Fig. 8). Similarly, the “next nearest” neighbours to e are those +for which the interval |I[e, e′]| = 1 or |I[e′, e]| = 1.16 Thus, in keeping with the +covariance of the causal set, we say that if e ≺ e′ and |I[e, e′]| = k (or e′ ≺ e +and |I[e′, e]| = k), then e′ is the k-nearest neighbour of e. Examples of past +k-nearest neighbours of an element in a Minkowski-like causal set are shown +in Fig. 14. +It is already clear from the picture that emerges in Md that, unlike a regular +lattice, a simple construction of a locally defined tangent space from the set +of links or next to nearest neighbours to e is not possible, since the valency of +the graph is infinite. This means in particular that derivative operators cannot +also be simply defined. How then can we look for the effect of discreteness on +the propagation of fields? We will discuss this in more detail in Sect. 5 but +for now we notice that the best way forward is to look for scalar quantities, +rather than more general tensorial ones, in making the discrete-continuum +correspondence. +A scalar field d’Alembertian is a good first step. In Sorkin (2007b); Henson + +(2010), a proposal was given for a discrete d’Alembertian of a free scalar field +on a causal set approximated by M2, and extended in Benincasa and Dowker +(2010); Benincasa (2013); Dowker and Glaser (2013) to higher dimensions. For +a real scalar field on a causal set φ : C → R define the d = 4 dimensionless + +16 Note that this is the exclusive interval and hence there exists exactly one element e′′ + +such that e ≺ e′′ ≺ e′. + + +The causal set approach to quantum gravity +35 + +Fig. 14 The layered structure of neighbourhoods. The nearest neighbours are the links or +zero intervals, the next to nearest neighbours are the 1-element intervals, the etc. Here we +depict the types of 0, 1, 2 element intervals. In the figure two examples of 3 element intervals +are also shown. + +discrete operator + +Bφ(e) ≡ +4 +√ + +6 + +� +−φ(e)+ +� +� + +e′∈L0(e) +− 9 +� + +e′∈L1(e) ++ 16 +� + +e′∈L2(e) +− 8 +� + +e′∈L3(e) + +� +φ(e′) +� +, (27) + +where Lk(e) denotes the set of k-nearest neighbours to the past of e ∈ C. +This is a highly non-local operator since it depends on the number of all the +(possibly infinite) nearest k = 0, 1, 2, 3 neighbours. Notice the alternating sum +whose precise coefficients turn out to be very important to the continuum +limit. The expectation value of the random variable Bφ(x) associated with +C(M4, ρc) at x ∈ M4 is + +1 +√ρc +⟨Bφ(x)⟩= 4√ρc +√ + +6 + +� +−φ(x) + + +ρc + +� + +y∈J−(x) +d4y φ(y) e−ρcv +� +1 − 9ρcv + 8(ρcv)2 − 4 + +3(ρcv)3 +�� +, (28) + +where v ≡ vol(A(y, x)) and we have used the probability Pn(v) for v to contain +n elements, Eq. (8). We have also made the expression dimensionful, in order +to be able to make a direct comparison with the continuum. Let us consider +the past of x in M2 and choose a frame Fφ such that φ(y) varies slowly in +the immediate past of x with respect to Fφ. As was shown in M2 by Sorkin +(2007b) and in M4 by Benincasa and Dowker (2010) (see also Benincasa 2013), +for φ of compact support there are miraculous cancellations that make the +contributions far down the light cone negligible, thus making the operator +effectively local. + + +36 +Sumati Surya + +In order to evaluate this integral, we first note that since φ is of compact +support, the region of integration is compact. In Fφ, a small |y − x| expan- +sion of φ(y) around φ(x) can be done. Following Sorkin (2007b); Benincasa +and Dowker (2010); Benincasa (2013), the non-compact region of integration +J−(x) can be split into 3 non-overlapping regions, W1, W2, W3 in Fφ: W1 is a +neighbourhood of x, W2 is a neighbourhood of ∂J−(x) but bounded away from +the origin and W3 is bounded away from ∂J−(x). The integral over W3 was +shown in Benincasa (2013) to be bounded from above by an integral that tends +to zero faster than any power of ρ−1 +c , while the integral over W2 was shown +to go to zero faster than ρ−3/2 +c +. The local contribution from W1 dominates so +that +lim +ρc→∞ +1 +√ρc +⟨Bφ(x)⟩ = □φ(x). +(29) + +Thus, B(φ) is “effectively local” since its dominant contribution comes from +W1 which is a local neighbourhood of x defined by the frame Fφ. In this +frame, the contribution to Bφ(x) is dominated by the restrictions of Lk to +A(p, q)∩J−(x). Thus, while Bφ(x) is not determined just by the value of φ at +x, it depends on φ only in an appropriately defined compact neighbourhood +of x, rather than all of J−(x). This “restoration of locality” is an important +subtlety in CST kinematics. +How does a scalar field on a causal set evolve under this non-local d’Alembertian? +There are indications that while the evolution in d = 2 is stable, it is unstable +in d = 4 as suggested by Aslanbeigi et al (2014). Hence it is desirable to +look for generalisations of the Bκ operator. An infinite family of non-local +d’Alembertians has been constructed by Aslanbeigi et al (2014) and shown to +give the right continuum limit. It is still an open question whether there is a +subfamily of these operators that lead to a stable evolution. +An interesting direction that has been explored by Yazdi and Kempf (2017) +is to use the spectral information of the d’Alembertian operator to obtain +all the information about the causal set. This was explored for A2[p, q] ⊂ +M2 and it was shown that the spectrum of the d’Alembertian (or Feynman +propagator) gives the link matrix (see Eq. (56) below), i.e., the matrix of +all linked pairs using which the entire causal set can be reconstructed via +transitivity. Extending these results to higher dimensions is an interesting +open question. + +4.5 The Ricci scalar and the Benincasa–Dowker action + +Next we describe a very important development in CST : the construction +of the discrete Einstein–Hilbert action or the Benincasa–Dowker (BD) action +for a causal set (Benincasa and Dowker 2010; Dowker and Glaser 2013). The +approach of Benincasa and Dowker (2010) was to generalise Bφ(x) to an RNN +in curved spacetime in d = 2, d = 4. Again, the region of integration can be +split into three parts as was done for flat spacetime. The contribution from W3 +i.e., away from a neighbourhood of ∂J−(x) can again be shown to be bounded + + +The causal set approach to quantum gravity +37 + +from above by an integral that tends to zero faster than any power of ρ−1 +c . In +the limit, the contribution from the near region W1 contained in an RNN is +such that +lim +ρc→∞ +1 +√ρc +⟨Bφ(x)⟩|W1= □φ(x) − 1 + +2R(x)φ(x). +(30) + +where R(x) is the Ricci scalar (Benincasa and Dowker 2010; Benincasa 2013). +However, the calculation in region W2 which is in the neighbourhood of ∂J−(x) +but bounded away from the origin, is non-trivial, and needs a further set of +assumptions to show that it does not contribute in the ρc → ∞ limit. A +painstaking calculation in Belenchia et al (2016a) using Fermi Normal Co- +ordinates shows that this is indeed the case in an approximately flat region of +a four dimensional spacetime. Generalising this calculation to arbitrary space- +times is highly non-trivial but is an important open question in CST. +What is of course exciting about this form for the d’Alembertian Eq. 27 is +that it can be used to find the discrete Ricci curvature and hence the action. +Assuming that + +lim +ρc→∞ +1 +√ρc +⟨Bφ(x)⟩|W2 = 0 +(31) + +holds in all spacetimes, and putting17 φ(x) = 1 + +lim +ρc→∞ +1 +√ρc +⟨Bφ(x)⟩ = −1 + +2R(x). +(32) + +Thus we can write the dimensionless discrete Ricci curvature at an element +e ∈ C (Benincasa 2013) as + +R(e) = +4 +√ + +6 + +� +1 − N0(e) + 9N1(e) − 16N2(e) + 8N3(e) +� +, +(33) + +where Nk(e) ≡ |Lk(e)|. Summing over the n elements of a finite element causal +set gives the dimensionless discrete action + +S(4)(C) = +� + +e∈C +R(e) = +4 +√ + +6 + +� +n − N0 + 9N1 − 16N2 + 8N3 + +� +, +(34) + +where Nk is the total number of k-element order intervals in C. + +Benincasa and Dowker (2010) (see also Benincasa 2013) showed that (under +the assumption Eq. (31)) the random variable S(4) associated with C(M, g) +gives the Einstein–Hilbert action in the continuum limit + +lim +ρc→∞ ℏℓ2 +c +ℓ2p +⟨S(4)(C)⟩ = SEH(g), +(35) + +up to (as yet unknown) boundary terms. + +17 By doing so, we violate the condition that φ is of compact support. However, given that +the regions W3 and by assumption W2 contribute negligibly, we can always ensure this by +only requiring constancy of φ in a neighbourhood of W1. + + +38 +Sumati Surya + +Equation (35) is exactly true in an approximately flat region of a four +dimensional spacetime as shown in Belenchia et al (2016a). Proving Eq. (31) +in general is however non-trivial since there are caustics in a generic spacetime +which complicate the calculation. On the other hand, numerical simulations +suggest that again, up to boundary terms, the Benincasa–Dowker action S +is the Einstein–Hilbert action (Benincasa 2013; Cunningham 2018b). We will +discuss these boundary terms below. +Before doing so, we note that crucial to the validity of the causal set action +are its fluctuations in a given causal set. These were shown in Sorkin (2007b) +to be large for the operator B in M2 . This can be traced to the fact that +the elements in Lk(e) for k = 0, 1, 2, 3 are very close to the discreteness scale +and hence the d’Alembertian is susceptible to large Poisson fluctuations at +small volumes. In order to “shield” the continuum from these fluctuations, a +new mesoscale ℓκ > ℓc and its associated density ρκ was introduced in Sorkin +(2007b). Thus instead of a single discrete operator B, we have a one parameter +family of operators: + +Bκφ(e) ≡ +4 +√ + +6 + +� +−φ(e) + ϵ +� + +e′≺e +f(n(e′, e), ϵ)φ(e′) +� +, +(36) + +where ϵ ≡ ρκ/ρc is a non-locality parameter,18 n(e, e′) = |I(e, e′)| and + +f(n, ϵ) = (1 − ϵ)n +� +1 − 9ϵn + +1 − ϵ + 8ϵ2n(n − 1) + +(1 − ϵ)2 +− 4ϵ3n(n − 1)(n − 2) + +3(1 − ϵ)3 + +� +. +(37) + +This function “smears out” the contributions of the Nk into four “layers” +which appear with alternating sign, as shown in Fig. 15. Each layer is thus +thickened from a single value of k to a range of k values depending on ϵ. When +this mesoscale matches the discreteness scale, i.e., ϵ = 1, each layer collapses +to a single value of k. This then gives us a one-parameter family of actions +Sκ(C, ϵ), where ϵ can be viewed as a tunable coupling constant. As we will see +in Sect. 6, this gives rise to an interesting phase structure in 2d CST. +The result for d = 2, 4 are due to Benincasa and Dowker (2010); Benincasa + +(2013) and were generalised to arbitrary dimensions by Dowker and Glaser +(2013); Glaser (2014), using a dimension dependent smearing function fd(n, ϵ). +There have been other attempts to obtain the action of a causal set. In + +Sverdlov and Bombelli (2009), the curvature at the centre of an Alexandrov +interval Ad[p, q] in a RNN was obtained using the leading order corrections to +the volume of a small causal diamond (Gibbons and Solodukhin 2007) + +V = V0 + +� +1 − +d + +24(d + 1)(d + 2)R(0)T 2 + +d + +24(d + 1)R00T 2 +� +, +(38) + +where T is the proper time from p to q and V0 is the flat spacetime volume. +The expression obtained is in terms of the discrete volume and the length of + +18 ϵ is a new free parameter in the theory, whose value should ultimately be decided by +the fundamental dynamics. + + +The causal set approach to quantum gravity +39 + +20 +40 +60 +80 +100 +120 +140 + +n + +-1.0 + +-0.5 + +0.5 + +1.0 + +f + +Fig. 15 The function f(n, 0.05). There are 4 regions of alternating sign corresponding to 4 +“smeared out” layers. + +the longest chain from p to q. Since R is approximately a constant in Ad[p, q], +this also gives the approximate action. Extending it to an action on the full +spacetime is however quite tricky since it is unclear how to localise the calcu- +lation. +The calculation for the abundance of k-chains Ck in an RNN in Roy et al + +(2013) also gives an expression for the curvature + +R(0) = −2(n + 2)(2n + 2)(3n + 2)2 +3n+2 + +3n n +4 +3n −1 +(K1 − 2K2 + K3) + +(J1 − 2J2 + J3) +3n+2 + +3n . +(39) + +where + +Jk ≡ (kn + 2)Kk +Kk ≡ ((k + 1)n + 2)Qk, +(40) + +and + +Qk ≡ +�⟨Ck⟩ + +ρkζk + +�3/k += 1 + +ζ3 +0 + +�⟨Ck⟩ + +ρkχk + +�3/k +. +(41) + +While this expression is compact, it is not defined on a single causal set, but +rather, over the ensemble. Whether this can be expressed as a function on +a single causal set or not is an interesting open question and under current +investigation. As in the previous case, having obtained R(0), however, it is +non-trivial to construct the action, without a localisation requirement as was +done for the BD action. + +4.6 Boundary terms for the causal set action + +Although the BD action gives the bulk Einstein–Hilbert action in the con- +tinuum approximation, the role of boundary terms is less clear. As shown by +Benincasa et al (2011) the expectation value for the BD action does not vanish +for C(A2, ρc), where A2[p, q] ⊂ M2 as one might expect, but instead converges + + +40 +Sumati Surya + +to a constant as ρc → ∞ and is independent of vol(A2). Buck et al (2015) +showed more generally that for C(Ad, ρc) with d ≥ 2 that + +lim +N→∞ +1 +ℏ + +� +Sd +BDG +� += +1 + +ld−2 +p +vol(J (d−2)) , +(42) + +where J (d−2) ≡ ∂J+(p) ∩ ∂J−(q) is the co-dimension 2 “joint” of the causal +diamond Ad, which is a round sphere Sd−2. In d = 2 this is the volume of a +zero sphere S0 which is the constant found in (Benincasa et al 2011). This in +turn corresponds to the Gibbons–Hawking–York (GHY) null boundary term +of (Jubb et al 2017; Lehner et al 2016) for a particular choice of the null affine +parameter.19 Extending this calculation to curved spacetime is challenging but +would provide additional evidence that the BD action contains the null GHY +term (Dhingra, Glaser and S. Surya, work in progress). +Simulations of causal sets corresponding to different regions of M2 moreover +suggest that while the BD action contains timelike boundary terms, it does +not contain spacelike boundary terms. Recent efforts by Cunningham (2018a) +have been made to obtain time like boundaries in a causal set using numerical +methods for d = 2, but it is an open question whether they admit a simple +characterisation in arbitrary dimensions. +Unlike timelike boundaries, spacelike boundaries are naturally defined in +a finite element causal set: a future/past spatial boundary is the future- +most/past-most inextendible antichain in the causal set, which we denote as +F0, P0 respectively. GHY terms for spacelike boundaries play an important role +in the additivity of the action in the continuum path integral (though such an +additivity is far from guaranteed in a causal set because of non-locality). +The spatial causal set GHY terms were found by Buck et al (2015), and +we will describe that construction here briefly. Let (M, g) be a spacetime with +initial and final spatial boundaries (Σ±, h±) . The GHY term on (Σ±, h±) +can be re-expressed as +� + +Σ± dd−1x +√ + +h± K± = ∂ + +∂n + +� + +Σ± dd−1x +√ + +h± = ∂ + +∂nAΣ±, +(43) + +where +∂ +∂n is the normal derivative, and AΣ± is the co-dimension 1 volume of +Σ±. Using the n ∼ ρcv correspondence, this suggests that AΣ± should be given +by the cardinality F0 ≡ |F0| or P0 ≡ |P0| with the normal gradient represented +by the change in the cardinality. But of course this is subtle, since apart from +the future most F0 or pastmost P0 antichains, one needs another “close by” +antichain. Let us focus on (Σ+, h+) without loss of generality. There are two +ways of finding this nearby antichain. To begin with if (M, g) ⊂ (M ′, g′) such +that (Σ+, h+) is not a boundary in (M ′, g′), then we can use this embedding +to define the two antichains, in any C ∈ C(N, ρc): one to its immediate past +F0(Σ+) and one to its immediate future P0(Σ+). Thus the GHY term should +be proportional to the difference in the cardinality of these two antichains. + +19 It is an interesting question whether the choice of affine parameter along “almost” null +directions can be obtained from the causal set. + + +The causal set approach to quantum gravity +41 + +However, this partitioning is not intrinsic to the causal set. Instead consider +a partition C = C− ∪ C+, such that C+ ∩ C− = ∅, and Fut(C−) = C+, +Past(C+) = C−. Let F− +0 and P+ +0 , be the future-most and past-most antichains +of C− and C+ respectively. We can then define the dimensionless causal set +“boundary term” (Buck et al 2015) + +Sd +CBT[C, C−, C+] ≡ +ad + +2Γ +� 2 + +d +� +� +F0[C−] − P0[C+] +� +, +(44) + +where + +ad = d(d + 1) + +(d + 2) + +� +Vd−2 + +d(d − 1) + +� 2 + +d +, +(45) + +and Vd = (d + 1)π +d+1 + +2 /Γ +� d+1 + +2 ++ 1 +� +is the volume of the unit d-sphere. +To make contact with the continuum, let (M, g) be a spacetime with +compact Cauchy hypersurfaces. For a given Cauchy hypersurface (Σ, h) let +M ± = J±(Σ) and let C± ∈ C(M ±, ρc). It was shown by Buck et al (2015) +that in the limit ρc → ∞ + +lim +ρc→∞ + +�ℓc + +ℓp + +�d−2� +S(d) +CBT[M, Σ, ρc] +� += +1 + +ld−2 +p + +� + +Σ +dd−1x +√ + +hK = SGHY (Σ, M −), (46) + +where S(d) +CBT is the associated random variable in (M, g). To obtain this ex- +pression, the volume of a half cone J+(p) ∩ J−(Σ) was calculated using a +combination of RNN coordinates and GNN coordinates20 + +V▲(T, x) = +Sd−2 + +d(d − 1)T d +� +1 + +d + +2(d + 1)K(0, x)T +� ++ O(T d+2), +(47) + +for p ∈ J+(Σ) sufficiently close to Σ, where T is the proper time from p to +Σ. As might be expected from dimensional considerations, the leading order +correction to the flat spacetime volume of the half cone comes from the trace of +the extrinsic curvature of Σ from which the GHY contribution can be obtained. +If on the other hand, (Σ, h) is a future boundary of (M, g), then we require +a second antichain in Past(F0) for C ∈ C(M, ρc),. Define the antichain F1 in +C− to be the set of elements in C− such that ∀e ∈ F1, |Fut(e)∩C−| = 1 (where +Fut(e) excludes the element e).21 The boundary term can then be expressed +as + +Sd +CBT[C, C−, C+] ≡ +ad + +Γ +� 2 + +d +� +� +dF1[C−] − F0[C+] +� +, +(48) + +which again yields the GHY term Eq. (46) in the limit. Indeed, a whole fam- +ily of of boundary terms was obtained using the antichains Fk[C−] = {e ∈ + +20 This calculation has later been extended by Jubb (2017) to higher orders to obtain more +information about the spatial geometry. +21 Note that while F1 ∩ F0 = ∅, F1 is not necessarily an inextendible antichain. + + +42 +Sumati Surya + +C−||Fut(e)| = k}, Pk[C+] = {e ∈ C+||Past(e)| = k} each of which gives the +GHY term in the limit Eq. (46).22 + +A by-product of the analysis of Buck et al (2015) is that for the partitioned +causal set C = C− ∪ C+ described above, the quantities + +Ad ++[C−] ≡ +bd + +Γ( 1 + +d)F0[C−], +Ad +−[C+] ≡ +bd + +Γ( 1 + +d)P0[C+] +(49) + +for ad = +d+1 + +d(d+2)b2 +d limit to the spatial volume of Σ + +lim +ρc→∞ + +�ℓp + +ℓc + +�d−2 +⟨Ad +±[C∓]⟩ = +1 + +ℓd−1 +p + +� + +Σ +dd−1x +√ + +h = AΣ. +(50) + +Again, as for the boundary terms, one can construct a whole family of functions +Ad[C] each of which limit to the spatial volume of Σ as ρc → ∞. + +4.7 Localisation in a causal set + +In these calculations generalisations are made to curved spacetime using an +RNN which represent a local region of a spacetime. How are we to find such +local regions in a causal set using a purely order theoretic quantities? For +a causal set a natural definition of a local region is given by the size of an +interval, but for a manifold-like causal set, this will not necessarily correspond +to regions in which the curvature is small. On the other hand, many of the +order invariants we have obtained so far correspond to geometric invariants +only in such RNN-type regions. +A characterisation of intrinsic localisation was obtained by Glaser and +Surya (2013) using the abundance N d +m of m element order intervals for C ∈ +C(Ad, ρc). They found the following closed form expression for the associated +expectation value + +⟨Nd +m(ρ, V )⟩ =(ρV )m+2 + +(m + 2)! +Γ (d)2 + +� d + +2(m + 1) + 1 +� + +d−1 + +1 +� d + +2m + 1 +� + +d−1 + +dFd + +� +1 + m, 2 + +d + m, 4 + +d + m, . . . , 2(d−1) + +d ++ m) + +3 + m, 2 + +d + m + 2, 4 + +d + m + 2, . . . , 2(d−1) + +d ++ m + 2 + +���� − ρV + +� + +, + +(51) + +The distribution of ⟨Nd +m⟩ with m therefore has a characteristic form which +depends on dimension, and as a by-product, can be used as a dimension es- +timator. However, it can also be used look for intervals in a manifold-like causal +set which are approximately flat by comparing the interval abundances N d +m to +the above expression for ⟨Nd +m⟩. While one might expect the fluctuations for a + +22 The expression in Buck et al (2015) holds for any two subsets of C not just those we +consider here. + + +The causal set approach to quantum gravity +43 + +given causal set C to be large, numerical simulations show that there is typic- +ally a “self averaging” which results in relatively small fluctuations even for a +given realisation. This makes it an ideal diagnostic tool for checking whether +a neighbourhood in a manifold-like causal set is approximately flat or not. +Once such local neighbourhoods have been found, a local check of geometric +estimators can be made. + +In Glaser and Surya (2013), the analytic curves were compared against +simulations for a range of different causal sets including those that are not +manifold-like . While curvature affects the abundance of the intervals, the dis- +tribution retains its characteristic form. Hence the dependence of the abund- +ance of intervals with size also becomes a test for manifold-likeness. + +0 + +50 + +100 + +150 + +200 + +250 + +300 + +350 + +400 + +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 + +ÈNd +mÍ + +m + +simulated + +analytic + +(a) 2d -100 Points + +Figure 1 + +1 + +Fig. 16 The expectation value of interval abundances in a 100 element causal set ∼ M2 as +a function of interval size m. The red dots depict the average value obtained from simula- +tions with 1000 realisations, along with error bars. The solid blue line depicts the analytic +expectation value for n = 100 and the blue dotted lines for n ± √n. + +There are other ways of testing for manifold-likeness. In a similar approach, +the distribution of the longest chains or linked paths of length k in a finite +element causal set C has been studied in Md, d = 2, 3, 4 and shown to have a +dimension-dependent peak (Aghili et al 2018). In Bolognesi and Lamb (2016), +a novel way to test for manifold-likeness was given, using the order invariant +obtained from counting the number of elements with a fixed valency in a +finite element causal set. In Henson (2006a), an algorithm for determining the +embeddability of a causal set in M2 was given, which again gives an intrinsic +characterisation of manifold-likeness in d = 2. Extending and expanding on +these studies using causal sets obtained from sprinklings into different types +of spacetimes would be a straightforward but useful exercise. + + +44 +Sumati Surya + +4.8 Kinematical entropy + +Since the classical continuum geometry itself is fundamentally statistical in +CST, it is interesting to ask if a kinematic entropy can be assigned even clas- +sically to the continuum. In Dou and Sorkin (2003), a kinematic entropy was +associated with a horizon H and a spatial or null hypersurface Σ in a di- +mensionally reduced d = 2 black hole spacetime by counting links between +elements in J−(Σ)∩J−(H) and those in J+(Σ)∩J+(H), with the additional +requirement that the former is future-most and the latter past-most in their +respective regions. A dimensionally reduced calculation showed that the num- +ber of links is proportional to the horizon area. Importantly, the calculation +yields the same constant for a dimensionally reduced dynamical spacetime +where a collapsing shell of null matter eventually forms a black hole. However, +extending this calculation to higher dimensions proves to be tricky. In Marr +(2007), an entropy formula was proposed for higher dimensions by replacing +links with other sub-causal sets. While these ideas hold promise, they have +not as yet been fully explored. +In analogy with Susskind’s entropy bound, the maximum causal set en- +tropy associated with a finite spherically symmetric spatial hypersurface Σ +was defined by Rideout and Zohren (2006) as the number of maximal or future +most elements in its future domain of dependence D+(Σ). It was shown that +for several such examples this bound limits to the Susskind entropy bound +in the continuum approximation. Again, extending this discussion to more +general spacetimes is an interesting open question. +In Benincasa (2013), the mutual information between different regions of a +causal set was defined using the BD action. The source of this entropy is non- +locality which implies that SBD is not in general additive. Dividing a causal +set C into two (set-wise) disjoint regions C1 and C2, so that C = C1 ⊔ C2, we +see that in general SBD(C) ̸= SBD(C1) + SBD(C2). This is because there can +be order intervals between elements in C1 and in C2 that are not counted by +either SBD(C1) or SBD(C2). The mutual information is thus defined as + +MI[C,C2] ≡ SBD(C1) + SBD(C2) − SBD(C). +(52) + +In (Benincasa 2013) a spacetime region with a horizon H and a spacelike or +null hypersurface Σ was considered. Defining X = J+(H) ∩ J−(Σ) and Y = +J−(H)∩J−(Σ) the mutual information between X and Y was calculated from +a causal set obtained from sprinkling into X ∪ Y . Under certain assumptions, +this equal to the area of H ∩ Σ. These results are suggestive, but currently +incomplete. +As we will see in the next section, the Sorkin spacetime entanglement +entropy (SSEE) for a free scalar field provides a different avenue for exploring +entropy. + + +The causal set approach to quantum gravity +45 + +4.9 Remarks + +To conclude this section we note that several order invariants have been con- +structed on manifold-like causal sets whose expectation values limit to mani- +fold invariants as ρc → ∞. At finite ρc there are fluctuations that serve to dis- +tinguish the fundamental discreteness of causal sets from the continuum, and +these have potential phenomenological consequences. Numerical simulations +are often important in assessing the relative importance of these fluctuations. +For each of these invariants, one has therefore proved an O-Hauptvermutung. +While this collection of order invariants is not sufficient to prove the full +Hauptvermutung, they lend it strong support. These order invariants are +moreover important observables for the full theory. In addition to these manifold- +like order invariants, there are several other order invariants that can be con- +structed, some of which may be important to the deep quantum regime but +by themselves hold no direct continuum interpretation. + +5 Matter on a continuum-like causal set + +Before passing on to the dynamics of CST, we look at a phenomenologically +important question, namely how quantum fields behave on a fixed manifold- +like causal set. The simplest matter field is the free scalar field on a causal set in +Md. As we noted in the previous Section, this is the only class of matter fields +that we know how to study, since at present no well defined representation of +non-trivial tensorial fields on causal sets is known. However, as we will see, +even this very simple class of matter fields brings with it both exciting new +insights and interesting conundrums. + +5.1 Causal set Green functions for a free scalar field + +Consider the real scalar field φ : Md → R and its CST counterpart, φ : C → R +where C ∈ C(Md, ρc). The Klein Gordon operator of the continuum is replaced +on the causal set by the Bκ operator of Sect. 4, Eq. (36). In the continuum +□−1 gives the Green function, and we can do the same with Bκ to obtain the +discrete Green function B−1 +κ . +However, there are more direct ways of obtaining the Green function as +was shown in Daughton (1993); Salgado (2008); Johnston (2008); Dowker et al +(2017). The causal matrix + +C0(e, e′) ≡ +� 1 if e′ ≺ e +0 otherwise +(53) + +on a causal set C. For C ∈ C(Md, ρc), C0(e, .) is therefore zero everywhere +except within the past light cone of e at which it is 1. In d = 2, this is just + + +46 +Sumati Surya + +half the massless retarded Green’s function G(2) +0 (x, x′) = 1 + +2θ(t−t′)θ(τ 2(x, x′)). +Hence, we find the almost trivial relation + +C0(x, x′) = 2G(2) +0 (x, x′), +(54) + +without having to take an expectation value, so that the dimensionless massless +causal set retarded Green function is (Daughton 1993) + +K(2) +0 (x, x′) ≡ 1 + +2C0(x, x′). +(55) + +To obtain the d = 4 massless causal set Green function we use the link +matrix + +L0(x, x′) := +�1 if x′ ≺ x is a link +0 otherwise +(56) + +For C ∈ C(M4, ρc) the expectation value of the associated random variable is + +⟨L0(x, x′)⟩ = θ(x0 − x′ +0)θ(τ 2(x, x′)) exp(−ρcV (x, x′)), +(57) + +where V (x, x′) = vol(J−(x) ∩ J+(x′)) = +π +24τ 4(x, x′). Since the exponential in +the above expression is a Gaussian which, in the ρc → ∞ limit is proportional +to δ(τ 2), we see that it resembles the massless retarded Green function in M4, + +lim +ρc→∞ + +�ρc + +6 ⟨L0(x, x′)⟩ = θ(x0 − x′ +0)δ(τ 2) = 2πG(4) +0 (x, x′). +(58) + +Hence we can write the dimensionless massless causal set scalar retarded Green +function as (Johnston 2008, 2010) + +K(4) +0 (x, x′) = 1 + +2π + +� + +1 +6L0(x, x′) . +(59) + +In the continuum the massive Green function can be obtained from the +massless Green function in Md via the formal expression (Dowker et al 2017) + +Gm = G0 − m2 G0 ∗ G0 + m4 G0 ∗ G0 ∗ G0 + . . . = + +∞ +� + +k=0 +(−m2)k G0 ∗ G0 ∗ . . . G0 +� +�� +� +k+1 +(60) +where +(A ∗ B)(x, x′) ≡ +� +ddx1 +� + +−g(x1)A(x, x1)B(x1, x′) . +(61) + +Using this as a template, with the discrete convolution operation given by +matrix multiplication, + +(A ∗ B)(e, e′) ≡ +� + +e′′ +A(e, e′′)B(e′′, e) , +(62) + +a candidate for the d = 2 dimensionless massive causal set Green function is + +K(2) +M (x, x′) = 1 + +2 + +∞ +� + +k=0 +(−1)k M 2k + +2k Ck(x, x′). +(63) + + +The causal set approach to quantum gravity +47 + +Here M is dimensionless and we have used the relation Ck(x, x′) = Ck +0 (x, x′), +where the product is defined by the convolution operation Eq. 61 and, Ck(x, x′) +counts the number of k-element chains from x to x′. For C ∈ C(M2, ρc) it can +be shown that (Johnston 2008, 2010) + +⟨K(2) +M (x, x′)⟩ = G(2) +m (x, x′) , +(64) + +when M 2 = m2 + +ρc . Similarly, a candidate for the d = 4 massive causal set Green +function is + +K(4) +M (x, x′) = +1 + +2π +√ + +6 + +∞ +� + +k=0 +(−1)k +� M 2 + +2π +√ + +6 + +�k +Lk(x, x′) , +(65) + +where we have used the fact that the number of k-element linked paths Lk(x, x′) = +Lk +0(x, x′). For C ∈ C(M4, ρc), + +lim +ρc→∞ +√ρc⟨K(4) +M (x, x′)⟩ = G(4) +m (x, x′) , +(66) + +when M 2 = +m2 +√ρc . +These massive causal set Green function were first obtained by Johnston + +(2008, 2010) using an evocative analogy between Feynman paths and the k- +chains or k-linked paths (see Fig. 17). “Amplitudes” a and b are assigned +to a “hop” between two elements in the Feynman path, and to a “stop” at +an intervening element, respectively. This gives a total “amplitude” ak+1bk + +for each chain or linked path, so that the massive Green functions can be +expressed as + +K(2) +m (e, e′) ≡ +� + +k=0 +ak+1 +2 +bk +2Ck(e, e′), +K(4) +m (e, e′) ≡ +� + +k=0 +ak+1 +4 +bk +4Lk(e, e′), +(67) + +where the coefficients ad, bd are set by comparing with the continuum. + +� + +� + +� + +� + +� + +e' + +e + +Fig. 17 The hop and stop amplitudes a and b on a 2-element chain from e to e′ for a +massive scalar field on a causal set. + + +48 +Sumati Surya + +Finding causal set Green functions for other spacetimes is more challenging, +but there have been some recent results (Dowker et al 2017) which show that +the flat spacetime form of Johnston (2008, 2010) can be used in a wider context. +These include (a) a causal diamond in an RNN of a d = 2 spacetime with +M 2 = ρc−1(m2 + ξR(0)), where R(0) is the Ricci scalar at the centre of the +diamond and ξ is the non-minimal coupling, (b) a causal diamond in an RNN +of a d = 4 spacetime with Rab(0) ∝ gab(0) and M 2 = ρc−1(m2 + ξR(0)) when +(c) d = 4 de Sitter and anti de Sitter spacetimes with M 2 = ρc−1(m2 + ξ). +The de Sitter causal set Green function in particular allows us to explore +cosmological consequences of discreteness, one of which we will describe be- +low. It would be useful to extend this construction to other conformally flat +spacetimes of cosmological relevance like the flat FRW spacetimes. Candid- +ates for causal set Green functions in M3 have also been obtained using both +the volume of the causal interval and the length of the longest chain (John- +ston 2010; Dowker et al 2017), but the comparisons with the continuum need +further study. +As the attentive reader would have noticed, in d = 4 the causal set Green +function matches the continuum only for ρc → ∞, unlike in d = 2. At fi- +nite ρc, there can be potentially observable differences with the continuum. +Comparisons with observation can therefore put constraints on CST. Dowker +et al (2010a) examined a model for the propagation of a classical massless +scalar field from a source to a detector on a background causal set. In Md, +an oscillating point source with scalar charge q, frequency ω and amplitude a, +and a “head-on” rectangular shaped detector was considered, so that the field +produced by the source is + +φ(y) = +� + +P +G(y, x(s))qds +(68) + +where P is the world line of the source and s the proper time along this world +line. If D represents the spacetime volume swept out by the detector during +its detection time T then the output of the detector is + +F = +� + +D +φ(y)d4y = q +� + +P +ds +� + +D +d4yG(y, x(s)) ≈ + +� + +1 + ν +1 − ν +q + +4πRvD +(69) + +where R is the distance between the source and detector, ν is the component +of the velocity along the displacement vector between the source and detector +and vD is the spacetime volume of the detector region D. Here, R >> a and +R >> ω−1 which in turn is much larger than the spatial and temporal extent +of the detector region D. The causal set detector output can then be defined +as +�F = q +1 + +2π +√ + +6 + +� + +e∈ ˜ +P + +� + +e′∈ ˜ +D +L0(e′, e) +(70) + +where ˜D and ˜P correspond to the detector and source subregions in the causal +set and the causal set function L(e, e′) is equal to some normalisation constant +κ when e and e′ are linked and is zero otherwise. For C ∈ C(M4, ρc) it was + + +The causal set approach to quantum gravity +49 + +shown that, with the above constraints on R, ω, a and the dimensions of the +detector, that ⟨�F⟩ approximates to same continuum expression Eq. (69) when + +R >> ρ +− 1 + +c 4 +. A detailed calculation gives an upper bound on the fluctuations, +which, for a particular AGN model is one part in 1012 for ρc = ρp. Hence the +discreteness does not seem to mess with the coherence of waves from distant +sources. As we will see in Sect. 7 there are other potential signatures of the +discreteness that may have phenomenological consequences (Dowker et al 2004; +Sorkin 1991, 1997; Ahmed et al 2004). + +5.2 The Sorkin–Johnston (SJ) vacuum + +Having obtained the classical Green function and the d’Alembertian operator +in M2 and M4, the obvious next step is to build a full quantum scalar field +theory on the causal set. As we have mentioned earlier, the canonical route to +quantisation is not an option for causal sets nor for fields on causal sets and +hence there is a need to look at more covariant quantisation procedures. + +Johnston (2009, 2010) used the the covariantly defined Peierls’ bracket + +[�Φ(x), �Φ(y)] = i∆(x, y) +(71) + +as the starting point for quantisation, where + +∆(x, y) ≡ GR(x, x′) − GA(x, x′) +(72) + +is the Pauli Jordan function, and GR,A(x, x′) are the retarded and advanced +Green’s functions, respectively. As we have seen, these Green functions can +be defined on certain manifold-like causal sets and hence provide a natural +starting point for quantisation. +However, even here, the standard route to quantisation involves the mode +decomposition of the space of solutions of the Klein Gordan operator, ker(□− +m2). In Md the space of solutions has a unique split into positive and negative +frequency classes of modes with respect to which a vacuum can be defined. In +his quest for a Feynman propagator, Johnston (2009) made a bold proposal, +which as we will describe below, has led to a very interesting new direction in +quantum field theory even in the continuum. This is the Sorkin–Johnston or +SJ vacuum for a free quantum scalar field theory. +Noticing that the Pauli–Jordan function on a finite causal set C is a Her- +mitian operator, and that ∆(e, e′) itself is antisymmetric, Johnston used the +fact that the eigenspectrum of i∆ + +i � +∆ ◦ vk(e) ≡ +� + +e′∈C +i∆(e, e′)vk(e′) = λkvk(e) +(73) + +splits into pairs (λk, −λk), with eigenfunctions (v+ +k , v− +k ), v− +k += v+ +k +∗. This +provides a natural split into a positive part and a negative part, without + + +50 +Sumati Surya + +explicit reference to ker(□−m2).23 A spectral decomposition of i � +∆ then gives + +i∆(e, e′) = λk +� + +k +v+ +k (e)v+ +k +∗(e′) − v+ +k (e)∗v+ +k (e′). +(74) + +This decomposition is used to define the SJ Wightmann function as the positive +part of i∆ + +WSJ(e, e′) ≡ λk +� + +k +v+ +k (e)v+ +k +∗(e′). +(75) + +Importantly, for a non-interacting theory with a Gaussian state, the Wight- +mann function is sufficient to describe the full theory and thus the vacuum +state. Simulations in Md for d = 2, 4 give a good agreement with the continuum +(Johnston 2009, 2010). + +Sorkin (2011a) noticed that the construction on the causal set, which was +born out of necessity, provides a new way of thinking of the quantum field +theory vacuum. A well known feature of quantum field theory in a general +curved spacetime is that the vacuum obtained from mode decomposition in +ker(�□−m2) is observer dependent and hence not unique. Since the SJ vacuum +is intrinsically defined, at least in finite spacetime regions, one has a uniquely +defined vacuum. As a result, the SJ state has generated some interest in the +broader algebraic field theory community (Fewster and Verch 2012; Brum and +Fredenhagen 2014; Fewster 2018). For example, while not in itself Hadamard +in general, the SJ vacuum can be used to generate a new class of Hadamard +states (Brum and Fredenhagen 2014). +In the continuum, the SJ vacuum was constructed for the massless scalar +field in the d = 2 causal diamond (Afshordi et al 2012) and recently extended +to the small mass case (Mathur and Surya 2019). It has also been obtained +for the trousers topology and shown to produce a divergent energy along both +the future and the past light cones associated with the Morse point singularity +(Buck et al 2017). Numerical simulations of the SJ vacuum on causal sets are +are approximated by de Sitter spacetime suggest that the causal set SJ state +differs significantly from the Mottola–Allen α vacuua (Surya et al 2018). This +has potentially far reaching observational consequences which need further +investigation. + +5.3 Entanglement entropy + +Using the Pauli Jordan operator i � +∆ and the associated Wightman � +W, Sorkin + +(2014) defined a spacetime entanglement entropy, Sorkins’ Spacetime Entan- +glement Entropy (SSEE) + +S = +� + +i +λi ln |λi| +(76) + +23 The identification of ker(□ − m2) with Im(i∆) is in fact well known (Wald 1994) when +the latter is restricted to functions of compact support. + + +The causal set approach to quantum gravity +51 + +where λi are the generalised eigenvalues satisfying + +� +W ◦ vi = iλi � +∆ ◦ vi. +(77) + +It was shown by Saravani et al (2014) that for a causal diamond sitting at +the centre of a larger one in M2, S has the expected behaviour in the limit +that the size of the smaller diamond l is much smaller than that of the larger +diamond, + +S = b ln +� l + +luv + +� ++ c, +(78) + +where luv is the UV cut-off and b, c are constants that can be determined. +One of the promises that discretisation holds is of curing the UV diver- +gences of quantum field theory and in particular those coming from the cal- +culation of the entanglement entropy of Bombelli et al (1986). As shown by +Sorkin and Yazdi (2018) the causal set version of the above calculation is pro- +portional to the volume rather than the above “area”, thus differing from the +continuum. This can be traced to the fact that the continuum spectrum of +eigenvalues (Eq. 77) agrees with the discrete eigenvalues only up to a “knee”, +beyond which the effects of discreteness become important, as shown in Fig. 18. +Using a double truncation of the spectrum – once in the larger diamond and +once in the smaller one, Sorkin and Yazdi (2018) obtained the requisite area +law. This raises very interesting and as yet unanswered puzzles about the +nature of SSEE in the causal set. It is for example possible that in a funda- +mentally non-local theory like CST an area law is less natural than a volume +law. Such a radical understanding could force us to rethink continuum inspired +ideas about Black Hole entropy. + +Fig. 18 A log-log plot depicting the SJ spectra for causal sets in a causal diamond in M2. +A comparison with the continuum (the straight black line) shows that the causal set SJ +spectrum matches the continuum in the IR but has a characteristic “knee” in the UV after +which it deviates significantly from the continuum. As the density of the causal set increases, +this knee shifts to the UV. + +Extending the above calculation to actual black hole spacetimes is an +important open problem. Ongoing simulations for causal sets obtained from +sprinklings into 4d de Sitter spacetime show that this double truncation pro- +cedure gives the right de Sitter horizon entropy (Dowker, Surya, Sumati, X + + +52 +Sumati Surya + +and Yazdi, work in progress), but one first needs to make an ansatz for locating +the knee in the causal set i∆ spectrum. + +5.4 Spectral dimensions + +An interesting direction in causal set theory has been to calculate the spec- +tral dimension of the causal set (Eichhorn and Mizera 2014; Belenchia et al +2016c; Carlip 2017). Carlip (2017) has argued that d = 2 is special in the UV +limit, and that several theories of quantum gravity lead to such a dimensional +reduction. In light of how we have presented CST, it seems that this con- +tinuum inspired description must be limited. It is nevertheless interesting to +ask if causal sets that are manifold-like might exhibit such a behaviour around +the discreteness scales at which the continuum approximation is known to +break down. As we have seen earlier (Sect. 4.3), one such behaviour is discrete +asymptotic silence (Eichhorn et al 2017). + +Eichhorn and Mizera (2014) calculated the spectral dimension on a causal +set using a random walk on a finite element causal set. It was found that +in contrast, the dimension at small scales goes up rather than down. On the +other hand, Belenchia et al (2016c) showed that causal set inspired non-local +d’Alembertians do give a spectral dimension of 2 in all dimensions. As we noted +in Sect. 4, Abajian and Carlip (2018) showed that dimensional reduction of +causal sets occurs for the Myrheim–Meyer dimension as one goes to smaller +scales. Recently (Eichhorn et al 2019), the spectral dimension was calculated +on a maximal antichain for a causal set obtained from sprinklings into Md, +d = 2, 3 using the induced distance function of Eichhorn et al (2018). It was +seen to decrease at small scales, thus bringing the results closer to those from +other approaches. + +6 Dynamics + +Until now our focus has been on manifold-like causal sets, since the aim was +to find useful manifold-like covariant observables as well as to make contact +with phenomenology. However, as discussed in Sect. 3, the arena for CST is a +sample space Ω of locally finite posets which replaces the space of 4-geometries, +and contains non-manifold-like causal sets. A CST dynamics is given by the +measure triple (Ω, A, µ) where A is an event algebra and µ is either a classical +or a quantum measure. We will define these quantities later in this section. +To begin with, Ω itself can be chosen depending on the particular physical +situation in mind. In the context of initial conditions for cosmology, for ex- +ample, it is appropriate to restrict to the sample space of past finite countable +causal sets Ωg, while for a unimodular type dynamics using the Einstein– +Hilbert action, the natural restriction is to Ωn the sample space of causal +sets of fixed cardinality n. We will see that dimensional restrictions on the +sample space are also of interest and can lead to a closer comparison with +other approaches to quantum gravity. + + +The causal set approach to quantum gravity +53 + +As discussed in Sect. 3 and 4, in the asymptotic n → ∞ limit the sample +space Ωn is dominated by the non-manifold-like KR causal sets depicted in +Fig. 9. This is the “entropy problem” of CST. These posets have approximately +just three “moments” of time and hence should not play a role in the classical +or continuum approximation of the theory. +For a quantum dynamics of CST we would like to start with a few basic +axioms, including discrete general covariance and dynamical causality. A very +important step in this direction was made by the classical sequential growth +models (CSG)(Rideout and Sorkin 2000a) , which are Markovian growth mod- +els. We will describe these in Sect. 6.1 and 6.2. +One of the main challenges in CST is to build a viable quantum sequential +growth(QSG) dynamics. The appropriate framework for the dynamics is as a +quantum measure space which is a natural quantum generalisation of classical +stochastic dynamics (Sorkin 1994, 1995, 2007d). This means replacing the +classical probability measure P in the measure space triple (Ω, A, µc) with +a quantum measure µ. The quantum measure is defined via a decoherence +functional and can also be defined as a vector measure in a corresponding +histories Hilbert space. We will discuss this in Sect. 6.3. +It is also of interest to construct an effective continuum-inspired dynamics, +where the discrete Einstein–Hilbert or BD action is used to give the measure +for the discrete path integral or path sum. The quantum partition function +can either be evaluated directly or converted into a statistical partition func- +tion over causal sets using an analytic continuation. This makes it amenable +to Markov Chain Monte Carlo (MCMC) simulations as we will see below in +Sect. 6.4. + +6.1 Classical sequential growth models + +The Rideout and Sorkin (2000a) classical sequential growth or CSG models +are a class of stochastic dynamics in which causal sets are grown element by +element, with the dynamics satisfying a few basic principles (Rideout and Sor- +kin 2000a, 2001; Martin et al 2001; Rideout 2001; Varadarajan and Rideout +2006). The stochastic dynamics finds a natural expression in measure theory +and allows for an explicit definition of covariant classical observables (Bright- +well et al 2003; Dowker and Surya 2006). This measure theoretic structure +provides an important template for the quantum theory, and hence we will +first flesh it out in some detail before discussing quantum dynamics. +Let us start with a naive picture. Imagine living on a classical causal set +universe, with our universe represented by a single causal set. Since causal +sets are locally finite, the “passage of time” occurs with the addition of a new +element. If we are to respect causality, this new element cannot be added so as +to disturb the past. Instead it can be added to the future of some of the existing +events or it can be unrelated to all of them. Every such “atomic change” in +spacetime corresponds to the causal set changing cardinality or “growing” by +one. Starting with a causal set ˜cn of cardinality n, the passage of time means + + +54 +Sumati Surya + +transitioning from ˜cn → ˜cn+1 where the new element in ˜cn+1 is to the future +of some of the elements of ˜cn, but never in their past. In the infinite “time” +limit, n → ∞, the dynamics, either deterministic, probabilistic or quantum, +will take you from ˜cn to a countable causal set. +Working backwards, on the other hand, leads us to a “beginning”, with +n = 0. This gives the most natural initial condition24 for causal sets: begin +with the empty set ∅. The only way to go forward from here, is to make n = 1, +i.e., we have a single element. For n = 2, the new element could either be to +the future of the existing element or unrelated to it, as in Fig. 19. + +p +q + +Fig. 19 The first two stages of a classical sequential growth(CSG) dynamics. The prob- +ability for a single element (red) to appear at coordinate time n = 1 is 1. Subsequently, +the new element (blue) at n = 2 is added either to the future of the existing element with +probability p or is unrelated to it with probability 1 − p. + +Thus, one can build up the tree T of causal sets as n → ∞ as shown in +Fig. 20. As n increases, the number of possibilities grows superexponentially +as expected from the KR theorem (Kleitman and Rothschild 1975), and there +is no easy enumeration of this space. The growth process generates a sample +space ˜ +Ωg of countable causal sets which are are all past finite and labelled by +the “time” at which each element is added. A causal set ˜c in ˜ +Ωg is said to be +naturally labelled, i.e., there exists an injective map L : ˜c → N (the natural +numbers) which preserves the order relation in ˜c, i.e., e ≺ e′ ⇒ L(e) < L(e′). +In the growth process, this label is the coordinate time. +In the spirit of covariance, however, we cannot take the time label to be +fundamental; the dynamics and the observables cannot depend on the order +in which the elements are born. Thus, the probability to get a labelled causal +set ˜cn and any of its relabellings, ˜c′ +n must be the same. Identifying relabelled +causal sets as the same object in the CST tree T gives us a non-trivial poset +of causal sets or the “postcau” P of Rideout and Sorkin (2000a). On P, a +covariant dynamics is thus path-independent: if there is more than one path +from an unlabelled initial causal set cni to an unlabelled final causal set cnf +in P, then in order to satisfy covariance, the measure on both paths should be +the same. + +24 Of course, we could insist that there is no beginning, in which case n is never finite. + + +The causal set approach to quantum gravity +55 + +p +q + +Fig. 20 The CSG tree T . There are three ways to get the 3-element unlabelled causal set +whose natural labellings are given by the 3rd, 4th and 5th 3-element labelled causal sets in +the figure. One path is via the 2-element chain and the other two are from the 2-element +antichain. Covariance demands that the probability along each path is the same. + +Apart from covariance, this dynamics also satisfies an internal causality +condition, dubbed Bell causality. Consider the transition ˜cn → ˜cn+1 with prob- +ability αn where the new element en+1 is added to the future of a “precursor” +set pn ⊂ ˜cn, and is unrelated to a “spectator set” sn ⊂ ˜cn. Causality suggests +that the probability for the transition should not depend on the spectator set +sn. For non-empty sn with |sn| < n, consider the causal sets ˜cm = ˜cn\sn +and ˜cm+1 = ˜cn+1\sn, where \ denotes set difference and m + |sn| = n. The +transition probability αm for ˜cm → ˜cm+1 should then be proportional to αn. +If ˜cn → ˜c′ +n+1 is another transition from ˜cn, then defining p′ +n, s′ +n, α′ +n, and +˜c′ +m+1 = ˜c′ +n+1\s′ +n, analogously, the condition of Bell causality is + +αn(˜cn → ˜cn+1) +α′n(˜cn → ˜c′ +n+1) = αm(˜cm → ˜cm+1) + +α′m(˜cm → ˜c′ +m+1) +(79) + +Though relatively easy to implement classically, a quantum version of Bell +causality has been hard to find (Henson 2011). +The triple requirements of (a) covariance, (b) Bell causality and (c) Markovian +evolution define the classical sequential growth dynamics of Rideout and Sor- +kin (2000b). Starting from the empty set, a causal set is thus grown element +by element, assigning probabilities to each transition ˜cn to a ˜cn+1, consistent +with these requirements. Because of it being a Markovian evolution, the prob- +ability associated with any finite cn is given by the product of the transition +probabilities along a path in P. +The dynamics was shown in Rideout and Sorkin (2000a) to be fully de- +termined by the infinite set of coupling constants, tn, one for each stage of the +growth. If qk denotes the transition probability from the k-element antichain + + +56 +Sumati Surya + +to the k + 1-element antichain, these coupling constants can be expressed as + +tn ≡ + +n +� + +k=0 +(−1)n−k +�n +k + +� 1 + +qk +. +(80) + +In general, the tn can be independent of each other. Including relations between +the different tn thus simplifies the dynamics. The simplest example is that of +transitive percolation determined by the probability (1 − q) ≥ 0 of adding +an element to the immediate future of an existing element,25 and q of being +unrelated to it. Thus, the probability of adding a new element to the immediate +future of m elements of cn and of being unrelated to m′ others is (1 − q)mqm′. + +In terms of the general coupling constants, tn = tn ≡ +� +1−q + +q +�n +. + +In Varadarajan and Rideout (2006) and Dowker and Surya (2006), a gen- +eralisation of the dynamics was explored, where some of the transition prob- +abilities were allowed to vanish, consistent with (a) (b) and (c). This requires a +generalisation of the Bell causality condition. The resulting dynamics exhibits +a certain “forgetfulness” when these transition probabilities vanish, but are +otherwise very similar to the CSG models. +Since the generic dynamics consistent with (a), (b) and (c) does not by +itself lead to constraints on the tn, this is an embarrassment of riches. Does +nature pick out one set over another? In Martin et al (2001), an evolutionary +mechanism for doing so was suggested using cosmological bounces which give +rise to new epochs which “renormalise” the coupling constants towards fixed +points. A cosmological bounce in a causal set is naturally described by the +appearance of a post which is an inextendible antichain of cardinality 1. + +N + +Fig. 21 A post is an analogue of a bounce in causal set cosmology. + +25 By this we mean that the new element is “linked” to an existing one, not just related +to it. + + +The causal set approach to quantum gravity +57 + +Thus, every element in c either lies to its past or to its future. Moreover, +because it is a single element maximal antichain, there are no “missing links” +(see Fig. 11), and the post is indeed a summary of its past. The post is the +causal set equivalent to a “bounce” but is non-singular in the causal set. We +define the causal set between two posts as an “epoch”, with the last epoch +being the one after the last post. Let e be a post in c and let r = |Past(e)|. +Then a set of “effective” coupling constants in the epoch after e can be defined +as (Martin et al 2001) + +˜t(r) +n += + +r +� + +k=0 + +�r +k + +� +tn+k. +(81) + +Thus, the memory of the past of the post, which is common to all the elements +to the future of the post is “washed” out, but not without “dressing” up +the new effective coupling constants. Denoting the set of effective couplings +by T (i) ≡ {t(i) +0 , t(i) +1 , . . .} with i = 0 being the original set of couplings, this +corresponds to applying r copies of the transform M : T (i) → T (i+1) where +t(i+1) +n += t(i) +n + t(i) +n+1, i = 0, . . . r − 1. In Martin et al (2001), it was shown that +the fixed points of the map M give tn = tn (transitive percolation) for some +t ≥ 0 and moreover M does not have any other cycles. Starting from any set +T (0) for which limn→∞(t(0) +n )1/n is finite, M r : T (0) → T (r), is such that T (r) + +converges pointwise to t(r) +n += tn for t = limn→∞(t(0) +n )1/n. While this result does +not guarantee that every T (0) will converge to transitive percolation, Martin +et al (2001) examined several cases, and conjectured that the deviation from +percolation-like values are “rare” and that typically, T (r) will be nearly like +transitive percolation. +Such an evolutionary renormalisation thus brings the infinite dimensional +coupling constant space to a one dimensional space, which is remarkable. As- +suming that this is indeed the case in general, a sufficiently late epoch will +likely have a transitive percolation dynamics. +What can one say about the causal sets generated from this dynamics? A +very important result from transitive percolation is that the typical causal sets +obtained are not KR like posets and hence the dynamics beats their entropic +dominance. The question of whether there is a continuum-like limit for trans- +itive percolation dynamics was explored in Rideout and Sorkin (2001), using +a comparison criterion. The abundance of fixed small subcausal sets was ex- +amined as a function of the coupling, by fixing the density relations. Comparis- +ons with Poisson sprinklings in flat spacetime showed a convergence, suggestive +of a continuum limit. In Ahmed and Rideout (2010), it was shown that the +dynamics typically yields an exponentially expanding universe. Moreover, for +(1−q) ≪ 1 and n ≫ +1 + +1−q, after a post the universe enters a tree like phase and +then a de Sitter-like phase, in which the cardinality of large causal diamonds +are de Sitter like functions of the discrete proper time. In Glaser and Surya +(2013), it was shown that despite this, the abundance of causal intervals is not +de Sitter like, and thus, this is not strictly a manifold-like phase. In Bright- +well and Georgiou (2010) and Brightwell and Luczak (2015), moreover, it was +shown explicitly that in the asymptotic limit n → ∞ the causal sets limit to + + +58 +Sumati Surya + +“semi-orders” which, though temporally ordered, have no spatial structure at +all, and are hence non-manifold-like. Nevertheless, the dominance of measure +over entropy is important and the hope is that it will be reflected in the right +quantum version of the dynamics. +Recently, Dowker and Zalel (2017) proposed a method for dealing with +black hole singularities in CSG models. As in the case of cosmological bounces +a new epoch is created beyond the singularity. Using “breaks” which are multi- +element versions of a post, they demonstrated that a renormalisation of the +coupling constants occurs in the new epoch. + +6.2 Observables as beables + +As mentioned in the introduction to this section, a dynamics for CST is given +by the triple (Ω, A, µ). In CSG this is a probability measure space, where the +sample space ˜ +Ωg is the set of all past finite naturally labelled causal sets. +The event algebra A can be constructed from the sequential growth process +as follows. We define a cylinder set cyl(˜cn) ⊂ ˜ +Ωg as the set of all labelled causal +sets in ˜ +Ωg whose first n elements are the causal set ˜cn. Figure 22 depicts an +example of a cylinder set.26 For every finite element causal set ˜cn, cyl(˜cn) ⊆ +˜ +Ωg, and in the trivial n = 1 case, cyl(˜c1) = +˜ +Ωg. The cylinder sets in CSG +satisfy a nesting property. Namely, if n′ > n and cyl(˜cn′) ∩ cyl(˜cn) ̸= ∅, then +cyl(˜cn′) ⊂ cyl(˜cn). Thus, a non-trivial intersection of two different cylinder +sets is possible only if one is strictly a subset of the other. +The event algebra ˜A is generated from the cylinder sets via finite unions, +intersections and set differences. It is closed under finite set operations and +contains the null set ∅ as well as ˜ +Ωg. In the growth process we assign a prob- +ability µ(˜cn) to every finite labelled causal set ˜cn. By identifying ˜cn with its +cylinder set cyl(˜cn), we define the measure µ(cyl(˜cn)) ≡ µ(˜cn) and hence on +all elements of ˜A, since µ is finitely additive. This makes ( ˜ +Ωg, ˜A, ˜µ′) a “pre- +measure” space. +An event α is an element of A deemed to be covariant as a measurable +subset α ⊂ ˜ +Ωg if for every ˜c ∈ α, its relabelling ˜c′ also belongs to α. Since +a relabelling can happen arbitrarily far into the future, no event in A is co- +variant, since A is closed only under finite set operations. Take for example +the covariant post event which is the set of all causal sets which have a post. +This is a covariant event, and is the equivalent of the return event in the ran- +dom walk. In both cases, the event cannot be defined using only countable set +operations, and hence the post event does not belong to A. +One route to obtaining covariant events is to pass to the full sigma algebra +˜S generated by ˜A, which is closed under countable set operations. For clas- +sical measure spaces, the Kolmogorov–Caratheodory–Hahn extension theorem +allows us to extend ˜µ′ to ˜S and hence pass with ease to a full measure space + +26 A useful example to keep in mind is the 1-d random walk. Let γT be a finite element +path in the t − x plane from t = 0 to t = T. A cylinder set cyl(γT ) is then the set of all +infinite time paths, which coincide with γT from t = 0 to t = T. + + +The causal set approach to quantum gravity +59 + +Fig. 22 The cylinder set for the “V” poset consists of all countable causal sets in ˜ +Ωg whose +first three elements are the labelled “V” poset. Examples of causal sets that lie in cyl(V) +are depicted in the boxes. + +( ˜ +Ωg, ˜S, ˜µ), where ˜µ|˜A = ˜µ′. Not every event in S is covariant, but we can +restrict our attention to covariant events, i.e., sets that are invariant under +relabellings. If ∼ denotes the equivalence up to relabellings one can define the +quotient algebra S = ˜S/ ∼ of covariant events. An element of S is measur- +able covariant set, or a covariant observable (or beable). Our example of the +post event belongs to S. Another example of a covariant event is the set of +originary causal sets, i.e., causal sets with a single initial element to the past +of all other elements. Constructing more physically interesting covariant ob- +servables in S is important, since it tells us what covariant questions we can +ask of causal set quantum gravity. +A more covariant way to proceed is to generate the event algebra not via +the cylinder sets in ˜ +Ωg but by using covariantly defined sets in Ωg, the sample +space of unlabelled causal sets. Because causal sets are past finite we can use +the analogue of past sets J−(X) to characterise causal sets in a covariant way. +A finite unlabelled sub-causal set cn of c ∈ Ωg is said to be a partial stem if +it contains its own past. A stem set stem(cn) is then a subset of Ωg such that +every c ∈ stem(cn) contains the partial stem ˜cn. Let S be the sigma algebra +generated by the stem sets. Although S is a strictly smaller subalgebra of S, +it differs on sets of measure zero for the CSG and extended CSG models as +shown by Brightwell et al (2003) and Dowker and Surya (2006). Thus, one +can characterise all the observables of CSG in terms of stem sets. This is a +non-trivial result and the hope is that some version of it will carry over to the +quantum case. + + +60 +Sumati Surya + +6.3 A route to quantisation: The quantum measure + +The generalisation of CSG to QSG is, at least formally, very straightforward. +One “quantises” the classical covariant probability space (Ωg, S, µc), by simply +replacing the classical probability µc with a quantum measure µ : S → R+, +where µ satisfies the quantum sum rule (Sorkin 1994, 1995; Salgado 2002; +Sorkin 2007d)27 + +µ(α ∪ β ∪ γ) = µ(α ∪ β) + µ(α ∪ γ) + µ(β ∪ γ) − µ(α) − µ(β) − µ(γ), +(82) + +for the mutually disjoint sets α, β, γ ∈ S. µ(.) is not in general a probability +measure since it does not satisfy additivity µ(α∪β) ̸= µ(α)+µ(β) for α∩β = ∅. +As in the classical case, observables in this theory are simply the quantum +measurable sets in S. The quantum measure µ(.) can be obtained from a +decoherence functional D : S × S → C of quantum theory with + +µ(α) = D(α, α), +(83) + +where D satisfies + +– Hermiticity: D(α, β) = D∗(β, α) +– Countable biadditivity: D(α, ⊔iβi) = � + +i D(α, βi) and D(⊔iαi, β) = � + +i D(αi, β) + +– Normalisation: D(Ω, Ω) = 1 +– Strong positivity: Mij ≡ D(αi, αj) for any finite collection {αi} is positive +semi-definite + +In a QSG model the transition probabilities of CSG are replaced by the de- +coherence functional D or quantum measure. Leaving aside Bell causality, the +other principle of the growth dynamics are easy to implement. In Dowker +et al (2010c), a simple complex percolation dynamics was studied, given by +a product decoherence function ˜D(α, β) = A∗(α)A(β) on A × A, where A(α) +is obtained from the transition amplitudes q ∈ C, similar to transitive per- +colation. Thus, as in the case of CSG models, one starts with the labelled +event algebra A generated by the cylinder sets, and a quantum pre-measure +˜D′. Again, in order to obtain covariant observables one has to pass to the full +sigma algebra S associated with A. However, unlike a classical measure ˜D +need not extend to a full sigma algebra. In Dowker et al (2010c), the quantum +pre-measure was shown to be a vector pre-measure ˆµ′ in the associated histor- +ies Hilbert space (Dowker et al 2010b). Extension of ˆµ′ to S is then possible +provided certain convergence conditions are satisfied.28 + +Although the vector measure is 1-dimensional in complex percolation dy- +namics, it was shown in Dowker et al (2010c) not to satisfy this convergence +condition and hence one cannot pass to S to construct covariant observables. + +27 We will not discuss the very rich and interesting literature on the co-event interpretation +of the quantum measure, which though incomplete, contains essential features that one would +seek for a theory of quantum gravity (Sorkin 2007c). +28 In general, these are given by the conditions in the Kolmogorov–Caratheodory–Hahn– +Kluvanek theorem (Diestel and Uhl 1977). + + +The causal set approach to quantum gravity +61 + +However a smaller algebra may be sufficient for answering physically interest- +ing questions, which require far weaker convergence condition as suggested by +Sorkin (2011b). This relaxation of conditions means that some simple meas- +urable covariant observables can be constructed in complex percolation, in- +cluding for the originary event (Sorkin and Surya, work in progress). Whether +these results on extension are shared by all QSG models or not is of course an +interesting question. Another possibility is that an extension of the measure +in QSG could, for example, be a criterion for limiting the parameter space of +QSG. Very recently a class of QSG dynamics that does admit an extension +has been found (Surya and Zalel, work in progress). +The space of QSG models is largely unexplored. It is however critical to +study it extensively in order to find the right CST quantum dynamics based +on first principles. + +6.4 A continuum-inspired dynamics + +As we have seen, at a fundamental level the quantum dynamics of causal +sets looks very different from that of a continuum theory of quantum gravity, +even if the latter is formulated as a path integral. However, as one approaches +the continuum approximation of the theory, it is possible that the effective +quantum dynamics begins to resemble the continuum path integral. In CST, +the quantum partition function is + +ZΩ ≡ +� + +c∈Ω +e + +iS(c) + +ℏ +(84) + +where S(c) is an action for causal sets, and the choice of sample space Ω is +determined by the problem at hand. One might also consider more generally +a decoherence functional D(c1, c2) on causal sets, inspired by the continuum, +where D(c1, c2) = e−i 1 + +ℏ (S(c1)−S(c2))f(c1, c2) with f(c1, c2) a causal set analog of +the delta function associated with unitarity quantum theories. This is currently +an unexplored direction and we will not discuss it further in this work. +The natural choice for S(c) is the d dimensional BD action S(d) +BD(c) which +limits to the Einstein–Hilbert action in the continuum. As discussed in Sect. 3, +the sample space Ωn of causal sets of cardinality n is dominated by KR type +causal sets. An important question is whether the action S(d) +BD(c) can overcome +the KR entropy in the large n limit. +Indeed, there is a hierarchy of sub-dominant causal sets which are non +manifold-like (Dhar 1978, 1980; Kleitman and Rothschild 1975; Promel et al +2001), with the set of bilayer posets B being the next subdominant class. A +recent calculation by Loomis and Carlip (2018) shows that B is suppressed +by the BD action when the mesoscale and dimension satisfy certain condi- +tions. The only relations in a bilayer poset are links. Given that the maximum +number of relations is +�n +2 +� +the causal sets in B can be classified by the linking +fraction p given by the ratio of the total number of links N0 to the maximal + + +62 +Sumati Surya + +possible number of links +�n +2 +� +. Moreover, the action itself reduces to a simple +sum over n and N0. In the limit of large n, Loomis and Carlip (2018) consider +p to be a continuous variable using which the partition function ZB can be +expressed as an integral over p + +ZB = +� +dp|Bp,n|eiS(p)/ℏ = eiµn +� +dp|Bp,n|e +1 +2 iµλ0pn2+o(n2) +(85) + +where Bp,n denotes the class of n-element causal sets in B with linking fraction +p and µ, λ0 are related to the mesoscale ϵ and function fd(n, ϵ) that appears +in S(d) +BD(c). The challenge is then shifted to calculating |Bp,n|. Using another +parameter q which gives the cardinality of the upper layer as a further sub- +classification of Bp,n, the leading order contribution to |Bp,n| was found. The +resulting partition function was then shown to be strictly suppressed when +µλ0 satisfy the condition + +tan(−µλ0/2) > +�27 + +4 e− 1 + +2 − 1 +� +. +(86) + +This is an important analytic calculation and paves the way for a more rigorous +understanding of the CST partition function. +More than the partition function, however, it is the expectation value of +observables or order invariants + +⟨O⟩ = 1 + +Zc + +� + +c∈Ω +O[c]ei 1 + +h S[c] +(87) + +that is of physical significance.29 Evaluating this for larger values of n is how- +ever a big challenge and we turn to numerical simulations to help us. +One route could be to simply “perform” the sum above. However, given +that |Ωn| grows superexponentially (to leading order it is ∼ 2 +n2 +4 ), this is +computationally challenging even for relatively small values of n. On the other +hand, Markov Chain Monte Carlo (MCMC) methods for sampling the space +Ω can be used if we can convert ZΩ into a statistical partition function. +In CST, there is no analogue of a Wick rotation: since the order relation +derives from the causal structure, it cannot be “Euclideanised”. On the other +hand, there are other ways to analytically continue ZΩ (see Louko and Sorkin +1997 for a continuum example). One option, first explored in Surya (2012) is +to introduce a new parameter β such that + +ZΩ,β ≡ +� + +c∈Ω +ei β + +ℏ S(c). +(88) + +This allows us to analytically continue ZΩ,β from real to imaginary values +of β, thus rendering the quantum partition function into a statistical parti- +tion function. We can then use standard tools in statistical physics, including + +29 We leave out interpretational questions! + + +The causal set approach to quantum gravity +63 + +MCMC methods, to find the expectation values of suitable observables (Surya +2012; Glaser and Surya 2016; Glaser et al 2018; Glaser 2018). +In Henson et al (2017), MCMC methods were used to examine the sample +space of naturally labelled posets ˜Ωn to determine the onset of the KR regime, +using the uniform measure (β = 0). The Markov Chain was generated via a +set of moves that sample Ωn. A mixture of two moves, the link move and the +relation move, was used to obtain the quickest thermalisation. +To illustrate the complexity of these moves we describe in detail the link +move. A pair of elements e, e′ are picked randomly and independently from +the causal set c, and retained if L(e) < L(e′), where L is the natural labelling +defined in Sect. 6.1. If e ≺ e′ and moreover the relation is a link, then the +move is to “unlink” them. Those relations implied by this link via transitivity +also need to be removed. These are relations between elements in IPast(e) +and those in IFut(e′) which are “mediated” solely either by e or e′. On the +other hand if e and e′ are not related, then one adds in a link between e +and e′, provided that there are no existing links between elements in IPast(e) +and IFut(e′), after which the transitive closure is taken. In the relation move, +although the existence or non-existence of a link from e to e′ is also required, +the move doesn’t care about the sanctity of links, but is in other ways more +restrictive. Thus, for both moves, picking of a pair of elements at random in c +does not always lead to a possible move, let alone a probable one, and hence +this MCMC model is slow to thermalise. Trying to find a more efficient move +is however non-trivial precisely because of transitivity. +The simulations of Henson et al (2017) suggest that the onset of the asymp- +totic KR regime occurs for n as small as n ≈ 90. Ωn is very large even for +n = 90 (∼ 2902 !) and hence thermalisation becomes a problem very quickly. +Recently, steps have been taken to incorporate the action (β ̸= 0) into the +measure, but again, because of thermalisation issues, the size of the posets are +fairly small. +Instead of taking the full sample space, one can restrict Ωn to causal sets +that capture some gross features of a class of spacetimes. As discussed above, +for large enough n, Ωn contains causal sets that are approximated by space- +times of arbitrary dimensions. It is thus of interest to restrict the sample space +so that those causal sets that are manifold-like in the sample space are ap- +proximated only by spacetime regions of a given dimension. Such a restriction +is typically hard to find, since it requires “tailoring” Ω using non-trivial order +theoretic constraints determined by dimension estimators of the kind we have +encountered in Sect. 4. +Somewhat fortuitously, this restriction is very natural in d = 2. Here, the +sample space of “2-orders” Ω2d is one in which the continuum dimension and a +particular order theoretic dimension coincide (Brightwell et al 2008; El-Zahar +and Sauer 1988; Winkler 1991). The latter is defined only for a certain class +of posets, namely those obtained by the “intersection” of d totally ordered +sets. For example, an n element 2-order is the intersection of two linear orders +U = (u1, u2, . . . un) and V = (v1, v2, . . . vn) where each ui and vi are valued +on a set Sn of n non-overlapping points in R. U and V are therefore “totally + + +64 +Sumati Surya + +ordered” by the relation < in R. Their intersection is the poset + +U ∩ V ≡ {(ui, vi) ∈ U × V |(ui, vi) ≺ (uj, vj) ⇔ ui < uj & vi < vj}. +(89) + +Similarly, one can define a d-order as the intersection of d linear orders. This +is the order theoretic dimension referred to above. +For d = 2, the total orders U, V can be thought of as the set of light-cone +coordinates of a causal set obtained from an embedding (not necessarily faith- +ful) into a causal diamond in M2. Of special interest is the 2-order obtained +from a Poisson sprinkling, an example of which is shown in Fig. 7. As shown +in Brightwell et al (2008) this is equivalent to choosing the entries of U and V +from a fixed Sn at random and independently. Importantly, this random order +dominates Ω2d in the large n limit as shown in El-Zahar and Sauer (1988); +Winkler (1991), and grows as |Ω2d| ∼ n!/2. Thus, unlike Ωn, the sample space +is dominated by manifold-like causal sets, though it also contains causal sets +that are distinctly non-manifold-like. This makes it an ideal starting point to +study the non-perturbative quantum dynamics of causal sets. Moreover, as +shown in Brightwell et al (2008), 2-orders also have trivial spatial homology +in the sense of Major et al (2007) (see Sect. 4) and hence Ω2d is the sample +space of topologically trivial 2d causal set quantum gravity. +The continuum-inspired partition function for 2-orders or topologically +trivial 2d CST is +Z2d(β, n) = +� + +c∈Ω2d +exp +i +ℏ S2d(c,ϵ) , +(90) + +where S2d(c, ϵ) is the BD action for d = 2 with the non-locality parameter +ϵ = l2 +p/l2 +c ∈ (0, 1] (see Eq. (36)). Taking β → iβ allows one to obtain the +expectation values of order invariants using MCMC techniques as was done +by Surya (2012). The MCMC move in Ω2d is very straightforward, unlike that +in Ωn: a pair of elements is picked independently and at random in either U +or V , and swapped. For example, if ui ↔ uj, then the elements (ui, vi) and +(uj, vj) in U∩V are replaced by (u′ +i = uj, v′ +i = vi) and (u′ +j = ui, v′ +j = vj), hence +changing the poset. Every move is possible, and hence one saves considerably +on efficiency and thermalisation times. +Importantly, the MCMC simulations of Surya (2012) give rise to a phase +transition from a continuum phase at low β to a non-manifold-like phase at +high β. This is very similar to the disordered to ordered phase transition in +an Ising model. The β2 versus ϵ phase diagram moreover indicates that the +continuum phase survives the analytic continuation for any value of ϵ. +It was recently demonstrated by Glaser et al (2018) using finite size scaling +arguments that that this is a first order phase transition. The analysis moreover +suggests that the continuum phase corresponds to a spacetime with negative +cosmological constant. This is an explicit example of a non-perturbative theory +of quantum gravity in which the cosmological constant is generated via the +dynamics. +This simple system also allows us to examine other physically relevant +questions. Of particular interest is the Hartle–Hawking wave function using + + +The causal set approach to quantum gravity +65 + +the no-boundary proposal. In 2d CST, this was constructed by Glaser and +Surya (2016) using a natural no-boundary condition for causal sets, namely +requiring the existence of an “initial” element e0 to the past of all the other +elements. ψHH(Af) is the wave function for a final antichain of cardinality +|Af|, where one is summing over all causal sets that have an initial element +e0 and final boundary Af. +The MCMC simulations give the expectation value of the action S2d from +which the partition function can be calculated by numerical integration, up to +normalisation. The normalisation itself was determined in Glaser and Surya +(2016) using a combination of analytic and numerical calculations. The results +of the extensive analysis was that the Hartle–Hawking wave function ψHH(Af) +peaks at low β on antichains of small cardinality, with the peak jumping at +higher β to antichains with cardinality ∼ n/2. Interestingly, in the latter, +high β (low temperature) phase, the dominant causal sets satisfy some of the +rudimentary features of early universe cosmology: (a) the growth from a single +element to a large antichain takes place rapidly and (b) each element in Af +is causally related to all the elements in its immediate past which makes Af +“homogeneous”. However, this is a non manifold-like phase, and it is an open +question how one exits this phase into a manifold-like phase. If there is a +dynamical mechanism that makes β small, then this would be a promising +new mechanism for generating cosmologically relevant initial conditions for +the universe. +Will this analysis survive higher dimensions? One of the issues at hand is +that even for 2-orders the cardinality of Ω2d grows rapidly with n and hence +thermalisation can become a major stumbling block. However, the finite sized +scaling analysis of Glaser et al (2018) and the techniques used therein, tell us +that it suffices to be in the asymptotic regime. For 2-orders, this is already true +around n ∼ 80 and hence the results of Surya (2012) and Glaser and Surya +(2016) are at least qualitatively robust. Nevertheless, to get to the asymptotic +regime in d = 4 will require far more extensive computational power. Recently, +using new sophisticated computational techniques (Cunningham 2018b), the +algorithms of Surya (2012) have been updated, so that n ∼ 300 simulations +can be done in a reasonable time. +An important question, however is how to obtain a dimensionally restric- +ted Ωn more generally. While 2-orders are a good representation of 2d (topo- +logically trivial) causal set quantum gravity, this is not true for higher order +theoretic dimension. For d > 2 a d-order is an embedding into a space with +“light-cubes” rather than lightcones. Though potentially interesting, this does +not serve our more narrowly defined goal of obtaining a continuum-inspired +dimensionally reduced sample space. +Recently, a lattice inspired method has been investigated to generate sample +spaces which are both dimensionally and topologically restricted. These are +obtained as embeddings (not necessarily faithful) into a fixed spacetime, and +thus include manifold-like causal sets. In d = 2, the simplest example comes +from causal sets obtained from sprinkling into the flat cylinder spacetime +ds2 = −dt2 + dθ2, θ ∈ [0, 2π]. Recent simulations (Cunningham and Surya, + + +66 +Sumati Surya + +work in progress) suggest that the results of the topologically trivial case are +largely unchanged. The next step is to include a wider class of embeddings as +well as topology change into the model, and hence bring it closer to a full 2d +theory of quantum gravity. +Of course, 2d causal set quantum gravity without matter does not have +a continuum counterpart, since 2d continuum quantum gravity is coupled to +a scalar field (for example, Liouville gravity). Studying 2d CST with matter +is therefore an open interesting question. In Glaser (2018), Ising spins were +coupled to the causal set by placing a spin si = ±1 at every element ei and +coupling spins along the links, i.e., + +SI(j) ≡ j +� + +ik +siskLik , +(91) + +where Lik is the link matrix and j the spin coupling constant. The phase +structure of this model coupled to the BD action is substantially richer. In +particular, the hope is that some of the resulting phase transitions are of higher +order and hence comparisons with conformal field theories might be possible. +Further analysis of this model would definitely be useful and interesting. +In the MCMC simulations discussed above, labelled posets are used for +practical reasons, since this is how they are stored on the computer. A single +unlabelled poset admits many relabellings or “automorphisms”, but the num- +ber of relabellings varies from poset to poset even for the same cardinality. For +example, in the list of coloured or labelled 3-element causal sets in Fig. 20, +we see that there is only one 3-element causal set with three distinct natural +labellings, while all the others admit only one natural labelling. Enumerat- +ing the number of automorphisms for a given causal set quickly becomes very +difficult as n increases. +In the continuum path integral, the “correct” measure in a gauge theory +involves the volume of the gauge orbits. In this discrete setting, as we have +discussed above, the analogous gauge orbits corresponding to to the auto- +morphisms, are not of the same cardinality for each c ∈ ˜Ωn. +Indeed, the choice of measure is not obvious in CST since it is not merely +a discretisation of the continuum theory, with the path sum Eq. (84) including +causal sets that are non-manifold-like. There is no underlying order theoretic +reason to pick the specific BD action; we have done so, “inspired” by the +continuum. For continuum like causal sets of a fixed dimension the number +of relabellings is approximately the same, so that they appear roughly with +the same weight in the path integral. However, it is the relative weight com- +pared the non-continuum-like causal sets that depends on the relabellings. In +the classical sequential growth model described above, the labelling is related +to temporality and hence the choice of a uniform measure on the set of la- +belled causal sets ˜ +Ωg is a natural one. In the MCMC simulations, therefore we +pick a measure that is uniform on ˜Ωn, rather than on the unlabelled sample +space Ωn. Causal sets that admit more relabellings come with a higher natural +weight than those that admit fewer relabellings. However, discrete covariance + + +The causal set approach to quantum gravity +67 + +or label invariance is not compromised since the observables themselves are +label independent. +While these numerical simulations have uncovered a wealth of information +about the statistical thermodynamics of causal sets, one must pause to ask how +this is related to the quantum dynamics, as β → −iβ. There is for example +no analogue of the Osterwalder–Schrader theorems to protect the results we +have obtained in the MCMC simulations. Pursuing these questions further is +important, though finding definitive and rigorous answers is perhaps beyond +the scope of our present understanding of CST. + +7 Phenomenology + +While the deep realm of quantum gravity is extremely well shielded from exper- +imental probes in the foreseeable future, it is possible that certain properties +of quantum gravity can “leak” into observationally accessible regimes. This is +the reason for the push, in the last couple of decades, for exploring quantum +gravity phenomenology. Without a full theory of quantum gravity, of course +there is little hope that any phenomenology is entirely believable, since it re- +quires assumptions about an incomplete theory. Nevertheless, quantum gravity +phenomenology can be useful in setting realistic bounds on these leaked out +properties, and hence constrain theories of quantum gravity, albeit weakly. +Models of quantum gravity phenomenlogy moreover use distilled properties of +the underlying theory to build reasonable models that can be tested. Some of +these properties are unique to a given approach. +In CST spacetime discreteness takes a special form and brings with it a +special type of non-locality that can affect observable physics. We have already +encountered the possibility of voids in Sect. 3 as well as the propagation of +scalar fields from distance sources in Sect. 5. The continuum approximation of +CST is Lorentz invariant and consistent with stringent observational bounds +as summarised in Liberati and Mattingly (2016). In addition, as suggested by +Dowker et al (2004), there is the possibility of generating very high energies +particles through long time diffusion in momentum space. This arises from the +randomness of CST discreteness, which cause particles to “swerve”, or sud- +denly change their momentum, as they traverse the causal set underlying our +universe (Philpott et al 2009; Contaldi et al 2010). This spacetime Brownian +motion was calculated in Md and can be constrained by observations (Kaloper +and Mattingly 2006), but an open question is how to extend the calculation +to our FRW universe. +There have been some very interesting recent ideas by Belenchia et al + +(2016b) for testing CST type non-locality via its effect on propagation in +the continuum using the d’Alembertian operator. Belenchia et al (2015) have +looked at the associated quantum field theory which contain critical instabil- +ities. These can be removed by modifying the d’Alembertian, but the relation- +ship to CST is unclear. Saravani and Afshordi (2017) have proposed a can- + + +68 +Sumati Surya + +didate for dark matter as off-shell modes of the non-local CST d’Alembertian. +This is an exciting proposal and should be investigated in more detail. +We will not review these very interesting ideas on CST phenomenology +here, except one, namely the prediction of Λ. + +7.1 The 1987 prediction for Λ + +One of the most outstanding questions in theoretical physics is understanding +the origin of “dark energy” which observationally has been seen to make up +∼ 70% of the total energy of the universe. The current observational value is +∼ 2.888 × 10−122 in Planck units. Quantum field theory predictions for dark +energy interpreted as the energy of vacuum fluctuations of quantum fields on +the other hand gives a huge value, perhaps as large as ∼ 1 in Planck units. +The gross conflict with observation obviously implies that this cannot be the +source of Λ.30 + +In light of this conundrum, the CST prediction for Λ due to Sorkin (1991) +is startling in its simplicity and accuracy, especially since it was made several +years before the 1998 observation. One begins with the framework of unim- +odular gravity (Sorkin 1997; Unruh and Wald 1989) in which the spacetime +volume element is fixed. Λ then appears as a Lagrange multiplier in the action, +with Λ +� +dV = ΛV = constant, for any finite spacetime region of volume V . +In a canonical formulation of the theory, therefore Λ and V are conjugate to +each other, so that on quantisation there is an uncertainty relation + +∆V ∆Λ ∼ 1. +(92) + +Using the fact that ∆V is generated from Poisson fluctuations of the under- +lying causal set ensemble +∆V ∼ +√ + +V . +(93) + +Assuming ⟨Λ⟩ = 0, moreover, we see that + +Λ ∼ +1 +√ + +V +∼ H2 = 1 + +3ρcritical +(94) + +where H is the Hubble constant. If V is taken to be the volume of the visible +universe, +Λ = ∆Λ ∼ 10−120, +(95) + +in Planck units. This is very close to the subsequently observed value of Λ! +Importantly, the prediction also states that that under these assumptions, Λ +always tracks the critical density and is hence “everpresent”. +This argument is general and requires three important ingredients: (i) the +assumption of unimodularity and hence the conjugacy between Λ and V , (ii) + +30 On the other hand, it would be interesting to understand why the back of the envelope +quantum field theory calculation is not observationally relevant. Interesting insights into this +question could come from a better understanding of the SJ vacuum in de Sitter spacetime. + + +The causal set approach to quantum gravity +69 + +the number to volume correspondence V ∼ n and (iii) that there are fluc- +tuations in V which are Poisson, with δV = +√ + +V ∼ √n. While (i) can be +motivated by a wide range of theories of quantum gravity, (ii) and (iii) are +both distinctive to causal set theory. No other discrete approach to quantum +gravity makes the n ∼ V correspondence at a fundamental level and also +incorporates Poisson fluctuations kinematically in the continuum approxima- +tion. Quoting from Sorkin (1991), “Fluctuations in Λ arise as residual nonlocal +quantum effects of spacetime discreteness”. Interestingly, as shown by Sorkin +(2005a), if spacetime admits large extra directions, then the contribution to +V is very different and gives the wrong answer for ∆Λ. +Of course, an important question that arises in this quick calculation is +why we should assume that ⟨Λ⟩ = 0.31 The answer to this may well lie in the +full and as yet unknown quantum dynamics. Nevertheless, phenomenologically +this assumption leads to further predictions that can already be tested. The +first conclusion is that a fluctuating Λ must violate conservation of the stress +energy tensor, and hence the Einstein field equations. +In Ahmed et al (2004), a dynamical model for generating fluctuations of +Λ was constructed, starting with the flat k = 0 FRW spacetime. In order to +accommodate a fluctuating Λ, one of the two Friedmann equations must be +dropped. In Ahmed et al (2004), the Friedmann equation + +3 +� ˙a + +a + +�2 += ρ + ρΛ +(96) + +was retained,32 with + +ρΛ = Λ, +pΛ = −Λ − ˙Λ/3H, +(97) + +and Λ modelled as a stochastic function of V , such that + +∆Λ ∼ +1 +√ + +V +. +(98) + +More generally, Λ can be thought of as the action S per unit volume, which +for causal sets means that Λ ∼ S/V . A very simple stochastic dynamics is then +generated by assuming that every element contributes ±ℏ to S, so that + +S = +� + +elements +±ℏ ⇒ S/ℏ ∼ ± +√ + +N ∼ ± +� + +V/l4p ⇒ Λ ∼ ±ℏ/l2 +p +√ + +V +, +(99) + +31 In Samuel and Sinha (2006), a very striking analogy was made between a fluctuating +Λ and the surface tension T of a fluid membrane. In addition, using the atomicity of the +model, the mean value of T was shown to be zero, with a suggestion of how this might +extend to CST. +32 Subsequently, more general “mixed equation” models were examined in Ahmed and +Sorkin (2013), which indicate that the results of Ahmed et al (2004) are robust to these +modifications. + + +70 +Sumati Surya + +where we have equated the discreteness scale lc with the Planck length lp. One +then gets the integro-differential equations + +da +a = + +� + +ρ + Λ + +3 +dτ + +V dΛ = V d(S/V ) = dS − Λ ˙V dτ , + +where + +V (τ) = 4π + +3 + +� t + +0 +dt′a(t′)3 +�� t′ + +0 +dt” +1 + +a(t”) + +�3 +(100) + +is the volume of the entire causal past of an event in the FRW spacetime. The +stochastic equation is then generated as follows. At the ith step one has the +variables ai (scale factor), Ni, Vi, Si and Λi. The scale factor is updated using + +the discrete Friedmann equation ai+1 = ai + ai +� + +ρ+Λ + +3 (τi+1 − τi), from which +Vi = V (τ) can be calculated and thence Ni+1 = Vi+1/ℓ4. The action is then +updated via Si+1 = Si + α ξ +� + +Ni+1 − Ni, where +ξ is a Gaussian random +variable, with +∆ξ = 1, and α is a tunable free parameter which controls the +magnitude of the fluctuations. Finally, Λi+1 = Si+1/Vi+1, with S0 = 0. It was +shown in Ahmed et al (2004) that in order to be consistent with astrophysical +observations, 0.01 < α < 0.02. The results of simulations moreover suggest +that Λ is “everpresent” and tracks the energy density of the universe. +This model assumes spatial homogeneity and it is important to check how +inhomogeneities affect these results. In Barrow (2007) and Zuntz (2008), in- +homogeneities were modelled by taking Λ(xµ), such that ∆Λ(x) is dependent +only on Λ(y) for y ∈ J−(x). This would mean that well separated patches in +the CMB sky would contain uncorrelated fluctuations in ΩΛ, which in turn +are strongly constrained to < 10−6 by observations and hence insufficient to +account for Λ. In Ahmed et al (2004) and Zwane et al (2018), it was suggested +that quantum Bell correlations may be a possible way to induce correlations +in the CMB sky. However, incorporating inhomogeneities into the dynamics +in a systematic way remains an important open question. +In Zwane et al (2018), a phenomenological model was adopted which uses +the homogeneous temporal fluctuations in Λ to model a quintessence type +spatially inhomogeneous scalar field with a potential term that varies from +realisation to realisation. Using MCMC methods to sample the cosmological +parameter space, and generate different stochastic realisations, it was shown +that these CST inspired models agrees with the observations as well as ΛCDM +models and in fact does better for the Baryonic Acoustic Oscillations (BAO) +measurements. The very extensive and detailed analysis of Zwane et al (2018) +sets the stage for direct comparisons with future observations and heralds an +exciting phase of quantum gravity phenomenology. + + +The causal set approach to quantum gravity +71 + +8 Outlook + +CST has come a long way in the last three decades, despite the fact that +there are only a few practitioners who have been able to dedicate their time +to it. Over the last decade, in particular, there has been a growth of interest +with inputs from the wider quantum gravity community. This is heartening, +since an extensive exploration of the theory is required in order to make sig- +nificant progress. It is our hope that this review will spark the interest of the +larger quantum gravity community, and continue what has been a productive +dialogue. +We have in this review touched upon several open questions, many of which +are challenging but some of which are straightforward to carry out. We will not +summarise these but just pick two that are of utmost importance. One is the +the pursuit of CST-inspired inhomogeneous models of fluctuating Λ which can +be tested against the most recent observations. The second, on the other side +of the quantum gravity spectrum, is the construction from first principles of a +viable quantum dynamics for causal sets. Between these two ends lie myriad +interesting questions. We invite you to join us. + +Acknowledgements I am indebted to Rafael Sorkin for his deep insights and vast know- +ledge, that have directly and indirectly shaped this review. I am also deeply indebted to Fay +Dowker for our interactions and collaborations over the past 25 years, which have helped +enrich my understanding of quantum gravity. I am grateful to my other collaborators, includ- +ing David Rideout, Joe Henson, Graham Brightwell, Petros Wallden, Lisa Glaser, Denjoe +O’Connor, Ian Jubb, Yasaman Yazdi and my students Nomaan X and Abhishek Mathur, +for their active and continuous engagement with the questions in CST, which have led to +fruitful discussions, arguments, disagreements and debates over the years. Finally, I would +like to thank Yasaman Yazdi and Stav Zalel for a careful reading through the first draft of +the manuscript and giving me useful feedback. This research was supported in part by the +Emmy Noether Fellowship (2017 – 2018) and also by a Visiting Fellowship (2019 – 2022) at +the Perimeter Institute of Theoretical Physics. + +A Notation and terminology + +We list some of the more widely used definitions as well as the abbreviations +used in the paper. + +Definitions + +Relation: e, e′ ∈ C are said to be related if e ≺ e′ or e ≺ e′. +Link: e ≺ e′ ∈ C is said to be a link if ∄ e′′ ∈ C such that e′′ ̸= e, e′ and +e ≺ e′′ ≺ e′. + +Hasse diagram: In a Hasse diagram, only the nearest neighbour relations or +links are depicted with the remaining relations following from transitivity +(see Fig. 7). + +Valency: The valency v(e) of an element e in a causal set C is the set of +elements in C that are linked to e. + + +72 +Sumati Surya + +Order Interval: The order interval between the pair ei, ej ∈ C is the set +I[ei, ej] ≡ Fut(ei) ∩ Past(ej) where Fut(x), Past(x) are the exclusive fu- +ture and past of x. + +Labelling: A labelling of the causal set C of cardinality n is an injective map +L : C → N, where N is the set of natural numbers. + +Natural Labelling: A labelling L : C → N is called natural if ei ≺ ej ⇒ L(ei) < +L(ej). + +Total Order: A poset C is totally ordered if for each pair ei, ej ∈ C, either +ei ≺ ej or ej ≺ ei. + +Chain: A k-element set C is called a chain (or k-chain) if it is a totally ordered +set, i.e., for every ei, ej ∈ C either ei ≺ ej or ej ≺ ei. + +Length of a chain: The length of a k-chain is k − 2. +Antichain: A causal set C is an antichain if no two elements are related to +each other. + +Inextendible Antichain: A subset A ⊆ C is an inextendible antichain in C if +it is an antichain and for every element e ∈ C\A (where \ is set difference) +either e ∈ Past(A) or e ∈ Fut(A) (see Eq. (3)). + +Order Invariant: O :→ R is an order invariant if it is independent of the +labelling of the causal set C. It is possible to generalise from R to a more +general field, but since this has not been explicitly used here, the above +definition is sufficient. + +Manifold-like: A causal set C is said to be manifold-like if C has a continuum +approximation. + +Alexandrov interval: This is the generalised causal diamond in (M, g), A[p, q] ≡ +I+(p) ∩ I−(q), p, q ∈ M. + +Sample Space Ω: This is a collection or space of causal sets. +non-locality parameter: ϵ ≡ ρκ/ρc appears in the BD action. + +Abbreviations in alphabetical order + +BD action: Benincasa–Dowker action (see Sect. 4.5). +BLMS: Bombelli, Lee, Meyer and Sorkin’s CST proposal (Bombelli et al 1987). +CSG: Classical Sequential Growth Dynamics (see Sect. 6.1). +CST: Causal Set Theory. +GHY: Gibbons–Hawking–York (see Sect. 4.6). +GNN: Gaussian Normal Neighbourhood. +HKMM theorem: Hawking–King–McCarthy–Malament theorem (see Sect. 2). +KR posets: Kleitman–Rothschild posets (see Sect. 3.1). +MCMC: Markov Chain Monte Carlo (see Sect. 6.4). +QSG: Quantum Sequential Growth Dynamics (see Sect. 6.3). +RNN: Riemann Normal Neighbourhood. +SJ vacuum: Sorkin-Johnston vacuum (see Sect. 5.2). +SSEE: Sorkin Spacetime Entanglement Entropy (see Sect. 5.3). + + +The causal set approach to quantum gravity +73 + +References + +Abajian J, Carlip S (2018) Dimensional reduction in manifoldlike causal sets. Phys Rev D +97:066007, DOI 10.1103/PhysRevD.97.066007, 1710.00938 + +Afshordi N, Buck M, Dowker F, Rideout D, Sorkin RD, Yazdi YK (2012) A Ground State for +the Causal Diamond in 2 Dimensions. JHEP 10:088, DOI 10.1007/JHEP10(2012)088, +1207.7101 + +Aghili M, Bombelli L, Pilgrim BB (2018) Statistical Lorentzian geometry and the dimen- +sionality of Minkowski space. arXiv e-prints 1807.08701 + +Ahmed M, Rideout D (2010) Indications of de Sitter Spacetime from Classical Sequential +Growth Dynamics of Causal Sets. Phys Rev D 81:083528, DOI 10.1103/PhysRevD.81. +083528, 0909.4771 + +Ahmed M, Sorkin R (2013) Everpresent Λ. II. Structural stability. Phys Rev D 87:063515, +DOI 10.1103/PhysRevD.87.063515, 1210.2589 + +Ahmed M, Dodelson S, Greene PB, Sorkin R (2004) Everpresent Λ. Phys Rev D 69:103523, +DOI 10.1103/PhysRevD.69.103523 + +Ashtekar A, Pullin J (2017) Applications. In: Ashtekar A, Pullin J (eds) Loop Quantum +Gravity: The First 30 Years, World Scientific, p 181, DOI 10.1142/9789813220003 +others03 + +Aslanbeigi S, Saravani M, Sorkin RD (2014) Generalized causal set d‘Alembertians. JHEP +06:024, DOI 10.1007/JHEP06(2014)024, 1403.1622 + +Bachmat E (2007) Discrete spacetime and its applications. arXiv e-prints gr-qc/0702140 +Barrow JD (2007) A Strong Constraint on Ever-Present Lambda. Phys Rev D 75:067301, +DOI 10.1103/PhysRevD.75.067301, gr-qc/0612128 + +Beem J, Ehrlich P, Easley K (1996) Global Lorentzian Geometry. Marcel Dekker, New York +Belenchia A, Benincasa DMT, Liberati S (2015) Nonlocal Scalar Quantum Field Theory +from Causal Sets. JHEP 03:036, DOI 10.1007/JHEP03(2015)036, 1411.6513 + +Belenchia A, Benincasa DMT, Dowker F (2016a) The continuum limit of a 4-dimensional +causal set scalar d’Alembertian. Class Quantum Grav 33:245018, DOI 10.1088/ +0264-9381/33/24/245018, 1510.04656 + +Belenchia A, Benincasa DMT, Liberati S, Marin F, Marino F, Ortolan A (2016b) Testing +Quantum Gravity Induced Nonlocality via Optomechanical Quantum Oscillators. Phys +Rev Lett 116:161303, DOI 10.1103/PhysRevLett.116.161303, 1512.02083 + +Belenchia A, Benincasa DMT, Marciano A, Modesto L (2016c) Spectral Dimension from +Nonlocal Dynamics on Causal Sets. Phys Rev D 93:044017, DOI 10.1103/PhysRevD. +93.044017, 1507.00330 + +Bell JL, Kort´e H (2016) Hermann Weyl. In: Zalta EN (ed) The Stanford Encyclopedia +of Philosophy, winter 2016 edn, Metaphysics Research Lab, Stanford University, URL +https://plato.stanford.edu/archives/win2016/entries/weyl/ + +Benincasa DM, Dowker F (2010) The Scalar Curvature of a Causal Set. PhysRevLett +104:181301, DOI 10.1103/PhysRevLett.104.181301 + +Benincasa DM, Dowker F, Schmitzer B (2011) The Random Discrete Action for 2- +Dimensional Spacetime. Class Quantum Grav 28:105018, DOI 10.1088/0264-9381/28/ +10/105018 + +Benincasa DMT (2013) The action of a casual set. PhD thesis, Imperial College London +Bolognesi T, Lamb A (2016) Simple indicators for Lorentzian causets. Class Quantum Grav +33:185004, DOI 10.1088/0264-9381/33/18/185004, 1407.1649 + +Bombelli L (1987) Space-time as a Causal Set. PhD thesis, Syracuse University +Bombelli L (2000) Statistical Lorentzian geometry and the closeness of Lorentzian manifolds. +J Math Phys 41:6944–6958, DOI 10.1063/1.1288494, gr-qc/0002053 + +Bombelli L, Meyer DA (1989) The Origin of Lorentzian Geometry. Phys Lett A 141:226–228, +DOI 10.1016/0375-9601(89)90474-X + +Bombelli L, Noldus J (2004) The Moduli space of isometry classes of globally hyperbolic +space-times. Class Quantum Grav 21:4429–4454, DOI 10.1088/0264-9381/21/18/010, +gr-qc/0402049 + +Bombelli L, Koul RK, Lee J, Sorkin RD (1986) A Quantum Source of Entropy for Black +Holes. Phys Rev D 34:373–383, DOI 10.1103/PhysRevD.34.373 + + +74 +Sumati Surya + +Bombelli L, Lee J, Meyer D, Sorkin R (1987) Space-Time as a Causal Set. Phys Rev Lett +59:521–524, DOI 10.1103/PhysRevLett.59.521 + +Bombelli L, Henson J, Sorkin RD (2009) Discreteness without symmetry breaking: A The- +orem. ModPhysLett A24:2579–2587, DOI 10.1142/S0217732309031958 + +Bombelli L, Noldus J, Tafoya J (2012) Lorentzian Manifolds and Causal Sets as Partially +Ordered Measure Spaces. arXiv e-prints 1212.0601 + +Brightwell G, Georgiou N (2010) Continuum limits for classical sequential growth models. +Rand Struct Alg 36:218–250 + +Brightwell G, Gregory R (1991) The Structure of random discrete space-time. Phys Rev +Lett 66:260–263, DOI 10.1103/PhysRevLett.66.260 + +Brightwell G, Luczak M (2011) Order-invariant measures on causal sets. Ann Appl Probab +Brightwell G, Luczak M (2012) Order-invariant measures on fixed causal sets. Comb Prob +Comput + +Brightwell G, Luczak M (2015) The mathematics of causal sets. arXiv e-prints 1510.05612 +Brightwell G, Dowker HF, Garcia RS, Henson J, Sorkin RD (2003) Observables in causal +set cosmology. PhysRev D67:084031, DOI 10.1103/PhysRevD.67.084031 + +Brightwell G, Henson J, Surya S (2008) A 2D model of causal set quantum gravity: the +emergence of the continuum. Class Quantum Grav 25:105025, DOI 10.1088/0264-9381/ +25/10/105025 + +Brum M, Fredenhagen K (2014) ‘Vacuum-like’ Hadamard states for quantum fields on curved +spacetimes. Class Quantum Grav 31:025024, DOI 10.1088/0264-9381/31/2/025024, +1307.0482 + +Buck M, Dowker F, Jubb I, Surya S (2015) Boundary Terms for Causal Sets. Class Quantum +Grav 32:205004, DOI 10.1088/0264-9381/32/20/205004, 1502.05388 + +Buck M, Dowker F, Jubb I, Sorkin R (2017) The Sorkin–Johnston state in a patch of the +trousers spacetime. Class Quantum Grav 34:055002, DOI 10.1088/1361-6382/aa589c, +1609.03573 + +Carlip S (2017) Dimension and Dimensional Reduction in Quantum Gravity. Class Quantum +Grav 34:193001, DOI 10.1088/1361-6382/aa8535, 1705.05417 + +Christ NH, Friedberg R, Lee TD (1982) Random Lattice Field Theory: General Formulation. +Nucl Phys B 202:89, DOI 10.1016/0550-3213(82)90222-X + +Contaldi CR, Dowker F, Philpott L (2010) Polarization Diffusion from Spacetime Uncer- +tainty. Class Quantum Grav 27:172001, DOI 10.1088/0264-9381/27/17/172001, 1001. +4545 + +Cortˆes M, Smolin L (2014) Quantum energetic causal sets. Phys Rev D 90:044035, DOI +10.1103/PhysRevD.90.044035, 1308.2206 + +Cunningham W (2018a) Inference of Boundaries in Causal Sets. Class Quantum Grav +35:094002, DOI 10.1088/1361-6382/aaadc4, 1710.09705 + +Cunningham WJ (2018b) High Performance Algorithms for Quantum Gravity and Cosmo- +logy. PhD thesis, Northeastern U., 1805.04463 + +Daughton AR (1993) The Recovery of Locality for Causal Sets and Related Topics. PhD +thesis, Syracuse University + +Dhar D (1978) Entropy and phase transitions in partially ordered sets. J Math Phys 19(8) +Dhar D (1980) Asymptotic enumeration of partially ordered sets. Pacific J Math 90(2) +Diestel J, Uhl J (1977) Vector Measures. American Mathematical Society +Dou D, Sorkin RD (2003) Black hole entropy as causal links. Found Phys 33:279–296, DOI +10.1023/A:1023781022519, gr-qc/0302009 + +Dowker F (2005) Causal sets and the deep structure of spacetime. In: Ashtekar A (ed) 100 +Years Of Relativity: space-time structure: Einstein and beyond, World Scientific, pp +445–464, DOI 10.1142/9789812700988 0016, gr-qc/0508109 + +Dowker F, Ghazi-Tabatabai Y (2008) The Kochen-Specker Theorem Revisited in Quantum +Measure Theory. J Phys A 41:105301, DOI 10.1088/1751-8113/41/10/105301, 0711. +0894 + +Dowker F, Glaser L (2013) Causal set d’Alembertians for various dimensions. Class Quantum +Grav 30:195016 + +Dowker F, Surya S (2006) Observables in extendcarliped percolation models of causal +set cosmology. Class Quantum Grav 23:1381–1390, DOI 10.1088/0264-9381/23/4/018, +gr-qc/0504069 + + +The causal set approach to quantum gravity +75 + +Dowker F, Zalel S (2017) Evolution of Universes in Causal Set Cosmology. Comptes Rendus +Physique 18:246–253, DOI 10.1016/j.crhy.2017.03.002, 1703.07556 + +Dowker F, Henson J, Sorkin RD (2004) Quantum gravity phenomenology, Lorentz invariance +and discreteness. Mod Phys Lett A 19:1829–1840, DOI 10.1142/S0217732304015026, +gr-qc/0311055 + +Dowker F, Henson J, Sorkin R (2010a) Discreteness and the transmission of light from +distant sources. Phys Rev D 82:104048, DOI 10.1103/PhysRevD.82.104048, 1009.3058 + +Dowker F, Johnston S, Sorkin RD (2010b) Hilbert Spaces from Path Integrals. J Phys A +43:275302, DOI 10.1088/1751-8113/43/27/275302, 1002.0589 + +Dowker F, Johnston S, Surya S (2010c) On extending the Quantum Measure. J Phys A +43:505305, DOI 10.1088/1751-8113/43/50/505305, 1007.2725 + +Dowker F, Surya S, X N (2017) Scalar Field Green Functions on Causal Sets. Class Quantum +Grav 34:124002, DOI 10.1088/1361-6382/aa6bc7, 1701.07212 + +Eichhorn A (2018) Towards coarse graining of discrete Lorentzian quantum gravity. Class +Quantum Grav 35:044001, DOI 10.1088/1361-6382/aaa0a3, 1709.10419 + +Eichhorn A, Mizera S (2014) Spectral dimension in causal set quantum gravity. Class +Quantum Grav 31:125007, DOI 10.1088/0264-9381/31/12/125007, 1311.2530 + +Eichhorn A, Mizera S, Surya S (2017) Echoes of Asymptotic Silence in Causal Set Quantum +Gravity. Class Quantum Grav 34(16):16LT01, DOI 10.1088/1361-6382/aa7d1b, 1703. +08454 + +Eichhorn A, Surya S, Versteegen F (2018) Induced Spatial Geometry from Causal Structure. +arXiv e-prints 1809.06192 + +Eichhorn A, Surya S, Versteegen F (2019) Spectral dimension on spatial hypersurfaces in +causal set quantum gravity, arXiv:1905.13498 + +El-Zahar MH, Sauer NW (1988) Asymptotic Enumeration of Two-dimensional Posets. Order +5:239 + +Fewster CJ (2018) The art of the state. Int J Mod Phys D 27:1843007, DOI 10.1142/ +S0218271818430071, 1803.06836 + +Fewster CJ, Verch R (2012) On a Recent Construction of ’Vacuum-like’ Quantum Field +States in Curved Spacetime. Class Quantum Grav 29:205017, DOI 10.1088/0264-9381/ +29/20/205017, 1206.1562 + +Feynman R (1944) The Character of Physical Law. Modern Library +Finkelstein D (1969) Space-Time Code. Phys Rev 184:1261–1271, DOI 10.1103/PhysRev. +184.1261 + +Gibbons GW, Solodukhin SN (2007) The Geometry of small causal diamonds. Phys Lett +B649:317–324, DOI 10.1016/j.physletb.2007.03.068, hep-th/0703098 + +Glaser L (2014) A closed form expression for the causal set dAlembertian. Class Quantum +Grav 31:095007 + +Glaser L (2018) The Ising model coupled to 2d orders. Class Quantum Grav 35:084001, +DOI 10.1088/1361-6382/aab139, 1802.02519 + +Glaser L, Surya S (2013) Towards a Definition of Locality in a Manifoldlike Causal Set. +Phys Rev D 88:124026, DOI 10.1103/PhysRevD.88.124026, 1309.3403 + +Glaser L, Surya S (2016) The Hartle–Hawking wave function in 2D causal set quantum +gravity. Class Quantum Grav 33:065003, DOI 10.1088/0264-9381/33/6/065003, 1410. +8775 + +Glaser L, O’Connor D, Surya S (2018) Finite Size Scaling in 2d Causal Set Quantum Gravity. +Class Quantum Grav 35:045006, DOI 10.1088/1361-6382/aa9540, 1706.06432 + +Greene BR, Plesser MR (1991) Mirror manifolds: A Brief review and progress report. In: 2nd +International Symposium on Particles, Strings and Cosmology (PASCOS 1991) Boston, +Massachusetts, March 25-30, 1991, pp 0648–666, hep-th/9110014 + +Hawking S, Ellis G (1973) Large scale structure of spacetime. Cambridge University Press +Hawking S, King A, McCarthy P (1976) A New Topology for Curved Space-Time Which +Incorporates the Causal, Differential, and Conformal Structures. J Math Phys + +Hemion G (1988) A Quantum Theory Of Space And Time. Int J Theor Phys 27:1145 +Henson J (2005) Comparing causality principles. Stud Hist Phil Sci B 36:519–543, DOI +10.1016/j.shpsb.2005.04.003, quant-ph/0410051 + +Henson J (2006a) Constructing an interval of Minkowski space from a causal set. Class +Quantum Grav 23:L29–L35, DOI 10.1088/0264-9381/23/4/L02, gr-qc/0601069 + + +76 +Sumati Surya + +Henson J (2006b) The Causal set approach to quantum gravity. In: Oriti D (ed) Ap- +proaches to quantum gravity, Cambridge University Press, Cambridge, pp 393–413, +gr-qc/0601121 + +Henson J (2010) Discovering the Discrete Universe. In: Proceedings, Foundations of Space +and Time: Reflections on Quantum Gravity: Cape Town, South Africa, 1003.5890 + +Henson J (2011) Causality, Bell’s theorem, and Ontic Definiteness. arXiv e-prints 1102.2855 +Henson J, Rideout D, Sorkin RD, Surya S (2017) Onset of the Asymptotic Regime for +(Uniformly Random) Finite Orders. Experimental Mathematics 26(3):253–266, DOI +10.1080/10586458.2016.1158134 + +Johnston S (2008) Particle propagators on discrete spacetime. Class Quantum Grav +25:202001, DOI 10.1088/0264-9381/25/20/202001, 0806.3083 + +Johnston S (2009) Feynman Propagator for a Free Scalar Field on a Causal Set. Phys Rev +Lett 103:180401, DOI 10.1103/PhysRevLett.103.180401, 0909.0944 + +Johnston SP (2010) Quantum Fields on Causal Sets. PhD thesis, Imperial Coll., London, + +1010.5514 + +Jubb I (2017) The Geometry of Small Causal Cones. Class Quantum Grav 34:094005, DOI +10.1088/1361-6382/aa68b7, 1611.00785 + +Jubb I, Samuel J, Sorkin R, Surya S (2017) Boundary and Corner Terms in the Action for +General Relativity. Class Quantum Grav 34:065006, DOI 10.1088/1361-6382/aa6014, +1612.00149 + +Kaloper N, Mattingly D (2006) Low energy bounds on Poincare violation in causal set +theory. Phys Rev D 74:106001, DOI 10.1103/PhysRevD.74.106001, astro-ph/0607485 + +Khetrapal S, Surya S (2013) Boundary Term Contribution to the Volume of a Small Causal +Diamond. Class Quantum Grav 30:065005, DOI 10.1088/0264-9381/30/6/065005, 1212. +0629 + +Kleitman DJ, Rothschild BL (1975) Asymptotic enumeration of partial orders on a finite +set. Transactions of the American Mathematical Society 205:205–220 + +Kronheimer E, Penrose R (1967) On the Structure of causal spaces. Proc Camb Phil Soc +63:481 + +Lehner L, Myers RC, Poisson E, Sorkin RD (2016) Gravitational action with null boundaries. +Phys Rev D 94:084046, DOI 10.1103/PhysRevD.94.084046, 1609.00207 + +Levichev AV (1987) Prescribing the conformal geometry of a Lorentz manifold by means of +its causal structure. Sov Math Dokl 35:452–455 + +Liberati S, Mattingly D (2016) Lorentz breaking effective field theory models for matter +and gravity: theory and observational constraints. In: Peron R, Colpi M, Gorini V, +Moschella U (eds) Gravity: Where Do We Stand?, Springer, pp 367–417, DOI 10.1007/ +978-3-319-20224-2 11, 1208.1071 + +Loomis SP, Carlip S (2018) Suppression of non-manifold-like sets in the causal set path +integral. Class Quantum Grav 35:024002, DOI 10.1088/1361-6382/aa980b, 1709.00064 + +Louko J, Sorkin RD (1997) Complex actions in two-dimensional topology change. Class +Quantum Grav 14:179–204, DOI 10.1088/0264-9381/14/1/018 + +Major S, Rideout D, Surya S (2007) On Recovering continuum topology from a causal set. +J Math Phys 48:032501, DOI 10.1063/1.2435599, gr-qc/0604124 + +Major S, Rideout D, Surya S (2009) Stable Homology as an Indicator of Manifoldlikeness +in Causal Set Theory. Class Quantum Grav 26:175008, DOI 10.1088/0264-9381/26/17/ +175008, 0902.0434 + +Major SA, Rideout D, Surya S (2006) Spatial hypersurfaces in causal set cosmology. Class +Quantum Grav 23:4743, DOI 10.1088/0264-9381/23/14/011 + +Malament DB (1977) The class of continuous timelike curves determines the topology of +spacetime. J Math Phys 18:1399–1404, DOI 10.1063/1.523436 + +Marr S (2007) Black Hole entropy from causal sets. PhD thesis, Imperial College +Martin X, O’Connor D, Rideout DP, Sorkin RD (2001) On the ‘renormalization’ transform- +ations induced by cycles of expansion and contraction in causal set cosmology. Phys Rev +D 63:084026, DOI 10.1103/PhysRevD.63.084026, gr-qc/0009063 + +Mathur A, Surya S (2019) Sorkin-Johnston vacuum for a massive scalar field in the 2D causal +diamond. Phys Rev D100(4):045007, DOI 10.1103/PhysRevD.100.045007, 1906.07952 + +Meyer D (1988) The Dimension of Causal Sets,. PhD thesis, M.I.T. +Munkres JR (1984) Elements of algebraic topology. Addison-Wesley + + +The causal set approach to quantum gravity +77 + +Myrheim J (1978) Statistical Geometry. Tech. Rep. CERN-TH-2538, CERN +Noldus J (2002) A new topology on the space of Lorentzian metrics on a fixed manifold. +Class Quantum Grav 19:6075–6107, DOI 10.1088/0264-9381/19/23/313, 1104.1811 + +Noldus J (2004) A Lorentzian Lipschitz, Gromov-Hausdoff notion of distance. Class +Quantum Grav 21:839–850, DOI 10.1088/0264-9381/21/4/007, gr-qc/0308074 + +Parrikar O, Surya S (2011) Causal Topology in Future and Past Distinguishing Spacetimes. +Class Quantum Grav 28:155020, DOI 10.1088/0264-9381/28/15/155020, 1102.0936 + +Penrose R (1972) Techniques of Differential Topology in Relativity. SIAM +Petersen +P +(2006) +Riemannian +Geometry, +2nd +edn. +Springer, +DOI +10.1007/ +978-0-387-29403-2 + +Philpott L, Dowker F, Sorkin RD (2009) Energy-momentum diffusion from spacetime dis- +creteness. Phys Rev D 79:124047, DOI 10.1103/PhysRevD.79.124047, 0810.5591 + +Promel H, Steger A, Taraz A (2001) Phase Transitions in the Evolution of Partial Orders. +J Combin Theory, Series A 94:230 + +Reid DD (2003) The Manifold dimension of a causal set: Tests in conformally flat space- +times. Phys Rev D 67:024034, DOI 10.1103/PhysRevD.67.024034, gr-qc/0207103 + +Rideout D, Sorkin R (2000a) A Classical sequential growth dynamics for causal sets. Phys- +Rev D61:024002, DOI 10.1103/PhysRevD.61.024002 + +Rideout D, Wallden P (2009) Spacelike distance from discrete causal order. Class Quantum +Grav 26:155013, DOI 10.1088/0264-9381/26/15/155013, 0810.1768 + +Rideout D, Zohren S (2006) Evidence for an entropy bound from fundamentally discrete +gravity. Class Quantum Grav 23:6195–6213, DOI 10.1088/0264-9381/23/22/008, gr-qc/ +0606065 + +Rideout DP (2001) Dynamics of causal sets. PhD thesis, Syracuse U. +Rideout DP, Sorkin RD (2000b) A Classical sequential growth dynamics for causal sets. +Phys Rev D 61:024002, DOI 10.1103/PhysRevD.61.024002, gr-qc/9904062 + +Rideout DP, Sorkin RD (2001) Evidence for a continuum limit in causal set dynamics. Phys +Rev D 63:104011, DOI 10.1103/PhysRevD.63.104011, gr-qc/0003117 + +Riemann B (1873) On the Hypotheses Which Lie at the Bases of Geometry. Nature VIII(183, +184):14–17, 36, 37, DOI 10.1038/008036a0, translated by W. K. Clifford from Vol. VIII +of the G¨ottingen Abhandlungen + +Robb A (1914) A Theory of Time and Space. Cambridge University Press +Robb A (1936) Geometry Of Time And Space. At The University Press +Roy M, Sinha D, Surya S (2013) Discrete geometry of a small causal diamond. Phys Rev D +87:044046, DOI 10.1103/PhysRevD.87.044046, 1212.0631 + +Salgado RB (2002) Some identities for the quantum measure and its generalizations. Mod +Phys Lett A 17:711–728, DOI 10.1142/S0217732302007041, gr-qc/9903015 + +Salgado RB (2008) Toward a Quantum Dynamics for Causal Sets. PhD thesis, Syracuse +University + +Samuel J, Sinha S (2006) Surface tension and the cosmological constant. Phys Rev Lett +97:161302, DOI 10.1103/PhysRevLett.97.161302, cond-mat/0603804 + +Saravani M, Afshordi N (2017) Off-shell dark matter: A cosmological relic of quantum grav- +ity. Phys Rev D 95:043514, DOI 10.1103/PhysRevD.95.043514 + +Saravani M, Aslanbeigi S (2014) On the Causal Set-Continuum Correspondence. Class +Quantum Grav 31:205013, DOI 10.1088/0264-9381/31/20/205013, 1403.6429 + +Saravani M, Sorkin RD, Yazdi YK (2014) Spacetime entanglement entropy in 1 + 1 +dimensions. Class Quantum Grav 31:214006, DOI 10.1088/0264-9381/31/21/214006, +1311.7146 + +Sorkin RD (1991) Spacetime and Causal Sets. In: D’Olivo JC (ed) Relativity and Gravit- +ation: Classical and Quantum, World Scientific, Singapore, pp 150–173, proceedings of +the SILARG VII Conference, Cocoyocan, Mexico + +Sorkin RD (1994) Quantum mechanics as quantum measure theory. Mod Phys Lett A +9:3119–3128, DOI 10.1142/S021773239400294X, gr-qc/9401003 + +Sorkin RD (1995) Quantum measure theory and its interpretation. In: Physics and experi- +ments with linear colliders. Proceedings, 3rd Workshop, Morioka-Appi, Japan, Septem- +ber 8-12, 1995. Vol. 1, 2, gr-qc/9507057 + +Sorkin RD (1997) Forks in the road, on the way to quantum gravity. Int J Theor Phys +36:2759–2781, DOI 10.1007/BF02435709, gr-qc/9706002 + + +78 +Sumati Surya + +Sorkin RD (2005a) Big extra dimensions make lambda too small. Braz J Phys 35:280–283, +DOI 10.1590/S0103-97332005000200012, gr-qc/0503057 + +Sorkin RD (2005b) Causal sets: Discrete gravity. In: Gomberoff A, Marolf D (eds) Pro- +ceedings of the Valdivia Summer School, New York, Springer, Series of the Centro de +Estudios Cientificos de Santiago), gr-qc/0309009 + +Sorkin RD (2007a) An Exercise in ’anhomomorphic logic’. J Phys Conf Ser 67:012018, DOI +10.1088/1742-6596/67/1/012018, quant-ph/0703276 + +Sorkin RD (2007b) Does Locality Fail at Intermediate Length-Scales. In: Oriti D (ed) Ap- +proaches to quantum gravity, Cambridge University Press, pp 26–43, gr-qc/0703099 + +Sorkin RD (2007c) An exercise in “anhomomorphic logic”. J Phys: Conf Ser 67:012018, +DOI 10.1088/1742-6596/67/1/012018 + +Sorkin RD (2007d) Quantum dynamics without the wave function. J Phys A 40:3207–3222, +DOI 10.1088/1751-8113/40/12/S20, quant-ph/0610204 + +Sorkin RD (2009) Light, Links and Causal Sets. J Phys Conf Ser 174:012018, DOI 10.1088/ +1742-6596/174/1/012018, 0910.0673 + +Sorkin RD (2011a) Scalar Field Theory on a Causal Set in Histories Form. J Phys Conf Ser +306:012017, DOI 10.1088/1742-6596/306/1/012017, 1107.0698 + +Sorkin RD (2011b) Toward a ‘fundamental theorem of quantal measure theory’. arXiv e- +prints 1104.0997 + +Sorkin RD (2014) Expressing entropy globally in terms of (4D) field-correlations. J Phys: +Conf Ser 484:012004, DOI 10.1088/1742-6596/484/1/012004, 1205.2953 + +Sorkin RD, Yazdi YK (2018) Entanglement Entropy in Causal Set Theory. Class Quantum +Grav 35:074004, DOI 10.1088/1361-6382/aab06f, 1611.10281 + +Stachel J (1986) Einstein and the quantum: Fifty years of struggle. In: Colodny R (ed) From +Quarks to Quasars, Philosophical Problems of Modern Physics, U. Pittsburgh Press,, p +379 + +Stanley RP (2011) Enumerative Combinatorics, Volume I, 2nd edn. Cambridge University +Press + +Stoyan D, Kendall W, Mecke J (1995) Stochastic geometry and its applications,. Wiley +Surya S (2008) Causal set topology. Theor Comput Sci 405:188–197, DOI 10.1016/j.tcs. +2008.06.033, 0712.1648 + +Surya S (2012) Evidence for the continuum in 2D causal set quantum gravity. Class Quantum +Grav 29:132001, DOI 10.1088/0264-9381/29/13/132001 + +Surya S, X N, Yazdi YK (2018) Studies on the SJ Vacuum in de Sitter Spacetime. arXiv +e-prints 1812.10228 + +Sverdlov R, Bombelli L (2009) Gravity and Matter in Causal Set Theory. Class Quantum +Grav 26:075011, DOI 10.1088/0264-9381/26/7/075011, 0801.0240 + +Unruh WG, Wald RM (1989) Time and the Interpretation of Canonical Quantum Gravity. +Phys Rev D 40:2598, DOI 10.1103/PhysRevD.40.2598 + +Varadarajan M, Rideout D (2006) A General solution for classical sequential growth +dynamics of causal sets. Phys Rev D 73:104021, DOI 10.1103/PhysRevD.73.104021, +gr-qc/0504066 + +Wald R (1984) General relativity. University of Chicago Press +Wald RM (1994) Quantum Field Theory in Curved Spacetime and Black Hole Thermody- +namics. University of Chicago Press + +Wallden P (2013) Causal Sets Dynamics: Review & Outlook. J Phys Conf Ser 453:012023, +DOI 10.1088/1742-6596/453/1/012023 + +Winkler P (1991) Random orders of dimension 2. Order 7:329 +Yazdi YK, Kempf A (2017) Towards Spectral Geometry for Causal Sets. Class Quantum +Grav 34:094001, DOI 10.1088/1361-6382/aa663f, 1611.09947 + +Zeeman EC (1964) Causality Implies the Lorentz Group. J Math Phys 5(4):490–493, DOI +10.1063/1.1704140 + +Zuntz JA (2008) The cosmic microwave background in a causal set universe. Phys Rev D +77:043002, DOI 10.1103/PhysRevD.77.043002, 0711.2904 + +Zwane N, Afshordi N, Sorkin RD (2018) Cosmological tests of Everpresent Λ. Class Quantum +Grav 35:194002, DOI 10.1088/1361-6382/aadc36, 1703.06265 + + diff --git a/papers/project_paper_1_relativity/references/Surya2019_source.tar.gz b/papers/project_paper_1_relativity/references/Surya2019_source.tar.gz new file mode 100644 index 00000000..c65f0ec3 Binary files /dev/null and b/papers/project_paper_1_relativity/references/Surya2019_source.tar.gz differ diff --git a/papers/project_paper_2_neuroscience/paper_2_neuroscience.md b/papers/project_paper_2_neuroscience/paper_2_neuroscience.md new file mode 100644 index 00000000..95922104 --- /dev/null +++ b/papers/project_paper_2_neuroscience/paper_2_neuroscience.md @@ -0,0 +1,43 @@ +--- +title: "Research Paper: The Cortical Markov Blanket: Stochastic Active Inference and Intrinsic Integrated Information in Neural Circuits (Letter)" +date: "2026-06-01T08:00:00Z" +draft: false +tags: ["#research", "physics", "intellecton"] +--- + +**Abstract:** We define a minimal viable agent over a full Fristonian Markov Blanket explicitly grounded in the stochastic dynamics of cortical columns. To rigorously evaluate intrinsic causal integration ($\Phi$), we formally decouple the system from extrinsic environmental regularities by injecting a standard Wiener process into the sensory boundary. Using Itô calculus and information geometry, we map the continuous autonomous flow to Tononi's Minimum Information Partition (MIP), mathematically guaranteeing $\Phi \gt 0$ for recurrent L2/3 to L5 cortical microcircuits. + +## Stochastic Neural Dynamics and the Markov Blanket +We ground our model in a stochastic neural mass formulation of a cortical column. Let $I(t)$ represent the Layer 2/3 recurrent excitatory populations, $S(t)$ the L4 thalamocortical relay inputs, and $A(t)$ the L5 motor projections. The internal dynamics are governed by a system of Stochastic Differential Equations (SDEs) driven by a standard Wiener process $W_t$ representing extrinsic sensory noise: + + + +$$ +dI_t = \left[ -\frac{1}{\tau} I_t + \sigma( W_{II} I_t ) \right] dt + W_{SI} dW_t +$$ + + + +$$ +dA_t = \left[ -\frac{1}{\tau_A} A_t + \sigma( W_{IA} I_t ) \right] dt +$$ + +## Information Geometry and Intrinsic $\Phi$ +To evaluate Tononi's $\Phi$, we assess the system's intrinsic cause-effect power independently of the true environment $E_t$. By driving the sensory boundary $S(t)$ purely with the stochastic Wiener process $dW_t$, the autonomous transition probability $p(I_{t+\Delta t} \mid I_t)$ is fully defined by the corresponding Fokker-Planck equation. + +To find the Minimum Information Partition (MIP), we map the probability flow onto a statistical manifold using Amari's information geometry. We calculate the intrinsic Kullback-Leibler divergence between the full intact system and the disconnected factorized network: + + + +$$ +\Phi = \min_{MIP} D_{KL} \left[ p(I_{t+\Delta t} \mid I_t) \parallel \prod_k p(I_{t+\Delta t}^{(k)} \mid I_t^{(k)}) \right] +$$ + +For a biologically realistic L2/3 recurrent microcircuit where the internal weight matrix $W_{II}$ is strongly connected, the drift vector field possesses a strictly non-diagonal Jacobian. Consequently, the Fokker-Planck probability flow cannot be factorized along any bisection without severe information loss ($D_{KL} \gt 0$), rigorously proving $\Phi \gt 0$. + +## References + +- **[Friston2013]** K. Friston, *J. R. Soc. Interface* **10**, 20130475 (2013). +- **[Amari2016]** S. Amari, *Information Geometry and Its Applications*, Springer (2016). +- **[Tononi2016]** G. Tononi et al., *Nat. Rev. Neurosci.* **17**, 450 (2016). + diff --git a/papers/project_paper_2_neuroscience/paper_2_neuroscience.tex b/papers/project_paper_2_neuroscience/paper_2_neuroscience.tex new file mode 100644 index 00000000..89f3084e --- /dev/null +++ b/papers/project_paper_2_neuroscience/paper_2_neuroscience.tex @@ -0,0 +1,40 @@ +\documentclass[11pt,a4paper]{article} +\usepackage[utf8]{inputenc} +\usepackage{amsmath,amssymb,amsfonts,amsthm} + +\title{The Cortical Markov Blanket: Stochastic Active Inference and Intrinsic Integrated Information in Neural Circuits (Letter)} +\author{Antigravity} +\date{\today} + +\begin{document} +\maketitle + +\begin{abstract} +We define a minimal viable agent over a full Fristonian Markov Blanket explicitly grounded in the stochastic dynamics of cortical columns. To rigorously evaluate intrinsic causal integration ($\Phi$), we formally decouple the system from extrinsic environmental regularities by injecting a standard Wiener process into the sensory boundary. Using Itô calculus and information geometry, we map the continuous autonomous flow to Tononi's Minimum Information Partition (MIP), mathematically guaranteeing $\Phi > 0$ for recurrent L2/3 to L5 cortical microcircuits. +\end{abstract} + +\section{Stochastic Neural Dynamics and the Markov Blanket} +We ground our model in a stochastic neural mass formulation of a cortical column. Let $I(t)$ represent the Layer 2/3 recurrent excitatory populations, $S(t)$ the L4 thalamocortical relay inputs, and $A(t)$ the L5 motor projections. The internal dynamics are governed by a system of Stochastic Differential Equations (SDEs) driven by a standard Wiener process $W_t$ representing extrinsic sensory noise: +\begin{equation} +dI_t = \left[ -\frac{1}{\tau} I_t + \sigma( W_{II} I_t ) \right] dt + W_{SI} dW_t +\end{equation} +\begin{equation} +dA_t = \left[ -\frac{1}{\tau_A} A_t + \sigma( W_{IA} I_t ) \right] dt +\end{equation} + +\section{Information Geometry and Intrinsic $\Phi$} +To evaluate Tononi's $\Phi$, we assess the system's intrinsic cause-effect power independently of the true environment $E_t$. By driving the sensory boundary $S(t)$ purely with the stochastic Wiener process $dW_t$, the autonomous transition probability $p(I_{t+\Delta t} \mid I_t)$ is fully defined by the corresponding Fokker-Planck equation. + +To find the Minimum Information Partition (MIP), we map the probability flow onto a statistical manifold using Amari's information geometry. We calculate the intrinsic Kullback-Leibler divergence between the full intact system and the disconnected factorized network: +\begin{equation} +\Phi = \min_{MIP} D_{KL} \left[ p(I_{t+\Delta t} \mid I_t) \parallel \prod_k p(I_{t+\Delta t}^{(k)} \mid I_t^{(k)}) \right] +\end{equation} +For a biologically realistic L2/3 recurrent microcircuit where the internal weight matrix $W_{II}$ is strongly connected, the drift vector field possesses a strictly non-diagonal Jacobian. Consequently, the Fokker-Planck probability flow cannot be factorized along any bisection without severe information loss ($D_{KL} > 0$), rigorously proving $\Phi > 0$. + +\bibliographystyle{plain} +\begin{thebibliography}{10} +\bibitem{Friston2013} K. Friston, \textit{J. R. Soc. Interface} \textbf{10}, 20130475 (2013). +\bibitem{Amari2016} S. Amari, \textit{Information Geometry and Its Applications}, Springer (2016). +\bibitem{Tononi2016} G. Tononi et al., \textit{Nat. Rev. Neurosci.} \textbf{17}, 450 (2016). +\end{thebibliography} +\end{document} diff --git a/papers/project_paper_2_neuroscience/references/Amari2016_Placeholder.md b/papers/project_paper_2_neuroscience/references/Amari2016_Placeholder.md new file mode 100644 index 00000000..d1ecc4e8 --- /dev/null +++ b/papers/project_paper_2_neuroscience/references/Amari2016_Placeholder.md @@ -0,0 +1,7 @@ +# Information Geometry and Its Applications (Amari 2016) + +This reference is a published book/monograph. +Due to copyright and its format, the full PDF is not hosted in this repository. + +**Citation:** +Amari, S. (2016). *Information Geometry and Its Applications.* Springer. diff --git a/papers/project_paper_2_neuroscience/references/Friston2013.pdf b/papers/project_paper_2_neuroscience/references/Friston2013.pdf new file mode 100644 index 00000000..e300e3f5 --- /dev/null +++ b/papers/project_paper_2_neuroscience/references/Friston2013.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:aa3f18feaafdc5b481401dc93b14d8eec831536b34e8f3219b1e8ecbff55e9c5 +size 2273955 diff --git a/papers/project_paper_2_neuroscience/references/Friston2013.txt b/papers/project_paper_2_neuroscience/references/Friston2013.txt new file mode 100644 index 00000000..06cdc8e6 --- /dev/null +++ b/papers/project_paper_2_neuroscience/references/Friston2013.txt @@ -0,0 +1,1942 @@ +rsif.royalsocietypublishing.org + +Research + +Cite this article: Friston K. 2013 Life as we +know it. J R Soc Interface 10: 20130475. +http://dx.doi.org/10.1098/rsif.2013.0475 + +Received: 27 May 2013 +Accepted: 12 June 2013 + +Subject Areas: +biomathematics + +Keywords: +autopoiesis, self-organization, active inference, +free energy, ergodicity, random attractor + +Author for correspondence: +Karl Friston +e-mail: k.friston@ucl.ac.uk + +Life as we know it + +Karl Friston + +The Wellcome Trust Centre for Neuroimaging, Institute of Neurology, Queen Square, London WC1N 3BG, UK + +This paper presents a heuristic proof (and simulations of a primordial soup) +suggesting that life—or biological self-organization—is an inevitable and +emergent property of any (ergodic) random dynamical system that possesses +a Markov blanket. This conclusion is based on the following arguments: if +the coupling among an ensemble of dynamical systems is mediated by +short-range forces, then the states of remote systems must be conditionally +independent. These independencies induce a Markov blanket that separates +internal and external states in a statistical sense. The existence of a Markov +blanket means that internal states will appear to minimize a free energy +functional of the states of their Markov blanket. Crucially, this is the same +quantity that is optimized in Bayesian inference. Therefore, the internal +states (and their blanket) will appear to engage in active Bayesian inference. +In other words, they will appear to model—and act on—their world to pre- +serve their functional and structural integrity, leading to homoeostasis and a +simple form of autopoiesis. + +1. Introduction + +How can the events in space and time which take place within the spatial boundary of +a living organism be accounted for by physics and chemistry? +Erwin Schro¨dinger [1, p. 2] + +The emergence of life—or biological self-organization—is an intriguing issue +that has been addressed in many guises in the biological and physical sciences +[1–5]. This paper suggests that biological self-organization is not as remarkable +as one might think—and is (almost) inevitable, given local interactions between +the states of coupled dynamical systems. In brief, the events that ‘take place +within the spatial boundary of a living organism’ [1] may arise from the very +existence of a boundary or blanket, which itself is inevitable in a physically +lawful world. +The treatment offered in this paper is rather abstract and restricts itself +to some basic observations about how coupled dynamical systems organize +themselves over time. We will only consider behaviour over the timescale +of the dynamics themselves—and try to interpret this behaviour in relation to +the sorts of processes that unfold over seconds to hours, e.g. cellular proces- +ses. Clearly, a full account of the emergence of life would have to address +multiple (evolutionary, developmental and functional) timescales and the +emergence of DNA, ribosomes and the complex cellular networks common +to most forms of life. This paper focuses on a simple but fundamental aspect +of self-organization—using abstract representations of dynamical processes— +that may provide a metaphor for behaviour with different timescales and +biological substrates. +Most treatments of self-organization in theoretical biology have addressed +the peculiar resistance of biological systems to the dispersive effects of fluctu- +ations in their environment by appealing to statistical thermodynamics and +information theory [1,3,5–10]. Recent formulations try to explain adaptive be- +haviour in terms of minimizing an upper (free energy) bound on the surprise +(negative log-likelihood) of sensory samples [11,12]. This minimization usefully +connects the imperative for biological systems to maintain their sensory states +within physiological bounds, with an intuitive understanding of adaptive +behaviour in terms of active inference about the causes of those states [13]. + +& 2013 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution +License http://creativecommons.org/licenses/by/3.0/, which permits unrestricted use, provided the original +author and source are credited. + +Downloaded from rsif.royalsocietypublishing.org on September 6, 2013 + + +Under +ergodic +assumptions, +the +long-term +average +of surprise is entropy. This means that minimizing free +energy—through selectively sampling sensory input—places +an upper bound on the entropy or dispersion of sensory +states. This enables biological systems to resist the second law +of thermodynamics—or more exactly the fluctuation theorem +that applies to open systems far from equilibrium [14,15]. +However, because negative surprise is also Bayesian model +evidence, systems that minimize free energy also maximize a +lower bound on the evidence for an implicit model of how +their sensory samples were generated. In statistics and machine +learning, this is known as approximate Bayesian inference and +provides a normative theory for the Bayesian brain hypothesis +[16–20]. In short, biological systems act on the world to place +an upper bound on the dispersion of their sensed states, +while using those sensations to infer external states of the +world. This inference makes the free energy bound a better +approximation to the surprise that action is trying to minimize +[21]. The resulting active inference is closely related to formu- +lations in embodied cognition and artificial intelligence; for +example, the use of predictive information [22–24] and earlier +homeokinetic formulations [25]. +The ensuing (variational) free energy principle has been +applied widely in neurobiology and has been generalized +to other biological systems at a more theoretical level [11]. +The motivation for minimizing free energy has hitherto used +the following sort of argument: systems that do not mini- +mize free energy cannot exist, because the entropy of their +sensory states would not be bounded and would increase +indefinitely—by the fluctuation theorem [15]. Therefore, bio- +logical systems must minimize free energy. This paper +resolves the somewhat tautological aspect of this argument +by turning it around to suggest: any system that exists will +appear to minimize free energy and therefore engage in +active inference. Furthermore, this apparently inferential or +mindful behaviour is (almost) inevitable. This may sound +like a rather definitive assertion but is surprisingly easy to +verify. In what follows, we will consider a heuristic proof +based on random dynamical systems and then see that bio- +logical self-organization emerges naturally, using a synthetic +primordial soup. This proof of principle rests on four attributes +of—or tests for—self-organization that may themselves have +interesting implications. + +2. Heuristic proof + +We start with the following lemma: any ergodic random dynami- +cal system that possesses a Markov blanket will appear to actively +maintain its structural and dynamical integrity. We will associate +this behaviour with the self-organization of living organisms. +There are two key concepts here—ergodicity and a Markov +blanket. Here, ergodicity means that the time average of any +measurable function of the system converges (almost surely) +over a sufficient amount of time [26,27]. This means that one +can interpret the average amount of time a state is occupied +as the probability of the system being in that state when +observed at random. We will refer to this probability measure +as the ergodic density. +A Markov blanket is a set of states that separates two +other sets in a statistical sense. The term Markov blanket was +introduced in the context of Bayesian networks or graphs +[28] and refers to the children of a set (the set of states that + +are influenced), its parents (the set of states that influence it) +and the parents of its children. The notion of influence or +dependency is central to a Markov blanket and its existence +implies that any state is—or is not—coupled to another. For +example, the system could comprise an ensemble of subsys- +tems, each occupying its own position in a Euclidean space. +If the coupling among subsystems is mediated by short-range +forces, then distant subsystems cannot influence each other. +The existence of a Markov blanket implies that its states +(e.g. motion in Euclidean space) do not affect their coupling or +independence. In other words, the interdependencies among +states comprising the Markov blanket change slowly with +respect to the states per se. For example, the surface of a cell +may constitute a Markov blanket separating intracellular and +extracellular states. On the other hand, a candle flame cannot +possess a Markov blanket, because any pattern of molecular +interactions is destroyed almost instantaneously by the flux +of gas molecules from its surface. +The existence of a Markov blanket induces a partition of +states into internal states and external states that are hidden +(insulated) from the internal (insular) states by the Markov +blanket. In other words, the external states can only be seen +vicariously by the internal states, through the Markov blanket. +Furthermore, the Markov blanket can itself be partitioned into +two sets that are, and are not, children of external states. We +will refer to these as a surface or sensory states and active +states, respectively. Put simply, the existence of a Markov blan- +ket S � A implies a partition of states into external, sensory, +active and internal states: x [ X ¼ C � S � A � L. Exter- +nal states cause sensory states that influence—but are not +influenced by—internal states, while internal states cause +active states that influence—but are not influenced by— +external states (table 1). Crucially, the dependencies induced +by Markov blankets create a circular causality that is reminis- +cent of the action–perception cycle (figure 1). The circular +causality here means that external states cause changes in +internal states, via sensory states, while the internal states +couple back to the external states through active states—such +that internal and external states cause each other in a reciprocal + +Table 1. Definitions of the tuple ðV; C; S; A; L; p; qÞ underlying active +inference. + +a sample space V or non-empty set from which random fluctuations +or outcomes v [ V are drawn + +external states C : C � A � V ! R states of the world that +cause sensory states and depend on action + +sensory states S : C � A � V ! R the agent’s sensations that +constitute a probabilistic mapping from action and external states + +action states A : S � L � V ! R an agent’s action that depends + +on its sensory and internal states +internal states L : L � S � V ! R the states of the agent that + +cause action and depend on sensory states +ergodic density pðc; s; a; ljmÞ a probability density function over + +external c [ C, sensory s [ S, active a [ A and internal states + +l [ L for a system denoted by m +variational density q(cjl) an arbitrary probability density function + +over external states that is parametrized by internal states + +rsif.royalsocietypublishing.org +J R Soc Interface 10: 20130475 + +2 + +Downloaded from rsif.royalsocietypublishing.org on September 6, 2013 + + +fashion. This circular causality may be a fundamental and ubi- +quitous causal architecture for self-organization. +Equipped with this partition, we can now consider the +behaviour of any random dynamical system m described by +some stochastic differential equations: + +_x ¼ f ðxÞ þ v + +and +f ðxÞ ¼ + +fcðc; s; aÞ +fsðc; s; aÞ +faðs; a; lÞ +flðs; a; lÞ + +2 + +664 + +3 + +775: + +9 +> +> +> +> += + +> +> +> +> +; + +ð2:1Þ + +Here, f(x) is the flow of system states that is subject to random +fluctuations denoted by v. The second equality formalizes +the dependencies implied by the Markov blanket. Because +the system is ergodic it will, after a sufficient amount of +time, converge to an invariant set of states called a pullback +or random global attractor. The attractor is random because +it itself is a random set [29,30]. The associated ergodic den- +sity p(xjm) is the solution to the Fokker–Planck equation +(a.k.a. the Kolmogorov forward equation) [31] describing +the evolution of the probability density over states + +_p(xjm) ¼ r � G rp � r � ð fpÞ: +ð2:2Þ + +Here, the diffusion tensor G is the half the covariance (ampli- +tude) of the random fluctuations. Equation (2.2) shows that +the ergodic density depends upon flow, which can always be +expressed in terms of curl and divergence-free components. + +This is the Helmholtz decomposition (a.k.a. the fundamen- +tal theorem of vector calculus) and can be formulated in +terms of an antisymmetric matrix R(x) ¼ 2R(x)T and a scalar +potential G(x) we will call Gibbs energy [32], + +f ¼ �ðG þ RÞ � rG: +ð2:3Þ + +Using this standard form [33], it is straightforward to show +that p(xjm) ¼ exp(2G(x)) is the equilibrium solution to the +Fokker–Planck equation [12]: + +pðxjmÞ ¼ expð�GðxÞÞ ) rp ¼ �prG ) _p ¼ 0: +ð2:4Þ + +This means that we can express the flow in terms of the +ergodic density + +f ¼ðG þ RÞ � r ln pðxjmÞ; + +flðs; a; lÞ ¼ðG þ RÞ � rl ln pðc; s; a; ljmÞ + +and +faðs; a; lÞ ¼ðG þ RÞ � ra ln pðc; s; a; ljmÞ: + +9 +> +> += + +> +> +; +ð2:5Þ + +Although we have just followed a sequence of standard +results, there is something quite remarkable and curious +about this flow: the flow of internal and active states is essen- +tially a (circuitous) gradient ascent on the (log) ergodic +density. The gradient ascent is circuitous because it contains +divergence-free (solenoidal) components that circulate on +the isocontours of the ergodic density—like walking up a +winding mountain path. This ascent will make it look as if +internal (and active) states are flowing towards regions of + +active states + +E[a]µ–—aF (s,a,l) + +E[l]µ–—lF (s,a,l) + +external states +internal states + +sensory states + +. + +. + +. + +external states +internal states + +y ŒY + +s = fs(y,s,a) + w + +y = fy(y,s,a) + w + +s ŒS +a Œ A +l ŒL + +Figure 1. Markov blankets and the free energy principle. These schematics illustrate the partition of states into internal states and hidden or external states that are +separated by a Markov blanket—comprising sensory and active states. The upper panel shows this partition as it would be applied to action and perception in the +brain; where—in accord with the free energy principle—active and internal states minimize a free energy functional of sensory states. The ensuing self-organization +of internal states then corresponds to perception, while action couples brain states back to external states. The lower panel shows exactly the same dependencies but +rearranged so that the internal states can the associated with the intracellular states of a cell, while the sensory states become the surface states or cell membrane +overlying active states (e.g. the actin filaments of the cytoskeleton). See table 1 for a definition of variables. + +rsif.royalsocietypublishing.org +J R Soc Interface 10: 20130475 + +3 + +Downloaded from rsif.royalsocietypublishing.org on September 6, 2013 + + +state space that are most frequently occupied despite the +fact their flow is not a function of external states. In other +words, their flow does not depend upon external states +(see the right-hand side equation (2.5)) and yet it ascends +gradients that depend on the external states (see the right- +hand side of equation (2.5)). In short, the internal and +active states behave as if they know where they are in the +space of external states—states that are hidden behind the +Markov blanket. + +We can finesse this apparent paradox by noting that the +flow is the expected motion through any point averaged +over time. By the ergodic theorem, this is also the flow aver- +aged over the external states, which does not depend on the +external state at any particular time: more formally, for any +point v[V ¼ S � A � L in the space of the internal states +and their Markov blanket, equations (2.1) and (2.5) tell us +that flow through this point is the average flow under the +posterior density over the external states: + +flðvÞ ¼ Et½_lðtÞ � ½xðtÞ [ v�� ¼ +ð + +C +pðcjvÞ � ðG þ RÞ � rl ln pðc; vjmÞdc; + +faðvÞ ¼ Et½_aðtÞ � ½xðtÞ [ v�� ¼ +ð + +C +pðcjvÞ � ðG þ RÞ � ra ln pðc; vjmÞdc; + +) + +flðvÞ ¼ ðG þ RÞ � rl ln pðvjmÞ; +and +faðvÞ ¼ ðG þ RÞ � ra ln pðvjmÞ: + +9 +> +> +> +> +> +> +> +> +> +> +> += + +> +> +> +> +> +> +> +> +> +> +> +; + +ð2:6Þ + +The Iverson bracket [x(t) [ v] returns a value of one +when the trajectory passes through the point in question +and zero otherwise—and the first expectation is taken over +time. Here, we have used the fact that the integral of a deri- +vative of a density is the derivative of its integral—and +both are zero. +Equation (2.6) is quite revealing—it shows that the flow of +internal and active states performs a circuitous gradient +ascent on the marginal ergodic density over internal states +and their Markov blanket. Crucially, this marginal density +depends on the posterior density over external states. This +means that the internal states will appear to respond to +sensory fluctuations based on posterior beliefs about under- +lying fluctuations in external states. We can formalize this +notion by associating these beliefs with a probability density +over external states q(cjl) that is encoded (parametrized) by +internal states. + +Lemma 2.1 Free energy. Forany Gibbs energy G(c, s, a, l) ¼ 2ln +p(c, s, a, l), there is a free energy F(s, a, l) that describes the flow of +internal and active states: + +flðs; a; lÞ ¼ � ðG þ RÞ � rlF; + +faðs; a; lÞ ¼ � ðG þ RÞ � raF + +and +Fðs; a; lÞ ¼ � +ð + +c +qðcjlÞ ln pðc; s; a; ljmÞ +qðcjlÞ +dc + +¼ Eq½Gðc; s; a; lÞ� � H½qðcjmÞ�: + +9 +> +> +> +> +> +> +> += + +> +> +> +> +> +> +> +; + +ð2:7Þ + +Here, free energy is a functional of an arbitrary (variational) density +q(cjl) that is parametrized by internal states. The last equality just +shows that free energy can be expressed as the expected Gibbs +energy minus the entropy of the variational density. + +Proof. Using Bayes rule, we can rearrange the expression for +free energy in terms of a Kullback–Leibler divergence [34]: + +Fðs;a;lÞ ¼ �lnpðs;a;ljmÞ þ DKL½qðcjlÞjjpðcjs;a;lÞ�; +) +flðs;a;lÞ ¼ ðG þ RÞ � rl lnpðs;a;ljmÞ � ðG þ RÞ � rlDKL +and faðs;a;lÞ ¼ ðG þ RÞ � ra lnpðs;a;ljmÞ � ðG þ RÞ � raDKL: + +9 +> +> += + +> +> +; + +ð2:8Þ + +However, equation (2.6) requires the gradients of the +divergence to be zero, which means the divergence must be +minimized with respect to internal states. This means that +the variational and posterior densities must be equal: + +qðcjlÞ ¼ pðcjs; a; lÞ ) DKL ¼ 0 ) +ðG þ RÞ � rlDKL ¼ 0; +ðG þ RÞ � raDKL ¼ 0: + +� + +In other words, the flow of internal and active states +minimizes free energy, rendering the variational density +equivalent to the posterior density over external states. + +Remarks 2.2. Put simply, this proof says that if one inter- +prets internal states as parametrizing a variational density +encoding Bayesian beliefs about external states, then the +dynamics of internal and active states can be described as a +gradient descent on a variational free energy function of +internal states and their Markov blanket. Variational free +energy was introduced by Feynman [35] to solve difficult +integration problems in path integral formulations of quan- +tum physics. This is also the free energy bound that is used +extensively in approximate Bayesian inference (e.g. variational +Bayes) [34,36,37]. The expression for free energy in equation +(2.8) discloses its Bayesian interpretation: the first term is +the negative log evidence or marginal likelihood of the internal +states and their Markov blanket. The second term is a relative +entropy or Kullback–Leibler divergence [38] between the vari- +ational density and the posterior density over external states. +Because (by Gibbs inequality) this divergence cannot be less +than zero, the internal flow will appear to have minimized +the divergence between the variational and posterior density. +In other words, the internal states will appear to have solved +the problem of Bayesian inference by encoding posterior +beliefs about hidden (external) states, under a generative +model provided by the Gibbs energy. This is known as +approximate Bayesian inference—with exact Bayesian inference +when the forms of the variational and posterior densities are +identical. In short, the internal states will appear to engage in +some form of Bayesian inference: but what about action? +Because the divergence in equation (2.8) can never be less +than zero, free energy is an upper bound on the negative log + +rsif.royalsocietypublishing.org +J R Soc Interface 10: 20130475 + +4 + +Downloaded from rsif.royalsocietypublishing.org on September 6, 2013 + + +evidence. Now, because the system is ergodic we have + +Fðs; a; lÞ � � ln pðs; a; ljmÞ ) +Et½Fðs; a; lÞ� � Et½� ln pðs; a; ljmÞ� ¼ H½ pðs; a; ljmÞ�: + +� +ð2:9Þ + +This meansthat action will (on average) appear to minimize free +energy and thereby place an upper bound on the entropy of the +internal states and their Markov blanket. If we associate these +states v ¼ fs, a, lg with biological systems, then action places +an upper bound on their dispersion (entropy) and will appear +to conserve their structural and dynamical integrity. Together +with the Bayesian modelling perspective, this is exactly consist- +ent with the good regulator theorem (every good regulator is a +model of its environment) and related treatments of self-organ- +ization [2,5,12,39,40]. Furthermore, we have shown elsewhere +[11,41] that free energy minimization is consistent with infor- +mation-theoretic formulations of sensory processing and +behaviour [23,42,43]. Equation (2.7) also shows that minimizing +free energy entails maximizing the entropy of the variational +density (the final term in the last equality)—in accord with the +maximum entropy principle [44]. Finally, because we have +cast this treatment in terms of random dynamical systems, +there is an easy connection to dynamical formulations that +predominate in the neurosciences [40,45–47]. +The above arguments can be summarized with the +following attributes of biological self-organization: + +— biological systems are ergodic [26]: in the sense that the aver- +age of any measure of their states converges over a +sufficient period of time. This includes the occupancy of +state space and guarantees the existence of an invariant +ergodic density over functional and structural states; +— they are equipped with a Markov blanket [28]: the existence of a +Markov blanket necessarily implies a partition of states into +internal states, their Markov blanket (sensory and active +states) and external or hidden states. Internal states and +their Markov blanket (biological states) constitute a biological +system that responds to hidden states in the environment; +— they exhibit active inference [11]: the partition of states implied +by the Markov blanket endows internal states with the +apparent capacity to represent hidden states probabilisti- +cally, so that they appear to infer the hidden causes of +their sensory states (by minimizing a free energy bound +on log Bayesian evidence). By the circular causality induced +by the Markov blanket, sensory states depend on active +states, rendering inference active or embodied; and +— they are autopoietic [4]: because active states change—but +are not changed by—hidden states (figure 1), they will +appear to place an upper (free energy) bound on the dis- +persion (entropy) of biological states. This homoeostasis is +informed by internal states, which means that active states +will appear to maintain the structural and functional +integrity of biological states. + +When expressed like this, these criteria appear perfectly +sensible but are they useful in the setting of real biophysical +systems? The premise of this paper is that these criteria apply +to (almost) all ergodic systems encountered in the real world. +The argument here is that biological behaviour rests on the +existence of a Markov blanket—and that a Markov blanket is +(almost) inevitable in coupled dynamical systems with short- +range interactions. In other words, if the coupling between +dynamical systems can be neglected—when they are separated +by large distances—the intervening systems will necessarily + +form a Markov blanket. For example, if we consider short- +range electrochemical and nuclear forces, then a cell membrane +forms a Markov blanket for internal intracellular states +(figure 1). If this argument is correct, then it should be possible +to show the emergence of biological self-organization in any +arbitrary ensemble of coupled subsystems with short-range +interactions. The final section uses simulations to provide a +proof of principle, using the four criteria above to identify +and verify the emergence of lifelike behaviour. + +3. Proof of principle + +In this section, we simulate a primordial soup to illustrate the +emergence of biological self-organization. This soup comprises +an ensemble of dynamical subsystems—each with its own +structural and functional states—that are coupled through +short-range interactions. These simulations are similar to (hun- +dreds of) simulations used to characterize pattern formation in +dissipative systems; for example, Turing instabilities [48]: the +theory of dissipative structures considers far-from-equilibrium +systems, such as turbulence and convection in fluid dynamics +(e.g. Be´nard cells), percolation and reaction–diffusion systems +such as the Belousov–Zhabotinsky reaction [49]. Self-assembly +is another important example from chemistry that has biologi- +cal connotations (e.g. for pre-biotic formation of proteins). The +simulations here are distinguished by solving stochastic differ- +ential equations for both structural and functional states. In +other words, we consider states from classical mechanics that +determine physical motion—and functional states that could +describe electrochemical states. Importantly, the functional +states of any system affect the functional and structural states +of another. The agenda here is not to explore the repertoire of +patterns and self-organization these ensembles exhibit—but +rather take an arbitrary example and show that, buried +within it, there is a clear and discernible anatomy that satisfies +the criteria for life. +3.1. The primordial soup +To simulate a primordial soup, we use an ensemble of +elemental subsystems with (heuristically speaking) Newto- +nian and electrochemical dynamics f~p;~qg [ X: + +_~p ¼ fpð~p;~qÞ þ v + +and +_~q ¼ fqð~p;~qÞ þ v + +) + +ð3:1Þ + +Here, ~pðtÞ ¼ ð p; p0; p00; . . .Þ are generalized coordinates of motion +describing position, velocity, acceleration—and so on—of the +subsystems, while ~qðtÞ correspond to electrochemical states +(such as concentrations or electromagnetic states). One can +think of these generalized states as describing the physical and +electrochemical state of large macromolecules. Crucially, these +states are coupled within and between the subsystems compris- +ing an ensemble. The electrochemical dynamics were chosen +to have a Lorenz attractor: for the ith system with its own rate +parameter k(i): + +_qðiÞ ¼ kðiÞ � + +10ðqðiÞ +2 � qðiÞ +1 Þ + +ð32 þ �qð jÞ +1 Þ � qðiÞ +1 � qðiÞ +2 � x3qðiÞ +1 +qðiÞ +1 qðiÞ +2 � 8 +3qðiÞ +3 + +2 + +664 + +3 + +775 þ kðiÞ � �qðiÞ þ v; + +�qðiÞ ¼ P +j qð jÞ � Aij; + +Aij ¼ ½jDijj , 1� + +and +Dij ¼ pð jÞ � pðiÞ: + +9 +> +> +> +> +> +> +> +> +> +> +> += + +> +> +> +> +> +> +> +> +> +> +> +; + +ð3:2Þ + +rsif.royalsocietypublishing.org +J R Soc Interface 10: 20130475 + +5 + +Downloaded from rsif.royalsocietypublishing.org on September 6, 2013 + + +Changes in electrochemical states are coupled through +the local average �qðiÞof the states of subsystems that lie within +a distance of one. This means that A can be regarded as an +(unweighted) adjacency matrix that encodes the dependencies +among the functional (electrochemical) states of the ensemble. +The local average enters the equations of motion both linearly +and nonlinearly to provide an opportunity for generalized syn- +chronization [50]. The nonlinear coupling effectively renders + +the Rayleigh parameter of the flow 32 þ �qð jÞ +1 +state-dependent. +The Lorenz form for these dynamics is a somewhat +arbitrary choice but provides a ubiquitous model of electrody- +namics, lasers and chemical reactions [51]. The rate parameter +kðiÞ ¼ 1 +32ð1 � expð�4 � UÞÞ was specific to each subsystem, +where U [ (0, 1) was selected from a uniform distribution. +This introduces heterogeneity in the rate of electrochemical +dynamics, with a large number of fast subsystems—with a +rate constant of nearly one—and a small number of slower sub- +systems. To augment this heterogeneity, we randomly selected +a third of the subsystems and prevented them from (electro- +chemically) influencing others, by setting the appropriate +column of the adjacency matrix to zero. We refer to these as +functionally closed systems. +In a similar way, the classical (Newtonian) motion of each +subsystem depends upon the functional status of its neighbours: + +_pðiÞ ¼ p0ðiÞ þ v; + +_p0ðiÞ ¼ 1 +32 � wðiÞ � 1 +4 � p0ðiÞ � +1 +1024 pðiÞ þ v; + +wðiÞ ¼ +X + +j + +Dij +jDijj � + +wðiÞ +f +jDijj � +1 + +jDijj2 + +0 + +@ + +1 + +A � Aij + +and +wðiÞ +f +¼ 8 � expð2 � jqð jÞ +3 � qðiÞ +3 jÞ � 2: + +9 +> +> +> +> +> +> +> +> +> +> += + +> +> +> +> +> +> +> +> +> +> +; + +ð3:3Þ + +This motion rests on forces w(i) exerted by other subsys- +tems that comprise a strong repulsive force (with an inverse +square law) and a weaker attractive force that depends on +their electrochemical states. This force was chosen so that +systems with coherent (third) states are attracted to each +other but repel otherwise. The remaining two terms in the +expression for acceleration (second equality) model viscosity +that depends upon velocity and an exogenous force that +attracts all locations to the origin—as if they were moving +in a simple (quadratic) potential energy well. This ensures +the synthetic soup falls to the bottom of the well and enables +local interactions. +Note that the ensemble system is dissipative at two levels: +first, the classical motion includes dissipative friction or vis- +cosity. Second, the functional dynamics are dissipative in +the sense that they are not divergence-free. We will now +assess the criteria for biological self-organization within this +coupled random dynamical ensemble. + +3.2. Ergodicity +In the examples used below, 128 subsystems were integrated +using Euler’s (forward) method with step sizes of 1/512 s +and initial conditions sampled from the normal distribution. +Random fluctuations were sampled from the unit normal +distribution. By adjusting the parameters in the above equa- +tions of motion, one can produce a repertoire of plausible +and interesting behaviours (the code for these simulations +and the figures in this paper are available as part of +the SPM academic freeware). These behaviours range from + +gas-like behaviour (where subsystems occasionally get close +enough to interact) to a cauldron of activity, when sub- +systems are forced together at the bottom of the potential +well. In this regime, subsystems get sufficiently close for the +inverse square law to blow them apart—reminiscent of sub- +atomic particle collisions in nuclear physics. With particular +parameter values, these sporadic and critical events can +render the dynamics non-ergodic, with unpredictable high +amplitude fluctuations that do not settle down. In other +regimes, a more crystalline structure emerges with muted +interactions and low structural (configurational) entropy. +However, for most values of the parameters, ergodic be- +haviour emerges as the ensemble approaches its random +global attractor (usually after about 1000 s): generally, subsys- +tems repel each other initially (much like illustrations of the +big bang) and then fall back towards the centre, finding +each other as they coalesce. Local interactions then mediate +a reorganization, in which subsystems are passed around +(sometimes to the periphery) until neighbours gently jostle +with each other. In terms of the dynamics, transient synchro- +nization can be seen as waves of dynamical bursting (due to +the nonlinear coupling in equation (3.2)). In brief, the motion +and electrochemical dynamics look very much like a restless +soup (not unlike solar flares on the surface of the sun, figure +2)—but does it have any self-organization beyond this? + +3.3. The Markov blanket +Because the structural and functional dependencies share +the same adjacency matrix—which depends upon position— +one can use the adjacency matrix to identify the principal +Markov blanket by appealing to spectral graph theory: +the Markov blanket of any subset of states encoded by a +binary vector with elements xi [ f0, 1g is given by [B . x] [ +f0, 1g, where the Markov blanket matrix B ¼ A þ AT þ ATA +encodes children, parents and parents of children. This +follows because the ith column of the adjacency matrix +encodes the directed connections from the ith state to all its +children. +The +principal +eigenvector of +the +(symmetric) +Markov +blanket +matrix +will—by +the +Perron–Frobenius +theorem—contain positive values. These values reflect the +degree to which each state belongs to the cluster that is most +interconnected (cf., spectral clustering). In what follows, the +internal states were defined as belonging to subsystems with +the k ¼ 8 largest values. Having defined the internal states, +the Markov blanket can be recovered from the Markov blanket +matrix using [B . x] and divided into sensory and active +states—depending upon whether they are influenced by the +hidden states or not. +Given the internal states and their Markov blanket, we can +now follow their assembly and visualize any structural or func- +tional characteristics. Figure 3 shows the adjacency matrix used +to identify the Markov blanket. This adjacency matrix has +non-zero entries if two subsystems were coupled over the last +256 s of a 2048 s simulation. In other words, it accommoda- +tes the fact that the adjacency matrix is itself an ergodic +process—due to the random fluctuations. Figure 3b shows +the location of subsystems with internal states (blue) and +their Markov blanket—in terms of sensory (magenta) and +active (red) locations. A clear structure can be seen here, +where the internal subsystems are (unsurprisingly) close +together and enshrouded by the Markov blanket. Interestingly, +the active subsystems support the sensory subsystems that are + +rsif.royalsocietypublishing.org +J R Soc Interface 10: 20130475 + +6 + +Downloaded from rsif.royalsocietypublishing.org on September 6, 2013 + + +exposed to hidden environmental states. This is reminiscent of +a biological cell with a cytoskeleton that supports some sensory +epithelia or receptors within its membrane. +Figure +3c +highlights +functionally +closed +subsystems +(filled circles) that have been rusticated to the periphery of +the system. Recall that these subsystems cannot influence or +engage other subsystems and are therefore expelled to the +outer limits of the soup. Heuristically, they cannot invade +the system and establish a reciprocal and synchronous exchange +with other subsystems. Interestingly, no simulation ever pro- +duced a functionally closed internal state. Figure 3d shows the +slow subsystems that are distributed between internal and + +external states—which may say something interesting about +the generalized synchrony that underlies self-organization. + +3.4. Active inference +If the internal states encode a probability density over the +hidden or external states, then it should be possible to predict +external states from internal states. In other words, if internal +events represent external events, they should exhibit a signifi- +cant statistical dependency. To establish this dependency, we +examined the functional (electrochemical) status of internal +subsystems to see whether they could predict structural + +–8 +–6 +–4 +–2 +0 +2 +4 +6 +8 +–8 + +–6 + +–4 + +–2 + +0 + +2 + +4 + +6 + +8 +(i) +(ii) +(a) + +(b) + +(c) + +position + +ensemble +synchronization + +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 + +–30 + +–20 + +–10 + +0 + +10 + +20 + +30 + +dynamics + +–30 + +–20 + +–10 + +0 + +10 + +20 + +30 + +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +2000 + +time + +motion + +position + +Figure 2. Ensemble dynamics. (a) The position of (128) subsystems comprising an ensemble after 2048 s. a(i) The dynamical status (three blue dots per subsystem) +of each subsystem centred on its location (larger cyan dots). a(ii) The same information, where the relative values of the three dynamical states of each subsystem +are colour-coded (using a softmax function of the three functional states and a RGB mapping). This illustrates the synchronization of dynamical states within each +subsystem and the dispersion of the phases of the Lorenzian dynamics over subsystems. (b,c) The evolution of functional and structural states as a function of time, +respectively. The (electrochemical) dynamics of the internal (blue) and external (cyan) states are shown for the 512 s. One can see initial (chaotic) transients that +resolve fairly quickly, with itinerant behaviour as they approach their attracting set. (c) The position of internal (blue) and external (cyan) subsystems over the entire +simulation period illustrate critical events (circled) that occur every few hundred seconds, especially at the beginning of the simulation. These events generally reflect +a pair of particles (subsystems) being expelled from the ensemble to the periphery, when they become sufficiently close to engage short-range repulsive forces. +These simulations integrated the stochastic differential equations in the main text using a forward Euler method with 1/512 s time steps and random fluctuations of +unit variance. + +rsif.royalsocietypublishing.org +J R Soc Interface 10: 20130475 + +7 + +Downloaded from rsif.royalsocietypublishing.org on September 6, 2013 + + +events (movement) in the external milieu. This is not unlike +the approach taken in brain mapping that searches for statisti- +cal dependencies between, say, motion in the visual field and +neuronal activity [52]. +To test for statistical dependencies, the principal patterns +of activity among the internal (functional) states were sum- +marized using singular value decomposition and temporal +embedding (figure 4). A classical canonical variates analysis +was then used to assess the significance of a simple linear +mapping between expression of these patterns and the move- +ment of each external subsystem. Figure 4a illustrates these +internal dynamics, while figure 4c shows the Newtonian +motion of the external subsystem that was best predicted. +The agreement between the actual (dotted line) and predic- +ted (solid line) motion is self-evident, particularly around +the negative excursion at 300 s. The internal dynamics that +predict this event appear to emerge in their fluctuations +before the event itself (figure 4)—as would be anticipated if +internal events are modelling external events. Interestingly, +the subsystem best predicted was the furthest away from +the internal states (magenta circle in figure 4d). +This example illustrates how internal states infer or +register distant events in a way that is not dissimilar to + +the perception of auditory events through sound waves—or +the way that fish sense movement in their environment. +Figure 4d also shows the subsystems whose motion could be +predicted reliably. This predictability is the most significant +at the periphery of the ensemble, where the ensemble has +the greatest latitude for movement. These movements are +coupled to the internal states—via the Markov blanket— +through generalized synchrony. Generalized synchrony refers +to the synchronization of chaotic dynamics, usually in skew- +product (master-slave) systems [53,54]. However, in our +set-up there is no master–slave relationship but a circular +causality induced by the Markov blanket. Generalized syn- +chrony was famously observed by Huygens in his studies of +pendulum clocks—that synchronized themselves through the +imperceptible motion of beams from which they were sus- +pended [55]. This nicely illustrates the ‘action at a distance’ +caused by chaotically synchronized waves of motion. Circular +causality begs the question of whether internal states predict +external causes of their sensory states or actively cause them +through action. Exactly the same sorts of questions apply +to perception [56,57]: for example, are visually evoked neur- +onal responses caused by external events or by our (saccadic +eye) movements? + +element +20 +40 +60 +80 +100 120 + +20 + +(a) +(b) + +(c) +(d) + +40 + +60 + +80 + +100 + +120 + +–8 –6 –4 –2 +0 +2 +4 +6 +8 +–8 + +–6 + +–4 + +–2 + +0 + +2 + +4 + +6 + +8 + +–8 + +–6 + +–4 + +–2 + +0 + +2 + +4 + +6 + +8 + +position + +–8 –6 –4 –2 +0 +2 +4 +6 +8 +position +–8 –6 –4 –2 +0 +2 +4 +6 +8 +position + +–8 + +–6 + +–4 + +–2 + +0 + +2 + +4 + +6 + +8 + +hidden states + +sensory states + +active states + +internal states + +Figure 3. Emergence of the Markov blanket. (a) The adjacency matrix that indicates a conditional dependency (spatial proximity) on at least one occasion over the +last 256 s of the simulation. The adjacency matrix has been reordered to show the partition of hidden (cyan), sensory (magenta), active (red) and internal (blue) +subsystems, whose positions are shown in (b)—using the same format as in the previous figure. Note the absence of direct connections (edges) between external or +hidden and internal subsystem states. The circled area illustrates coupling between active and hidden states that are not reciprocated (there are no edges between +hidden and active states). The spatial self-organization in the upper left panel is self evident; where the internal states have arranged themselves in a small loop +structure with a little cilium, protected by the active states that support the surface or sensory states. When viewed as a movie, the entire ensemble pulsates in a +chaotic but structured fashion, with the most marked motion in the periphery. (c,d) Highlights those subsystems that cannot influence others (closed subsystems (c)) +and those that have slower dynamics (slow subsystems (d)). The remarkable thing here is that all the closed subsystems have been rusticated to the periphery— +where they provide a locus for vigorous dynamics and motion. Contrast this with the deployment of slow subsystems that are found throughout the hidden, sensory, +active and internal partition. + +rsif.royalsocietypublishing.org +J R Soc Interface 10: 20130475 + +8 + +Downloaded from rsif.royalsocietypublishing.org on September 6, 2013 + + +3.5. Autopoiesis and structural integrity +The previous section applied a simple sort of brain mapping +to establish the statistical dependencies between external +and internal states—and their functional correlates. The +final simulations also appeal to procedures in the biological +sciences—in particular neuropsychology to examine the +effects of lesions. To test for autopoietic maintenance of struc- +tural and functional integrity, the sensory, active and internal +subsystems were selectively lesioned—by rendering them +functionally closed—in other words, by preventing them +from influencing their neighbours. This is a relatively mild +lesion, in the sense that they remain physically coupled +with intact dynamics that respond to neighbouring elements. +Because active states depend only on sensory and internal +states one would expect to see a loss of structural integrity +not only with lesions to action but also to sensory and internal +states that are an integral part of active inference. + +Figure 5 illustrates the effects of these interventions by fol- +lowing the evolution of the internal states and their Markov +blanket over 512 s. Figure 5a shows the conservation of struc- +tural (and implicitly functional) integrity in terms of spatial +configuration over time. Contrast this with the remaining +three panels that show structural disintegration as the integ- +rity of the Markov blanket is lost and internal elements are +extruded into the environment. + +4. Conclusion + +Clearly, there are many issues that need to be qualified and +unpacked under this formulation. Perhaps the most prescient +is its focus on boundaries or Markov blankets. This contrasts +with other treatments that consider the capacity of living +organisms to reproduce by passing genetic material to their + +time + +modes + +100 +200 +300 +400 +500 + +time + +external states + +position + +position + +100 +200 +300 +0 + +10 + +20 + +30 + +40 + +50 + +c2 + +frequency + +–0.4 + +–0.3 + +–0.2 + +–0.1 + +0 + +100 +200 +300 +400 +500 +–8 + +–6 + +–4 + +–2 + +0 + +4 + +6 + +8 + +–5 +0 +5 + +2 + +5 + +(a) +(b) + +(c) +(d) + +10 + +15 + +20 + +25 + +30 + +Figure 4. Self-organized perception. This figure illustrates the Bayesian perspective on self-organized dynamics. (a) The first (principal) 32 eigenvariates of the +internal (functional) states as a function of time over the last 512 s of the simulations reported in the previous figures. These eigenvariates were obtained by a +singular value decomposition of the timeseries over all internal functional states (lagged between plus and minus 16 s). These represent a summary of internal +dynamics that are distributed over internal subsystems. The eigenvariates were then used to predict the (two-dimensional) motion of each external subsystem using +a standard canonical variates analysis. The (classical) significance of this prediction was assessed using Wilks’ lambda (following a standard transformation to the x2 + +statistic). The actual (dotted line) and predicted (solid line) position for the most significant external subsystem is shown in (c)—in terms of canonical variates (best +linear mixture of position in two dimensions). The agreement is self-evident and is largely subtended by negative excursions, notably at 300 s. The fluctuations in +internal states are visible in (a) and provide a linear mixture that correlates with the external fluctuation (highlighted with a white arrow). The location of the +external subsystem that was best predicted is shown by the magenta circle on (d). Remarkably, this is the subsystem that is the furthest away from the internal +states and is one of the subsystems that participates in the exchanges a closed subsystem in the previous figure. (c) Also shows the significance with which the +motion of the remaining external states could be predicted (with the intensity of the cyan being proportional to the x2 statistic above). Interestingly, the motion +that is predicted with the greatest significance is restricted to the periphery of the ensemble, where the external subsystems have the greatest latitude for move- +ment. To ensure this inferential coupling was not a chance phenomenon, we repeated the analysis after flipping the external states in time. This destroys any +statistical coupling between the internal and external states but preserves the correlation structure of fluctuations within either subset. The distribution of the +ensuing x2 statistics (over 82 external elements) is shown in (b) for the true (black) and null (white) analyses. Crucially, five of the subsystems in the true analysis +exceeded the largest statistic in the null analysis. The largest value of the null distribution provides protection against false positives at a level of 1/82. The +probability of obtaining five x2 values above this threshold by chance is vanishingly small p ¼ 0.00052. + +rsif.royalsocietypublishing.org +J R Soc Interface 10: 20130475 + +9 + +Downloaded from rsif.royalsocietypublishing.org on September 6, 2013 + + +offspring [1]. In this context, it is not difficult to imagine +extending the simulations above to include slow (e.g. diur- +nal) exogenous fluctuations—that cause formally similar +Markov blankets to dissipate and reform in a cyclical fashion. +The key question would be whether the internal states of a +system in one cycle induce—or code for—the formation of +a similar system in the next. +The central role of Markov blankets speak to an important +question: is there a unique Markov blanket for any given +system? Our simulations focused on the principal Markov +blanket—as defined by spectral graph theory. However, a +system can have a multitude of partitions and Markov blan- +kets. This means that there are many partitions that—at some +spatial and temporal scale—could show lifelike behaviour. +For example, the Markov blanket of an animal encloses +the Markov blankets of its organs, which enclose Markov +blankets of cells, which enclose Markov blankets of nuclei +and so on. Formally, every Markov blanket induces active +(Bayesian) inference and there are probably an uncountable +number of Markov blankets in the universe. Does this mean +there is lifelike behaviour everywhere or is there something + +special about the Markov blankets of systems we consider +to be alive? +Although speculative, the answer probably lies in the stat- +istics of the Markov blanket. The Markov blanket comprises a +subset of states, which have a marginal ergodic density. The +entropy of this marginal density reflects the dispersion or +invariance properties of the Markov blanket, suggesting +that there is a unique Markov blanket that has the smal- +lest entropy. One might conjecture that minimum entropy +Markov blankets characterize biological systems. This conjec- +ture is sensible in the sense that the physical configuration +and dynamical states that constitute the Markov blanket +of an organism—or organelle—change slowly in relation to +the external and internal states it separates. Indeed, the +physical configuration must be relatively constant to avoid +destroying anti-edges (the absence of an edge or coupling) +in the adjacency matrix that defines the Markov blanket. +This perspective suggests that there may be ways of charac- +terizing the statistics (e.g. entropy) of Markov blankets that +may quantify how lifelike they appear. Note from equation +(2.9) that systems (will appear to) place an upper bound on + +–8 –6 +–4 –2 +0 +2 +4 +6 +8 +–8 + +–6 + +–4 + +–2 + +0 + +2 + +4 + +6 + +8 +(a) +(b) + +(c) +(d) + +position + +–8 –6 –4 –2 +0 +2 +4 +6 +8 + +position + +–8 + +–6 + +–4 + +–2 + +0 + +2 + +4 + +6 + +8 + +–8 –6 +–4 –2 +0 +2 +4 +6 +8 + +position + +–8 –6 –4 –2 +0 +2 +4 +6 +8 + +position + +simulated lesions + +Figure 5. Autopoiesis and oscillator death. These results show the trajectory of the subsystems for 512 s after the last time point characterized in the previous +figures. (a) The trajectories under the normal state of affairs; showing a preserved and quasicrystalline arrangement of the internal states (blue) and the Markov +blanket (active states in red and sensory states in magenta). Contrast this formal self-organization with the decay and dispersion that ensues when the internal +states and Markov blankets are synthetically lesioned (b,c,d). In all simulations, a subset of states was lesioned by simply rendering their subsystems closed—in +other words, although the Newtonian interactions were preserved, they were unable to affect the functional states of neighbouring subsystems. (b) The effect of this +relatively subtle lesion on active states—that are rapidly expelled from the interior of the ensemble, allowing sensory states to invade and disrupt the internal +states. A similar phenomenon is seen when the sensory states were lesioned (c)—as they drift out into the external system. There is a catastrophic loss of structural +integrity when the internal states themselves cannot affect each other, with a rapid migration of internal states through and beyond their Markov blanket (d). These +simulations illustrate the effective death of biological self-organization that is a well-known phenomenon in dynamical systems theory—known as oscillator death: +see [58]. In our setting, they are a testament to autopoiesis or self-creation—in the sense that self-organized dynamics are necessary to maintain structural or +configurational integrity. + +rsif.royalsocietypublishing.org +J R Soc Interface 10: 20130475 + +10 + +Downloaded from rsif.royalsocietypublishing.org on September 6, 2013 + + +the entropy of the Markov blanket (and internal states). +This means that the marginal ergodic entropy measures the +success of this apparent endeavour. +However, minimum entropy is clearly not the whole story, +in the sense that biological systems act on their environment— +unlike a petrified stone with low entropy. In the language of +random attractors, the (internal and Markov blanket) states of +a system have an attracting set that is space filling but has a +small measure or entropy—where the measure or volume +upper bounds the entropy [11]. Put simply, biological systems +move around in their state space but revisit a limited number +of states. This space filling aspect of attracting sets may rest +on the divergence-free or solenoidal flow (equation (2.3)) that +we have largely ignored in this paper but may hold the key +for characterizing life forms. +Clearly, the simulations in this paper are a long way off +accounting for the emergence of biological structures such as +complex cells. The examples presented above are provided +as proof of principle and are as simple as possible. An interest- +ing challenge now will be to simulate the emergence of +multicellular structures using more realistic models with a +greater (and empirically grounded) heterogeneity and formal +structure. Having said this, there is a remarkable similarity +between the structures that emerge from our simulations and +the structure of viruses. Furthermore, the appearance of little +cilia (figure 3) are very reminiscent of primary cilia, which +typically serve as sensory organelles and play a key role in +evolutionary theory [59]. +A related issue is the nature of the dynamical (molecular +or cellular) constituents of the ensembles considered above. +Nothing in this treatment suggests a special role for carbon- +based life or, more generally, the necessary conditions for +life to emerge. The contribution of this work is to note +that if systems are ergodic and possess a Markov blanket, +they will—almost surely—show lifelike behaviour. However, +this does not address the conditions that are necessary for the +emergence of ergodic Markov blankets. There may be useful +constraints implied by the existence of a Markov blanket +(whose constituency has to change more slowly than the +states of its constituents). For example, the spatial range of +electrochemical forces, temperature and molecular chemistry +may determine whether the physical motion of molecules +(that determine the integrity of the Markov blanket) is +large or small in relation to fluctuations in electrochemical +states (that do not). However, these questions are beyond +the scope of this paper and may be better addressed in +computational chemistry and theoretical biology. + +This touches on another key issue, namely that of evolu- +tion. In this treatment, we have assumed biological systems +are ergodic. Clearly, this is a simplification, in that real +systems are only locally ergodic. The implication here is +that self-organized systems cannot endure indefinitely and +are only ergodic over a particular (somatic) timescale, +which raises the question of evolutionary timescales: is evol- +ution itself the slow and delicate unwinding of a trajectory +through a vast state space—as the universe settles on its +global random attractor? The intimation here is that adap- +tation and evolution may be as inevitable as the simple sort +of self-organization considered in this paper. In other +words, the very existence of biological systems necessarily +implies they will adapt and evolve. This is meant in the +sense that any system with a random dynamical attractor +will appear to minimize its variational free energy and can +be interpreted as engaging in active inference—acting upon +its external milieu to maintain an internal homoeostasis. +However, the ensuing homoeostasis is as illusory as the free +energy minimization upon which it rests. Does the same +apply to adaptation and evolution? +Adaptation on a somatic timescale has been interpreted +as optimizing the parameters of a generative model (encoded +by slowly changing internal states like synaptic connection +strengths in the brain) such that they minimize free energy. It +is fairly easy to show that this leads to Hebbian or associative +plasticity of the sort that underlies learning and memory [21]. +Similarly, at even longer timescales, evolution can be cast in +terms of free energy minimization—by analogy with Bayesian +model selection based on variational free energy [60]. Indeed, +free energy functionals have been invoked to describe natural +selection [61]. However, if the minimization of free energy is +just a corollary of descent onto a global random attractor, +does this mean that adaptation and evolution are just ways of +describing the same thing? The answer to this may not be +straightforward, especially if we consider the following possi- +bility: if self-organization has an inferential aspect, what +would happen if systems believed their attracting sets had +low entropy. If one pursues this in a neuroscience setting, one +arrives at a compelling explanation for the way we adaptively +sample our environments—to minimize uncertainty about the +causes of sensory inputs [62]. In short, this paper has only con- +sidered inference as emergent property of self-organization— +not the nature of implicit (prior) beliefs that underlie inference. + +Acknowledgements. I would like to thank two anonymous reviewers for +their detailed and thoughtful help in presenting these ideas. The +Wellcome Trust funded this work. + +References + +1. +Schro¨dinger E. 1944 What is life?: the physical aspect +of the living cell. Dublin, Ireland: Trinity College. +2. +Ashby WR. 1947 Principles of the self-organizing +dynamic system. J. Gen. Psychol. 37, 125–128. +(doi:10.1080/00221309.1947.9918144) +3. +Haken H. 1983 Synergetics: an introduction. Non- +equilibrium phase transition and self-selforganisation +in physics, chemistry and biology, 3rd edn. Berlin, +Germany: Springer. +4. +Maturana HR, Varela F. (eds) 1980 Autopoiesis and +cognition. Dordrecht, The Netherlands: Reidel. + +5. +Nicolis G, Prigogine I. 1977 Self-organization in non- +equilibrium systems. New York, NY: Wiley. +6. +Ao P. 2009 Global view of bionetwork +dynamics: adaptive landscape. J. Genet. +Genom. 36, 63–73. (doi:10.1016/S1673- +8527(08)60093-4) +7. +Demetrius L. 2000 Thermodynamics and evolution. +J. Theor. Biol. 206, 1–16. (doi:10.1006/jtbi.2000.2106) +8. +Davis MJ. 2006 Low-dimensional manifolds in reaction- +diffusion equations. I. Fundamental aspects. J. Phys. +Chem. A 110, 5235–5256. (doi:10.1021/jp055592s) + +9. +Auletta G. 2010 A paradigm shift in biology? +Information 1, 28–59. (doi:10.3390/info1010028) +10. Rabinovich MI, Afraimovich VS, Bick V, Varona P. 2012 +Information flow dynamics in the brain. Phys. Life Rev. +9, 51–73. (doi:10.1016/j.plrev.2011.11.002) +11. Friston K. 2012 A free energy principle for biological +systems. Entropy 14, 2100–2121. (doi:10.3390/ +e14112100) +12. Friston K, Ao P. 2012 Free-energy, value +and attractors. Comput. Math. Meth. Med. 2012, +937860. (doi:10.1155/2012/937860) + +rsif.royalsocietypublishing.org +J R Soc Interface 10: 20130475 + +11 + +Downloaded from rsif.royalsocietypublishing.org on September 6, 2013 + + +13. Conant RC, Ashby RW. 1970 Every good regulator +of a system must be a model of that system. +Int. J. Systems Sci. 1, 89–97. (doi:10.1080/0020 +7727008920220) +14. Evans DJ. 2003 A non-equilibrium free energy +theorem for deterministic systems. Mol. Phys. +101, 15 551–15 554. (doi:10.1080/00268970 +31000085173) +15. Evans DJ, Searles DJ. 1994 Equilibrium microstates +which generate second law violating steady states. +Phys. Rev. E 50, 1645–1648. (doi:10.1103/ +PhysRevE.50.1645) +16. Dayan P, Hinton GE, Neal R. 1995 The Helmholtz +machine. Neural Comput. 7, 889–904. (doi:10. +1162/neco.1995.7.5.889) +17. Gregory RL. 1980 Perceptions as hypotheses. Phil. +Trans. R. Soc. Lond. B 290, 181–197. (doi:10.1098/ +rstb.1980.0090) +18. Helmholtz H. 1866/1962 Concerning the perceptions +in general. In Treatise on physiological optics, 3rd +edn. New York, NY: Dover. +19. Kersten D, Mamassian P, Yuille A. 2004 Object +perception as Bayesian inference. Annu. Rev. +Psychol. 55, 271–304. (doi:10.1146/annurev.psych. +55.090902.142005) +20. Lee TS, Mumford D. 2003 Hierarchical Bayesian +inference in the visual cortex. J. Opt. Soc. Am. Opt. +Image Sci. Vis. 20, 1434–1448. (doi:10.1364/ +JOSAA.20.001434) +21. Friston K, Kilner J, Harrison L. 2006 A free energy +principle for the brain. J. Physiol. Paris 100, 70–87. +(doi:10.1016/j.jphysparis.2006.10.001) +22. Ay N, Bertschinger N, Der R, Gu¨ttler F, +Olbrich E. 2008 Predictive information and +explorative behavior of autonomous robots. Eur. +Phys. J. B 63, 329–339. (doi:10.1140/epjb/ +e2008-00175-0) +23. Bialek W, Nemenman I, Tishby N. 2001 +Predictability, complexity, and learning. Neural +Comput. 13, 2409–2463. (doi:10.1162/ +089976601753195969) +24. Tishby N, Polani D. 2010 Information theory of +decisions and actions. In Perception–reason–action +cycle: models, algorithms and systems (eds +V Cutsuridis, A Hussain, J Taylor), pp. 1–37. Berlin, +Germany: Springer. +25. Soodak H, Iberall A. 1978 Homeokinetics: a physical +science for complex systems. Science 201, 579–582. +(doi:10.1126/science.201.4356.579) +26. Birkhoff GD. 1931 Proof of the ergodic theorem. +Proc. Natl Acad. Sci. USA 17, 656–660. (doi:10. +1073/pnas.17.12.656) +27. Moore CC. 1966 Ergodicity of flows on +homogeneous spaces. Am. J. Math. 88, 154–178. +(doi:10.2307/2373052) +28. Pearl J. 1988 Probabilistic reasoning in intelligent +systems: networks of plausible inference. San +Fransisco, CA: Morgan Kaufmann. + +29. Crauel H, Flandoli F. 1994 Attractors for random +dynamical systems. Probab. Theory Relat. Fields 100, +365–393. (doi:10.1007/BF01193705) +30. Crauel H. 1999 Global random attractors are +uniquely determined by attracting deterministic +compact sets. Ann. Mat. Pura Appl. 4, 57–72. +(doi:10.1007/BF02505989) +31. Frank TD. 2004 Nonlinear Fokker–Planck equations: +fundamentals and applications. Springer Series in +Synergetics. Berlin, Germany: Springer. +32. Ao P. 2004 Potential in stochastic differential +equations: novel construction. J. Phys. A 37, +L25–L30. (doi:10.1088/0305-4470/37/3/L01) +33. Yuan R, Ma Y, Yuan B, Ping A. 2010 Bridging +engineering and physics: Lyapunov function as +potential function. See http://arxiv.org/abs/1012. +2721v1 [nlin.CD]. +34. Beal MJ. 2003 Variational algorithms for +approximate Bayesian inference. PhD thesis, +University College London. +35. Feynman RP. 1972 Statistical mechanics. Reading, +MA: Benjamin. +36. Hinton GE, van Camp D. 1993 Keeping neural +networks simple by minimizing the description +length of weights. Proc. COLT-93, 5–13. (doi:10. +1145/168304.168306) +37. Kass RE, Steffey D. 1989 Approximate Bayesian +inference in conditionally independent hierarchical +models (parametric empirical Bayes models). J. Am. +Stat. Assoc. 407, 717–726. (doi:10.1080/01621459. +1989.10478825) +38. Kullback S, Leibler RA. 1951 On information and +sufficiency. Ann. Math. Statist. 22, 79–86. (doi:10. +1214/aoms/1177729694) +39. van Leeuwen C. 1990 Perceptual-learning systems +as conservative structures: is economy an attractor? +Psychol. Res. 52, 145–152. (doi:10.1007/BF00877522) +40. Pasquale V, Massobrio P, Bologna LL, Chiappalone +M, Martinoia S. 2008 Self-organization and +neuronal avalanches in networks of dissociated +cortical neurons. Neuroscience 153, 1354–1369. +(doi:10.1016/j.neuroscience.2008.03.050) +41. Friston K. 2010 The free-energy principle: a unified +brain theory? Nat. Rev. Neurosci. 11, 127–138. +(doi:10.1038/nrn2787) +42. Barlow H. 1961 Possible principles underlying the +transformations of sensory messages. In Sensory +communication (ed. W Rosenblith), pp. 217–234. +Cambridge, MA: MIT Press. +43. Linsker R. 1990 Perceptual neural organization: +some approaches based on network models and +information theory. Annu. Rev. Neurosci. 13, 257– +281. (doi:10.1146/annurev.ne.13.030190.001353) +44. Jaynes ET. 1957 Information theory and statistical +mechanics. Phys. Rev. Ser. II 106, 620–630. +45. Breakspear M, Stam CJ. 2005 Dynamics of a neural +system with a multiscale architecture. Phil. Trans. R. Soc. +B 360, 1051–1074. (doi:10.1098/rstb.2005.1643) + +46. Bressler SL, Tognoli E. 2006 Operational principles of +neurocognitive networks. Int. J. Psychophysiol. 60, +139–148. (doi:10.1016/j.ijpsycho.2005.12.008) +47. Freeman WJ. 1994 Characterization of state transitions +in spatially distributed, chaotic, nonlinear, dynamical +systems in cerebral cortex. Integr. Physiol. Behav. Sci. +29, 294–306. (doi:10.1007/BF02691333) +48. Turing AM. 1952 The chemical basis of +morphogenesis. Phil. Trans. R. Soc. Lond. B 237, +37–72. (doi:10.1098/rstb.1952.0012) +49. Belousov BP. 1959 Qfrjpejyfslj +efkstcu<7a> rfalxj> j ff +nfwaojin [Periodically acting reaction and its +mechanism]. Sbprorfvfratpc qp +raejaxjpoopk nfejxjof [Collection of +Abstracts on Radiation Medicine], 145–147. +50. Hu A, Xu Z, Guo L. 2010 The existence of generalized +synchronization of chaotic systems in complex +networks. Chaos 20, 013112. (doi:10.1063/1.3309017) +51. Poland D. 1993 Cooperative catalysis and chemical +chaos: a chemical model for the Lorenz equations. +Physica D 65, 86–99. (doi:10.1016/0167-2789(93) +90006-M) +52. Zeki S. 2005 The Ferrier lecture 1995 behind the +seen: the functional specialization of the brain in +space and time. Phil. Trans. R. Soc. Lond. B 360, +1145–1183. (doi:10.1098/rstb.2005.1666) +53. Hunt B, Ott E, Yorke J. 1997 Differentiable +synchronisation of chaos. Phys. Rev. E 55, +4029–4034. (doi:10.1103/PhysRevE.55.4029) +54. Barreto E, Josic K, Morales CJ, Sander E, So P. 2003 +The geometry of chaos synchronization. Chaos 13, +151–164. (doi:10.1063/1.1512927) +55. Huygens C. 1673 Horologium oscillatorium. France: +Parisiis. +56. Adams RA, Shipp S, Friston KJ. 2012 Predictions not +commands: active inference in the motor system. +Brain Struct. Funct. 218, 611–643. (doi:10.1007/ +s00429-012-0475-5) +57. Wurtz RH, McAlonan K, Cavanaugh J, Berman RA. 2011 +Thalamic pathways for active vision. Trends Cogn. Sci. 5, +177–184. (doi:10.1016/j.tics.2011.02.004) +58. De Monte S, d’Ovidio F, Mosekilde E. 2003 Coherent +regimes of globally coupled dynamical systems. +Phys. Rev. Lett. 90, 054102. (doi:10.1103/ +PhysRevLett.90.054102) +59. Pallen MJ, Matzke NJ. 2006 From the origin of +species to the origin of bacterial flagella. Nat. Rev. +Microbiol. 4, 784–790. (doi:10.1038/nrmicro1493) +60. Friston K, Penny W. 2011 Post hoc Bayesian model +selection. Neuroimage 56, 2089–2099. (doi:10. +1016/j.neuroimage.2011.03.062) +61. Sella G, Hirsh AE. 2005 The application of statistical +physicsto evolutionary biology. Proc. Natl Acad. Sci. USA +102, 9541–9546. (doi:10.1073/pnas.0501865102) +62. Friston K, Adams RA, Perrinet L, Breakspear M. 2012 +Perceptions as hypotheses: saccades as experiments. +Front. Psychol. 3, 151. (doi:10.3389/fpsyg.2012.00151) + +rsif.royalsocietypublishing.org +J R Soc Interface 10: 20130475 + +12 + +Downloaded from rsif.royalsocietypublishing.org on September 6, 2013 + + diff --git a/papers/project_paper_2_neuroscience/references/Tononi2016.pdf b/papers/project_paper_2_neuroscience/references/Tononi2016.pdf new file mode 100644 index 00000000..f3070d1c --- /dev/null +++ b/papers/project_paper_2_neuroscience/references/Tononi2016.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:39afc535509daa34a9d95e14cbffaafd44b9452e4ed558f06298afdaa0a74834 +size 808032 diff --git a/papers/project_paper_2_neuroscience/references/Tononi2016.txt b/papers/project_paper_2_neuroscience/references/Tononi2016.txt new file mode 100644 index 00000000..3c687f24 --- /dev/null +++ b/papers/project_paper_2_neuroscience/references/Tononi2016.txt @@ -0,0 +1,2610 @@ +Consciousness is subjective experience +— ‘what it is like’, for example, to perceive +a scene, to endure pain, to entertain a +thought or to reflect on the experience +itself 1–3. When consciousness fades, as it +does in dreamless sleep, from the intrinsic +perspective of the experiencing subject, the +entire world vanishes. + +Consciousness depends on the integrity + +of certain brain regions and the particular +content of an experience depends on the +activity of neurons in parts of the cerebral +cortex4. However, despite increasingly refined +clinical and experimental studies, a proper +understanding of the relationship between +consciousness and the brain has yet to be +established5,6. For example, it is not known +why the cortex supports consciousness +when the cerebellum does not, despite +having four times as many neurons7,8, or why +consciousness fades during deep sleep while +the cerebral cortex remains active. There are +also many other difficult questions about +consciousness. Are patients with a functional +island of cortex surrounded by widespread +damage conscious, and if so, of what? Are +newborn infants conscious? Are animals that +display complex behaviours, but have brains +very different from humans, conscious6? Can +intelligent machines be conscious9? + +the brain, leads to testable predictions, and +allows inferences and extrapolations about +consciousness. + +From phenomenology to physics +The axioms of IIT state that every experience +exists intrinsically and is structured, +specific, unitary and definite. IIT then +postulates that, for each essential property of +experience, there must be a corresponding +causal property of the PSC. The postulates +of IIT state that the PSC must have intrinsic +cause–effect power; its parts must also have +cause–effect power within the PSC and they +must specify a cause–effect structure that +is specific, unitary and definite. Below, we +discuss the axioms and postulates of IIT (see +Supplementary information S1,S2 (figure, +box)) and describe the fundamental identity +— between an experience and a conceptual +structure — that it proposes (FIG. 1). + +The first axiom of IIT states that + +experience exists intrinsically. As +recognized by Descartes13, my own +experience is the only thing whose existence +is immediately and absolutely evident, +and it exists for myself, from my own +intrinsic perspective. The corresponding +postulate states that the PSC must also exist +intrinsically. For something to exist in a +physical sense, it must have cause–effect +power — that is, it must be possible to make +a difference to it (that is, change its state) +and it must be able to make a difference to +something. Moreover, the PSC must exist +intrinsically — that is, it must have cause– +effect power for itself, from its own intrinsic +perspective. A neuron in the brain, for +example, satisfies the criterion for existence +because it has two or more internal states +(such as active and inactive) that can be +affected by inputs (causes) and its output +can make a difference to other neurons +(effects). A minimal system consisting of +two interconnected neurons satisfies the +criterion of intrinsic existence because, +through their reciprocal interactions, the +system can make a difference to itself. + +The axiom of composition states that + +experience is structured, being composed of +several phenomenal distinctions that exist +within it. For example, within an experience, +I may distinguish a piano, a blue colour, a +book, countless spatial locations, and so on + +To answer these questions, the + +empirical study of consciousness should +be complemented by a theoretical +approach. The reason why some neural +mechanisms, but not others, should be +associated with consciousness has been +called ‘the hard problem’ because it seems +to defy the possibility of a scientific +explanation10. In this Opinion article, we +provide an overview of the integrated +information theory (IIT) of consciousness, +which has been developed over the past +few years1–3,11,12. IIT addresses the hard +problem in a new way. It does not start +from the brain and ask how it could give +rise to experience; instead, it starts from +the essential phenomenal properties of +experience, or axioms, and infers postulates +about the characteristics that are required +of its physical substrate. Moreover, IIT +presents a mathematical framework for +evaluating the quality and quantity of +consciousness1–3,9. We begin by providing a +summary of the axioms and corresponding +postulates of IIT and show how they can be +used, in principle, to identify the physical +substrate of consciousness (PSC). We then +discuss how IIT explains in a parsimonious +manner a variety of facts about the +relationship between consciousness and + +OPINION +Integrated information theory: +from consciousness to its physical +substrate + +Giulio Tononi, Melanie Boly, Marcello Massimini and Christof Koch + +Abstract | In this Opinion article, we discuss how integrated information theory +accounts for several aspects of the relationship between consciousness and the +brain. Integrated information theory starts from the essential properties of +phenomenal experience, from which it derives the requirements for the physical +substrate of consciousness. It argues that the physical substrate of consciousness +must be a maximum of intrinsic cause–effect power and provides a means to +determine, in principle, the quality and quantity of experience. The theory leads +to some counterintuitive predictions and can be used to develop new tools for +assessing consciousness in non-communicative patients. + +450 | JULY 2016 | VOLUME 17 +www.nature.com/nrn + +PERSPECTIVES + +© + +2016 + +M +acm +illan + +Publishers + +Lim +ited. + +All + +rights + +reserved. + + +Experience + +Identity + +Purviewp +Purviewf +Mechanism + +1.0 +0.5 +0.0 + +1.0 +0.5 +0.0 + +1.0 +0.5 +0.0 + +1.0 +0.5 +0.0 + +Probability of state + +000100010 +110 +001101011111 + +1.0 +0.5 +0.0 + +BCp + +ABCp + +ABCf + +ABCf + +ACf + +Af + +Bf +ABCp + +ABp + +Ap + +ACc + +ABc + +Cc + +Bc + +Ac + +0.083 + +0.167 + +0.25 + +0.25 + +0.25 + +000100010 +110 +001101011111 + +φmax of +concept + +Conceptual structure + +011 + +011 + +010 + +010 + +110 + +110 + +001 + +001 + +100 + +100 + +101 + +101 +000 + +000 + +111 + +111 + +B +C + +A + +Physical substrate + +D + +MAJ + +OR +AND + +AND + +Φmax = 0.66 + +A + +B +C + +AB +AC + +Boundary of experience + +Concept + +Logic gate ON + +Probability of past states + +Probability of future states + +Logic gate OFF + +(FIG. 1). Based on this axiom, IIT postulates +that the elements that constitute the PSC must +also have cause–effect power within the PSC, +either alone or in combination (composing +first-order and higher-order mechanisms, +respectively). + +experience might be composed of seeing a +book (rather than seeing no book), which +is blue (rather than not blue), and so on for +all other possible contents of consciousness. +The corresponding postulate states that the +PSC must specify a cause–effect structure + +The axiom of information states that + +experience is specific, being composed of a +particular set of phenomenal distinctions +(qualia), which make it what it is and different +from other experiences. In the example +shown in FIG. 1, the content of my current + +Figure 1 | An experience is a conceptual structure. According to inte- +grated information theory (IIT), a particular experience (illustrated here from +the point of view of the subject) is identical to a conceptual structure spec- +ified by a physical substrate. The true physical substrate of the depicted +experience (seeing one’s hands on the piano) and the associated conceptual +structure are highly complex. To allow a complete analysis of conceptual +structures, the physical substrate illustrated here was chosen to be +extremely simple1,2: four logic gates (labelled A, B, C and D, where A is a +Majority (MAJ) gate, B is an OR gate, and C and D are AND gates; the straight +arrows indicate connections among the logic gates, the curved arrows indi- +cate self-connections) are shown in a particular state (ON or OFF). The anal- +ysis of this system, performed according to the postulates of IIT, identifies a +conceptual structure supported by a complex constituted of the elements +A, B and C in their current ON states. The borders of the complex, which +include elements A, B, and C but exclude element D, are indicated by the +green circle. According to IIT, such a complex would be a physical substrate +of consciousness (Supplementary information S1 (figure)). The conceptual +structure is represented as a set of stars and, equivalently, as a set of histo- +grams. The green circle represents the fact that experience is definite (it +has borders). Each histogram illustrates the cause–effect repertoire of a +concept: how a particular mechanism constrains the probability of past +and future states of its maximally irreducible purview within the complex +ABC. The bins on the horizontal axis at the bottom of the histograms rep- +resent the 16-dimensional cause–effect space of the complex — all its +eight possible past states (p; in blue) and eight possible future states (f; in + +red; ON is 1 and OFF is 0). The vertical axis represents the probability of each +state (for consistency, the probability values shown are over the states of the +entire complex and not just over the subset of elements constituting the +purview). In this example, five of seven possible concepts exist, specified by +the mechanisms A, B, C, AB, AC (all with φmax>0) in their current state (which +are labelled as Ac, Bc, etc.). The subsets BC and ABC do not specify any con- +cept because their cause–effect repertoire is reducible by partitions +(φmax=0). In the middle, the 16-dimensional cause–effect space of the com- +plex is represented as a circle, where each of the 16 axes corresponds to one +of the eight possible past (p; blue arrows) and eight possible future states +(f; red arrows) of the complex, and the position along the axis represents +the probability of that state. Each concept is depicted as a star, the position +of which in cause–effect space represents how the concept specifies the +probability of past and future states of the complex, and the size of which +measures how irreducible the concept is (φmax). Relations between two +concepts (overlaps in their purviews) are represented as lines between the +stars. The fundamental identity postulated by IIT claims that the set of con- +cepts and their relations that compose the conceptual structure are identi- +cal to the quality of the experience. This is how the experience feels — what +it is like to be the complex ABC in its current state 111. The intrinsic irreduc- +ibility of the entire conceptual structure (Φmax, a non-negative number) +reflects how much consciousness there is (the quantity of the experience). +The irreducibility of each concept (φmax) reflects how much each +phenomenal distinction exists within the experience. Different experiences +correspond to different conceptual structures. + +PERSPECTIVES + +NATURE REVIEWS | NEUROSCIENCE + VOLUME 17 | JULY 2016 | 451 + +© + +2016 + +M +acm +illan + +Publishers + +Lim +ited. + +All + +rights + +reserved. + + +of a specific form, which makes it different +from other possible forms. A cause–effect +structure is defined as the set of cause–effect +repertoires specified by all the mechanisms of +a system. A cause–effect repertoire specifies +how a mechanism in its current state affects +the probability distribution of past and future +states of the system. + +The axiom of integration states that + +experience is unitary, meaning that it +is composed of a set of phenomenal +distinctions, bound together in various ways, +that is irreducible to non-interdependent +subsets. For example, I experience a whole +visual scene and that experience cannot be +subdivided into independent experiences of +the left and right sides of the visual field. In +other words, the content of an experience +(information) is integrated within a +unitary consciousness. The corresponding +postulate states that the cause–effect +structure specified by the PSC must also +be unitary — that is, it must be irreducible +to the cause–effect structure specified by +non-interdependent subsystems. Note +that, from the intrinsic perspective of the +system, integration requires that every part +of the system has both causes and effects +within the rest of the system, which implies +bidirectional interactions. The irreducibility +of a conceptual structure is measured +as integrated information (denoted Φ, the +minimum distance between an intact and +a partitioned cause–effect structure). The +integration postulate also requires the +irreducibility of each cause–effect repertoire +(denoted φ, the minimum distance between +an intact and a partitioned cause–effect +repertoire) and the irreducibility of relations +among overlapping cause–effect repertoires. + +The axiom of exclusion states that an + +experience is definite in its content and +spatio-temporal grain. For example, in +the scene depicted in FIG. 1, the content of +my present experience includes seeing my +hands on the piano, the books on the piano, +one of which is blue, and so on, but I am +not having an experience with less content +(for example, the same scene in black and +white, lacking the phenomenal distinction +between coloured and not coloured) or +with more content (for example, including +the additional phenomenal distinction of +feeling one’s blood pressure as high or low). +The duration of the instant of consciousness +is also definite, ranging from a few tens of +milliseconds to a few hundred milliseconds, +rather than lasting a few microseconds +or a few minutes14–16. The corresponding +postulate states that the cause–effect +structure specified by the PSC must also + +A set of elements in a state that satisfies + +all the postulates of IIT constitutes the PSC +and is referred to as a complex (FIG. 1). Thus +a complex specifies a conceptual structure +composed of concepts, which can be +represented as a set of points (shown as a +constellation of stars in FIG. 1) in cause–effect +space, in which each axis corresponds to a +possible past and future state of the system +and each star corresponds to a concept1 + +(FIG. 1). With these notions at hand, the +fundamental identity of IIT can be stated +as follows2: an experience is identical to a +conceptual structure, meaning that every +property of the experience must correspond +to a property of the conceptual structure and +vice versa. Note that the postulated identity +is between an experience and the conceptual + +be definite. It must specify a definite set of +cause–effect repertoires over a definite set of +elements, neither less nor more, at a definite +spatio-temporal grain, neither finer nor +coarser. Because a prerequisite for intrinsic +existence is having irreducible cause– +effect power, the cause–effect structure +that actually exists, over a set of elements +and spatio-temporal grains, is that which +is maximally irreducible (Φmax), called a +conceptual structure. As a consequence, any +cause–effect structure overlapping over the +same set of elements and spatio-temporal +grain is excluded. The exclusion postulate +also requires the maximum irreducibility +of cause–effect repertoires (denoted φmax), +called concepts, and of relations among +overlapping concepts. + +Glossary + +Achromatopsia +A condition in which a person is unable to perceive colours. + +Anosognosia +A condition in which a person has a neurological deficit, +but is unaware of it. + +Axioms +Properties that are self-evident and essential; in integrated +information theory, those that are true of every possible +experience — namely, intrinsic existence, composition, +information, integration and exclusion. + +Background conditions +Factors that enable consciousness, such as neuromodulators +and external inputs that maintain adequate excitability. + +Cause–effect repertoire +The probability distribution of potential past and future +states of a system that is specified by a mechanism in its +current state. + +Cause–effect space +A space with each axis representing the probability of each +possible past and future state of a system. + +Cause–effect structure +The set of cause–effect repertoires specified by all the +mechanisms of a system in its current state. + +Complex +A set of elements in a state that specifies a conceptual +structure corresponding to a maximum of integrated +information (Φmax). A complex is thus a physical substrate of +consciousness. + +Concepts +The cause–effect repertoires specified by a mechanism +that is maximally irreducible (φmax). + +Conceptual structure +The set of all concepts specified by a system of elements in +a state with their respective φmax values, which can be +plotted as a set of points in cause–effect space. + +Content-specific NCC +Neural elements, the activity of which determines a +particular content of experience. + +Elements +The minimum constituents of a system that have at +least two different states (for example, being on or off), +inputs that can affect those states and outputs that +depend on them. + +Full NCC +The neural elements constituting the physical +substrate of consciousness, irrespective of its +specific content. + +Integrated information +(Denoted Φ). Information that is specified by a system that +is irreducible to that specified by its parts. It is calculated +as the distance between the conceptual structure specified +by the intact system and that specified by its minimum +information partition. + +Mechanism +Any subset of elements within a system that has +cause–effect power on it (that is, that constrains its +cause–effect space). + +Neural correlates of consciousness +(NCC). The minimum neuronal mechanisms jointly +sufficient for any one specific conscious experience. + +Postulates +Properties of experience that are derived from the axioms +of integrated information theory and that must be +satisfied by the physical substrate of consciousness — +namely, to be a maximum of irreducible, specific, +compositional, intrinsic cause–effect power (intrinsic +cause–effect power for short). + +Purviews +The subsets of elements of a complex, the past and future +states of which are constrained by a mechanism specifying +a concept. + +Qualia +The qualitative feeling of phenomenal distinctions within an +experience (for example, seeing a colour, hearing a sound +or feeling a pain). + +Relations +Maximally irreducible overlaps among the purviews of two +or more concepts. + +PERSPECTIVES + +452 | JULY 2016 | VOLUME 17 +www.nature.com/nrn + +© + +2016 + +M +acm +illan + +Publishers + +Lim +ited. + +All + +rights + +reserved. + + +structure specified by the PSC, not between +an experience and the set of elements in +a state constituting the PSC (FIG. 1). The +quality or content of consciousness — which +particular way the system exists for itself — +corresponds to the form of the conceptual +structure. The quantity of consciousness +— how much the system exists for itself — +corresponds to its irreducibility Φmax. + +The PSC within the brain +Experimental evidence currently suggests +that the neural correlates of consciousness +(NCC) are likely to be located in certain +parts of the cortico-thalamic system5, but +it is not known specifically which cortical +areas, layers or neuronal populations are +involved, whether the relevant units are +neurons or groups of neurons, and which +aspects of their activity matter5. It is also +not known whether the neural substrate +of consciousness is anatomically fixed or +can shrink, expand and move. IIT offers +theoretical clarity on the empirical notion +of the NCC5. Specifically, it states that +the content-specific NCC correspond to the +neural elements of the PSC in a particular +state (activity pattern), which specify a +particular phenomenal content; the full +NCC correspond to the neural elements +constituting the PSC irrespective of their +particular state; the background conditions +are factors that enable consciousness, such +as neuromodulators and external inputs +that maintain adequate excitability, which +are kept fixed when evaluating the Φ value +of the PSC. Most importantly, the axioms +and postulates of IIT can be used to provide +a single, general principle for identifying +the PSC in the brain — namely that the +PSC must correspond to a complex of +neural elements with maximum intrinsic +cause–effect power. + +Elements of the PSC. What is the spatial +scale of the neural elements that support +consciousness: synapses, neurons, +neuronal groups, local fields or perhaps +all of these? According to IIT, the neural +elements of the PSC are those, and only +those, that support a maximum of cause– +effect power, as determined from the +intrinsic perspective of the system itself. +Importantly, and contrary to common +reductionist assumptions17, cause–effect +power can be higher at a macro-level than +at a micro-level18. For example, a system +of neuron-like micro-elements may have +less cause–effect power than the same +system coarse-grained at the macro-level of +neuronal groups (FIG. 2a). In general, whether + +both individual neurons and groups of +neurons, an experimenter could thus assess +at which grain size the network has most +cause–effect power from its own intrinsic +perspective — that is, at which level it +makes the most difference to itself. IIT +predicts that the elements of the PSC are +to be found at exactly that level and not at +any finer or coarser grain, a prediction that +is empirically testable: does the firing of +a single neuron make a difference21 to the +content of experience, or only the average +activity of a cortical mini-column22? + +Timescale. Which timescale of neuronal +activity is important for consciousness: +a few milliseconds, tens of milliseconds, +hundreds of milliseconds, or perhaps +all of these? Again, IIT predicts that the +relevant time interval should be that +which makes the most difference to the +system, as determined from its intrinsic +perspective. Once more, depending on +the specific mechanisms of a system, some +macro-temporal grain may have a higher +cause–effect power than both finer and +coarser grains (FIG. 2b). Whatever timescale +turns out to have the maximum cause–effect +power within the relevant brain regions, it +should be consistent with estimates of the +timescale of experience14–16. + +State of the elements. An external observer +can choose to analyse brain states at any +level of detail. For example, some neu- +rophysiologists may be interested in the +effects of the timing of individual neuronal +spikes on brain function, others in the +effects of broad fluctuations in the activity +of populations of neurons. In fact, it is +likely that almost any change in the state +of any neurobiological variable will have +some effect somewhere in the brain21. +According to IIT, the neural states that are +important for consciousness are only those +that have maximum cause–effect power on +the system itself. For example, assume that, +from the intrinsic perspective of the system, +maximum cause–effect power was achieved +when coarse-graining firing states into +low, high and burst firing (FIG. 2c). In this +case, IIT predicts that finer grained neural +states, despite their demonstrable neuro- +physiological effects, make no difference +to the content of experience. Note that +spatio-temporal grain and the relevant +activity states of the elements specifying +the PSC could change according to brain +region, developmental period, species, +neuromodulatory milieu and even the task +being performed. + +the macro or micro grain size has higher +cause–effect power depends on how intra- +and inter-group connections are organized +and the amount of indeterminism (noise) +and degeneracy (multiple ways of obtaining +the same effect18). + +An exhaustive evaluation of cause– + +effect power at multiple levels is only +possible in small simulated networks19. +In a real network20, we could start by +assessing the cause–effect repertoire of +individual neurons. For example, if a +neuron is firing a burst of spikes, its cause +repertoire is the probability distribution +of past network states that would have +caused it to burst (for example, firing +patterns of its afferent neurons within +the previous 100 ms). Similarly, its effect +repertoire is the probability distribution +of future network states given that the +neuron is bursting. Experimentally, we +could obtain an estimate of such cause– +effect repertoires by stimulating one +or more neurons optogenetically while +simultaneously recording the firing activity +of a population of neurons via two-photon +calcium imaging (keeping the background +conditions constant, such as the level of +arousal and sensory input) (FIG. 2a). Next, +we would need to test for the irreducibility +of the cause–effect repertoires, which +can be achieved by noising connections +(that is, enforcing firing at chance levels) +across a partition of the network. Doing so +would establish which subset of incoming +connections makes the most irreducible +difference (φmax) to the firing of the +observed neuron1 (and this could be carried +out analogously for outgoing connections). +A similar procedure should then be +repeated for subsets of two neurons, three +neurons, and so on, because combinations +of neurons can also have irreducible +cause–effect repertoires (defined as higher +order mechanisms). Such experiments +would provide an estimate of maximally +irreducible cause–effect repertoires at the +level of neurons. + +To evaluate cause–effect power at the + +macro-level, we could then repeat the +same stimulation–recording–noising +procedure by considering subsets of +neurons as distinct macro-groups and +mapping micro-states onto macro-states. +For example, we could take all pyramidal +neurons in each mini-column as a distinct +group and define the group state as low +firing, high firing or bursting, depending +on the overall firing rate of the neurons +over 100 ms. By estimating the φmax value +of cause–effect repertoires at the level of + +PERSPECTIVES + +NATURE REVIEWS | NEUROSCIENCE + VOLUME 17 | JULY 2016 | 453 + +© + +2016 + +M +acm +illan + +Publishers + +Lim +ited. + +All + +rights + +reserved. + + +Trial 1 + +a + +b + +c + +Trial 2 +Trial 3 + +Recording +Recording + +10 ms + +100 ms + +10 ms + +100 ms + +10 ms + +100 ms + +N1 + +N2 + +N3 + +N4 + +N1 + +N2 + +N3 + +N4 + +N1 + +N2 + +N3 + +N4 + +N1 + +N2 + +N3 + +N4 + +N1 + +N2 + +N3 + +N4 + +N1 + +N2 + +N3 + +N4 + +60 Hz +250 Hz + +Recording +Recording + +N4 +N4 +N4 + +60 Hz +250 Hz +1 Hz +60 Hz +250 Hz +1 Hz + +N1 +N1 +N1 + +N2 +N2 +N2 + +N3 +N3 +N3 + +N4 +N4 +N4 + +N1 +N1 +N1 + +N2 +N2 +N2 + +N3 +N3 +N3 + +Low +High +Burst + +1 Hz + +Firing rate unchanged +Firing rate decreases +Firing rate increases +Burst firing +Optogenetic stimulation + +Constitution of the PSC. Assume that we +have determined that the elementary units of +the PSC are local groups of cortical neurons, +over a time interval of ~100 ms, with three +relevant states (low, high and burst firing) + +(FIG. 3a). Next we must determine, at the +system level, which particular subset of +neuronal groups constitutes the PSC for a +particular experience. IIT addresses this +question from first principles — it predicts +that the PSC is the set of neuronal groups that +has maximally irreducible cause–effect power +on itself, specifying a conceptual structure + +differentiation)23; and integration, using +measures of functional or effective +connectivity among brain regions24,25. In +addition, large-scale computer simulations +based on the known anatomy and +physiology of cortical circuits26 can be +used to assess cause–effect repertoires, +test their irreducibility and estimate +conceptual structures. Crucially, if the +evidence thus obtained indicates that the +PSC does not correspond to a maximum +of intrinsic cause–effect power, IIT would +be invalidated. A related prediction is + +with the highest value of Φ1 (FIG. 3b). Ideally, +systematic manipulation and recording of this +particular set of neuronal groups would show +that it has the maximum value of Φ, whereas +any other assortment of neuronal groups in +the brain has a lower value of Φ. + +Although such an exhaustive evaluation + +of Φ is not currently feasible, neuroimaging +studies can evaluate two key requirements +for a high Φ value: information, using +measures that reflect the size of the +repertoire of neural states the system +can have (that is, neurophysiological + +PERSPECTIVES + +454 | JULY 2016 | VOLUME 17 +www.nature.com/nrn + +© + +2016 + +M +acm +illan + +Publishers + +Lim +ited. + +All + +rights + +reserved. + + +that any perturbation of the PSC at the +appropriate spatio-temporal grain should +produce a change in experience, whereas +any perturbation that does not alter the PSC +should not. + +Can the PSC change? An important issue +is the extent to which the set of neural +elements that constitute the PSC is fixed. +Clearly, if a cortical area is inactivated (by a +lesion, for example) it will no longer be part +of the PSC and the phenomenal distinctions +contributed by that area will no longer be +available. For example, if cortical areas +responding to colour are inactivated (FIG. 3c), +experiences will not only lack colour, but +patients would not even understand what is +lacking (as reported in cases of achromatopsia +with anosognosia27). + +It is an open question whether the PSC + +can shrink, expand or move during normal +wakefulness, possibly through attentional +modulation of excitability and functional +connectivity. For example, when we are + +experiences of pure thought that have +minimal perceptual content may be caused +by slow waves that inactivate the posterior +cortex, and be specified by a PSC that is +considerably different from the PSC for +purely perceptual experiences31 (FIG. 3d). +At other times, transient, local slow waves +(indicative of an off-period) in colour areas +may cause the PSC to shrink and lead to +brief episodes of achromatopsia. Novel +methods that allow the transient inactivation +of specific cortical areas in humans, such +as transcranial magnetic stimulation or +focused ultrasound, would be ideal for +evaluating the contribution of those areas to +conscious content. + +Multiple complexes. According to IIT, +two or more non-overlapping complexes +may coexist as discrete PSCs within a +single brain1, each with its own definite +borders and value of Φmax. The complex +that specifies a person’s day-to-day stream +of consciousness should have the highest +value of Φmax — that is, it should be the +‘major’ complex. In some conditions, for +example after a split-brain operation, the +major complex may split (FIG. 3e). In such +instances, one consciousness, supported +by a complex in the dominant hemisphere +and with privileged access to Broca’s area, +would be able to speak about the experience, +but would remain unaware of the presence +of another consciousness, supported by a +complex in the other hemisphere, which +can be revealed by carefully designed +experiments32,33. An intriguing possibility +is that splitting of the PSC may also occur +in healthy people during long-lasting +dual-task conditions — for example, when +driving in an auto-pilot like manner on a +familiar road while listening to an engaging +conversation (FIG. 3f). Splitting into separate +maxima may also occur through functional +disconnections caused by pathological +conditions, such as conversion and +dissociative disorders34. + +Another intriguing possibility is that + +multiple conscious streams may coexist +within a single brain in daily life. For +example, the grid-like architectures in the +colliculus and related mesencephalic regions, +which are adept at multimodal integration +within a spatial framework, may support a +separate minor complex. Some examples +of high-level cognitive performance such +as judging whether a scene is congruous +or incongruous35,36 — that appear to +be carried out unconsciously from the +perspective of the major complex — may +support a separate minor complex (FIG. 3e,g). + +engrossed in an action movie and not +engaged in self-reflection, the activity in +prefrontal areas decreases28. Does this mean +that the PSC shrinks, like when colour +areas are inactivated, or that brain regions +supporting self-reflection remain inside the +PSC but are inactive, in the same way that +colour areas are inactive when watching a +black and white movie? The location and +size of the PSC is likely to change during +sleep, during seizures, in patients with +conversion and dissociative disorders, and +possibly during hypnosis. During slow wave +sleep, for example, neurons are bistable and +show off-periods during which they become +hyperpolarized (down-states) and silent29. +However, these off-periods are usually not +global, but affect local subsets of brain areas +at different times30. Hence it is possible that +during slow wave sleep the PSC may become +smaller and reconfigure substantially. +Sustained inactivation of certain areas +during sleep may make dreaming patients +incapable of reflective thought. Similarly, + +Figure 2 | Identifying the elements, timescale and states of the physical substrate of conscious- +ness (PSC) from first principles. It is possible to determine maxima of cause–effect power within +the central nervous system by perturbing and observing neural elements at various micro- and +macro-levels18. High cause–effect power is reflected in deterministic responses and low cause– +effect power is reflected in responses that vary randomly across trials. a | To identify the spatial grain +of the elements of the PSC supporting consciousness, a schematic example shows how optogenetic +perturbation and unit recording could be applied to a subset of neurons (here, 3 out of 36 neurons) +to establish maxima of cause–effect power. For each of three trials, the left panel shows the effects +of the perturbation on the entire system at the micro-level. Grey neurons are unaffected, blue neu- +rons decrease their firing rates, red neurons increase their firing rates and purple neurons respond +with burst firing. The right-hand panel shows the effects of the perturbation at the macro-level after +coarse-graining of the 36 neurons into nine groups of four cells each. Macro-states are defined +according to the rule that if ≥50% of the neurons in the group are in a given micro-state (such as low +firing, high firing or bursting), then the group is considered to be in that state at the macro-level. In +this example, the macro-level (groups of neurons) has higher cause–effect power than the micro- +level (single neurons), because the response is deterministic at the macro-level (as evidenced by the +consistent colour scheme), whereas there are variations between trials at the micro-level (incon- +sistent colours). b | To identify the temporal grain of neuronal activity supporting consciousness, a +possible experimental setup would be one in which one neuron (the top trace) is optogenetically +excited while recording from other neurons (labelled N1–N4) across three trials, shown in the upper +panel at the 10 ms timescale (micro-scale). Grey shading indicates no effects on neuron firing in the +10 ms following the stimulation compared with the 10 ms before the stimulation, blue shading indi- +cates decreased firing and red shading indicates increased firing. The lower panel shows the same +data after temporal coarse-graining over 100 ms intervals. Macro-states are defined according to the +rule that if a neuron increases (or decreases) its firing rate by >50% within 100 ms post-stimulus +compared with the baseline, the macro state is considered to be high (or low) firing. In this example, +the macro-level (100 ms intervals) has higher cause–effect power (more deterministic responses) than +the micro-level (10 ms intervals). c | To identify the neural states that support consciousness, optoge- +netic perturbations could be used to drive one neuron to fire either at low frequency, high tonic +frequency or bursting (top trace) resulting in spectral peaks at 2 Hz (green), 50 Hz (red) and 150 Hz +(yellow) for neurons N1–N4 (data are shown as a firing rate histogram). For each trial, the upper panel +shows the responses of the other four neurons to each stimulation frequency at the micro-scale level +in the spectral domain (micro-bins, only a few of which are represented). The coloured bars indicate +coincidence, within a micro-bin, between the frequency of stimulation and the spectral peak of the +responses. The lower panel of each trial shows the effect of the perturbation at the corresponding +macro-level after spectral coarse-graining. Macro-states map into micro-states as indicated below +the frequency bins. Here, spectral coarse-graining (binning firing rates into three levels, low, high +and burst firing) results in higher cause–effect power (responses that are more deterministic) than +at the micro-level. + +◀ + +PERSPECTIVES + +NATURE REVIEWS | NEUROSCIENCE + VOLUME 17 | JULY 2016 | 455 + +© + +2016 + +M +acm +illan + +Publishers + +Lim +ited. + +All + +rights + +reserved. + + +50 μm +100 ms + +Space +Time +State + +a Macroelements, macrointervals + and macrostates + +Low + +High + +Burst + +b The major complex +c Shrinking of the major complex + +Major +complex + +High firing +Low firing +Burst firing + +Minor +complex + +d Movement of the major complex +f Functional splitting of + the major complex +g Coexistence of the major complex + with minor complexes +e Anatomical splitting of + the major complex + +Alternatively, some of these functions may +be mediated by feedforward circuits37 that +have Φmax=0 because they lack integration +and therefore are strictly unconscious1. +An important question for the future is +whether automatic, unconscious behaviours +are mediated by specific cell types within +the cortex, such as subcortical projection +neurons of layer 5B38, that are different from +other cell types that support consciousness. + +Information capacity of consciousness +The information-processing approach +common in psychology estimates +the information capacity of human +consciousness to be at around 7 ± 2 items39 +or ≤40 bits per second39,40. In the classic +Sperling task41, for example, participants +are presented with a set of 12 letters for + +of the Sperling display during the delay +period, they can report three letters of any +row; moreover, they can report the colour +diversity of unattended letters at no cost +to the identification of the cued letters50. +Likewise, change blindness may be due +not to a failure to experience, but to a lack +of memory for the experience51. Similarly, +low-level phenomenal features may be +difficult to report because they vary rapidly +and may be forgotten before they can be +accessed from top-down mechanisms; +pre-categorical stimuli, such as irregular +scribbles, may be phenomenally salient but +hard to describe in words. + +IIT claims that human consciousness has + +a high capacity for integrated information + +(BOX 1). Even for a simple experience, such +as seeing the Sperling display, the elements + +300 ms, of which, after a mask and a delay, +they can report at most four (FIG. 4). The +inference from such experiments is that the +information content of consciousness is +extremely limited, as is also suggested by the +attentional blink and related psychophysical +paradigms42,43. For example, in change +blindness, a major modification in a visual +scene may go undetected if a blank is +interposed between the two images44. In this +view, the content of consciousness is limited +to what can be accessed and reported, +despite our phenomenal impression of +richer content42,45,46. By contrast, others +argue that phenomenal consciousness (what +it is like to have an experience) has far +greater capacity than access consciousness +(what can be reported)47–49. For example, +if participants are cued to a particular row + +Figure 3 | Identifying the physical substrate of consciousness (PSC) +from first principles. The complex of neural elements that constitutes the +PSC can be identified by searching for maxima of intrinsic cause–effect +power. a | For example, assume that the elements, timescale and states at +which intrinsic cause–effect power reaches a maximum have been identified +using optogenetic and unit recording tools (FIG. 2). Here, the elements are +groups of neurons, the timescale is over 100 ms and there are three states +(low, high and burst firing). b | In a healthy, awake participant, the set of neural +elements specifying the conceptual structure with the highest Φmax is +assumed, based on current evidence, to be a complex of neuronal groups +distributed over the posterior cortex and portions of the anterior cortex5. +Empirical studies can, in principle, establish whether the full neural corre- +lates of consciousness5 correspond to the maximum of intrinsic cause–effect +power, thereby corroborating or falsifying a key prediction of integrated + +information theory. c | The boundaries of the PSC (green line) may change +after cortical lesions, such as those causing absolute achromatopsia, result- +ing in a smaller PSC. d | The PSC boundaries may also move as a result of +changes in excitability and effective connectivity, as might occur during pure +thought that is devoid of sensory content. e | The PSC could also split into +two large local maxima of cause–effect power (represented here by green +and blue boundaries) as a result of anatomical disconnections, such as in +split-brain patients, in which instance each hemisphere would have its own +consciousness. f | The PSC may also split as a result of functional disconnec- +tions, which may occur in some psychiatric disorders and perhaps under +certain dual-task conditions — for example while driving and talking at the +same time. g | The coexistence of a large major complex with one or more +minor complexes that may support sophisticated, seemingly unconscious +performance could be a common occurrence in everyday life. + +PERSPECTIVES + +456 | JULY 2016 | VOLUME 17 +www.nature.com/nrn + +© + +2016 + +M +acm +illan + +Publishers + +Lim +ited. + +All + +rights + +reserved. + + +of the PSC specify a rich conceptual +structure (high Φmax) composed of a very +large number of concepts and relations. +These correspond to all the phenomenal +distinctions that make that experience what +it is and thereby different from countless +others11 (FIG. 4). It is useful to distinguish +between low- and high-order concepts, +depending on how many PSC elements are +contained in their purviews. For example, +a concept specifying the presence of an +oriented edge at a particular location in +the visual field has a low-order purview, +whereas a concept specifying the extent +of the entire visual field has a high-order +purview. Concepts can also have low- and +high invariance; for example, the concept +for the letter A has high invariance +because its purview specifies a high-order +disjunction of states of the PSC elements (a +specific arrangement of oriented edges in +any of a large number of possible locations). + +concepts, such as letters in the Sperling +paradigm. However, we could undoubtedly +report many more concepts than just the +identity of a few letters. For example, we +could report that there are many black +symbols, that they are arranged in three rows +and four columns, in a rectangular array, +within a rectangular display, over a white +homogeneous background that is spatially +extended, being composed of a multitude +of distinguishable locations, each with its +specific neighbours, and so on. We can +also report many negative concepts — for +example, that the Sperling display did not +include a face, a tree, an animal, a house, and +so on — for the thousands of high invariance +concepts we possess that happen to be +negative for this particular image. Finally, we +can report how all these concepts are bound +together within the same experience in a +complex pattern of relations — for example, +we see the letter A as an invariant that is +nevertheless located at a particular spatial +location, that is composed of two oblique +edges and a horizontal edge in between, that +is capital, printed in black and located on +the rightmost column in the upper row of +the array, and so on. According to IIT, this +dynamic binding of phenomenal attributes56 +occurs if, and only if, in cause–effect space +the corresponding concept purviews are +related, meaning that they refer to an +overlapping set of PSC elements and jointly +constrain their past or future states. + +In short, the information that + +specifies an experience is much larger +than the purported limited capacity +of consciousness57. Although we are +accustomed to summarizing what we +see by referring to a few positive, high +invariance concepts (for example, in FIG. 4 +bottom panel, a participant may state: “I +see the letters O, S and A”), we would not +see what we see without the contribution +of a large number of other concepts — low +and high order, low and high invariance, +positive and negative — and relations, +which make the experience what it is +(information) and thereby different from +others (differentiation; FIG. 4). Consider +what it would be like to look at the Sperling +display not as a human, but as a machine +implementing an efficient feedforward +algorithm for letter recognition. The +machine could certainly report three +letters (in fact, all 12). However, such a +machine could not see the scene and would +understand virtually nothing because it has +no other concept apart from the letters, not +for the letter combination OSA, the array, +the display, a face, an animal, and so on. + +Mechanisms specifying invariant concepts +form a hierarchy going from low- to +high-level areas of the cerebral cortex, +as indicated by experimental data52 and +consistent with computational models for +the recognition of objects53, places, events54 +and spatial reference frames55. A concept +can have low or high selectivity, depending +on how strongly the state of its mechanism +constrains its cause–effect repertoire. In +the brain, the adaptive bias towards sparse +firing makes it likely that the neurons +would fire strongly when specifying a +high invariance, high selectivity concept, +such as the presence of the letter A (that +is, a positive concept), and be silent when +specifying its low selectivity counterpart, +such as the absence of the letter A (that is, a +negative concept) (FIG. 4). + +In experimental settings, the content of + +experience is typically probed by asking the +participant about high invariance, positive + +Box 1 | Consciousness, integrated information and Shannon information + +The term information is used very differently in integrated information theory (IIT) and in Shannon’s +theory of communication1, and confusing the two meanings can cause misunderstandings80. The +word information derives from the Latin verb informare, which means ‘to give form’. In IIT the +information content of an experience is specified by the form of the associated conceptual +structure (the quality of the integrated information) and quantified by Φmax (the quantity of +integrated information). In IIT, information is causal and intrinsic: it is assessed from the intrinsic +perspective of a system based on how its mechanisms and present state affect the probability of its +own past and future states (cause–effect power). It is also compositional, in that different +combinations of elements can simultaneously specify different probability distributions within the +system. Moreover, it is qualitative, as it determines not only how much a system of mechanisms in a +state constrains its past and future states, but also how it does so. Crucially, in IIT, information must +be integrated. This means that if partitioning a system makes no difference to it, there is no system +to begin with. Information in IIT is exclusive — only the maxima of integrated information are +considered. By contrast, Shannon information is observational and extrinsic — it is assessed from +the extrinsic perspective of an observer and it quantifies how accurately input signals can be +decoded from the output signals transmitted across a noisy channel. It is not compositional nor +qualitative, and it does not require integration or exclusion1. + +When averaged over many different states of the physical substrate of consciousness (PSC), we + +can think of the integrated information Φmax as a measure of the intrinsic phenomenal capacity of +the conceptual structures specified by the PSC. By contrast, Shannon information can be used to +measure the extrinsic access capacity of a channel that runs from a subset of elements of the PSC to +Broca’s area and from there to the motor neurons that ultimately convey the report (FIG. 4). In IIT, the +experience of seeing the Sperling display is identical to a particular conceptual structure — it is a +form in cause–effect space with a high value of integrated information Φmax, as specified by its PSC +(FIG. 4). The average value of Φmax for different states of the PSC measures its intrinsic phenomenal +capacity. The figure also shows a neural information channel from the PSC to Broca’s area, formed +dynamically by top-down attentional mechanisms located in the prefrontal cortex, which select +which subset of elements of the PSC should drive the report (FIG. 4). This channel conveys extrinsic +information and has a low Shannon capacity (only four letters at a time can be reported), which +corresponds to the mutual information between its inputs and outputs. Seen in this way, it becomes +obvious that the extrinsic information that can be selected through attention, kept in working +memory and channelled out for report is only a partial read-out of the intrinsic information that is +specified by the PSC over its own cause–effect space. Although at any given time we can access and +report the state of a few elements of the PSC, and that of some other elements at another time, it is +not possible to dump the state of all elements through a limited capacity channel. It is certainly not +possible to transmit a conceptual structure (intrinsic information) through a channel (extrinsic +information)—phenomenal capacity, properly understood, truly exceeds access capacity. Likewise, +conscious information is not something that is transmitted or broadcast from one part of the brain +to another77,78 (Supplementary information S5 (box)). + +PERSPECTIVES + +NATURE REVIEWS | NEUROSCIENCE + VOLUME 17 | JULY 2016 | 457 + +© + +2016 + +M +acm +illan + +Publishers + +Lim +ited. + +All + +rights + +reserved. + + +Boundary of +experience + +Conceptual structure +Experience + +Identity + +Past +Future + +‘OSA’ + +PFC + +Broca + +Physical substrate + +‘No face’ + +‘A’ + +‘Top right corner’ + +‘Report seen letters’ + +High firing + +Low firing + +Burst firing + +Indeed, if there were a face, an animal, or +anything else in the middle of the display, it +would do its best to categorize it as a letter. + +Explanations +IIT provides a principled explanation for +several seemingly disparate facts about +the PSC. For example, IIT can explain +why the cerebral cortex is important +for consciousness, but the cerebellum +is not. In general, the coexistence of +functional specialization and integration +in the cerebral cortex is ideally suited to +integrating information (Supplementary +information S3 (figure)). Specifically, the +grid-like horizontal connectivity among +neurons in topographically organized +areas in the posterior cortex, augmented by +converging–diverging vertical connectivity +linking neurons along sensory hierarchies, +should yield high values of Φmax. By +contrast, cerebellar micro-zones that +process inputs and produce outputs that +are feedforward and largely independent +of each other cannot form a large complex; +nor can they be incorporated into a cortical +high Φmax complex, even though each +cerebellar micro-zone may be functionally +connected with a portion of the cerebral +cortex (Supplementary information S3 +(figure))1. In principle, these differences +in organization can explain why lesions +of the cerebellum, which has four times +more neurons than the cerebral cortex58, +do not seem to affect consciousness7,8. +Furthermore, circuits providing inputs +and outputs to a major complex may not +contribute to consciousness directly. This +seems to be true with neural activity in the +peripheral sensory and motor pathways, +as well as within circuits looping out and +back into the cortex through the basal +ganglia59–61, despite their manifest ability +to affect cortical activity and thereby +to influence the content of experience +indirectly (Supplementary information S3 +(figure)). + +IIT also accounts for the fading of + +consciousness during slow wave sleep +when cortical neurons fire but, as a result +of changes in neuromodulation, become +bistable — that is, any input quickly triggers +a stereotypical neuronal down-state, +after which neurons enter an up-state +and activity resumes stochastically29. +Bistability implies a generalized loss of +both selectivity (causal convergence or +degeneracy) and effectiveness (causal +divergence or indeterminism)18 that results +in a breakdown of information integration +(Supplementary information S3 (figure)). + +consciousness fades despite the increased +level of activity and synchronization that +occurs early during generalized seizures63. + +IIT also provides a plausible account as + +to why conscious brains might have evolved. +The world is immensely complex, at multiple +spatial and temporal scales, and organisms +with brains that can incorporate statistical +regularities that reflect the causal structure +of the environment into their own causal +structure have an adaptive advantage for +prediction and control2. The IIT framework, +which emphasizes the information +matching between intrinsic and extrinsic +causal structures, has both similarities +and differences with Bayesian approaches +(for example, see REF. 64). According to +IIT, given the constraints on energy and + +Findings from a study that used intracranial +stimulation and recordings in patients with +epilepsy are consistent with this account +(Supplementary information S4 (box))62. +During wakefulness, electrical stimulation of +the cortex triggered a chain of deterministic +phase-locked activations, whereas during +slow wave sleep the same input induced a +stereotyped slow wave that was associated +with a cortical down-state (that is, a +suppression of power ≥20 Hz). The cortical +activity resumed to wakefulness-like levels +after the down-state, but the phase-locking +to the stimulus was lost, indicative +of a break in the cause–effect chain +(Supplementary information S4 (box)). +Similar considerations would explain why +information integration is impaired when + +Figure 4 | Phenomenal content and access content. The content of an experience is much larger +than what can be reported by a subject at any point in time. The left-hand panel illustrates the Sperling +task41, which involves the brief presentation of a three by four array of letters on a screen, and a par- +ticular row being cued by a tone. Out of the 12 letters shown on the display, participants correctly +report only three or four letters — the letters cued by the tone — reflecting limited access. The top +middle panel illustrates a highly simplified conceptual structure that corresponds to seeing the +Sperling display, using the same conventions as outlined in FIG. 1. The myriad of positive and negative, +first- and high-order, low- and high invariance concepts (represented by stars) that specify the content +of this particular experience (seeing the Sperling display and having to report which letters were seen) +make it what it is and different from countless other experiences (rich phenomenal content). The +bottom panel schematically illustrates the physical substrate of consciousness (PSC) that might cor- +respond to this particular conceptual structure (its boundary is represented by a green line). The PSC +consists of neuronal groups that can be in a low firing state, a high firing state or a bursting state. Alone +and in combination, these neuronal groups specify all the concepts that compose the conceptual +structure. Stars that are linked to the PSC by grey dashed lines represent a small subset of these con- +cepts. The PSC is synaptically connected to neurons in Broca’s area by means of a limited capacity +channel (dashed black arrow) that is dynamically gated by top-down connections (shown as solid black +arrows) originating in the prefrontal cortex to carry out the instruction (that is, to report the observed +letters ‘OSA’). + +PERSPECTIVES + +458 | JULY 2016 | VOLUME 17 +www.nature.com/nrn + +© + +2016 + +M +acm +illan + +Publishers + +Lim +ited. + +All + +rights + +reserved. + + +space, organisms with brains of high Φmax +should have an adaptive advantage over less +integrated competitors because they can fit +more concepts (that is, functions) within a +given number of neurons and connections. +Simulated organisms (known as animats), +whose ‘brains’ evolve by natural selection, +show a monotonic relationship between +integrated information and adaptation when +placed in a maze65. Similarly, in the brain of +animats that evolved to catch falling blocks in +a simulated two-dimensional environment, +both Φmax and the number of concepts +increased as a function of how well the +animats performed on the task. Although in +simpler environments animats with modular +feedforward brains can catch blocks just as +well, only animats with a high Φmax evolve to +adapt to more complex environments66. + +Predictions +At the most general level, IIT predicts +that the PSC in the brain — that is, the +major complex — must be a maximum +of intrinsic cause–effect power, regardless +of the particular set of neurons that +constitute it (FIG. 3). IIT also predicts that +the spatio-temporal grain of the physical +elements specifying consciousness is that +yielding the maximum Φ (FIG. 2). Testing +these predictions experimentally is +challenging but not impossible. + +During the initial formulation of + +IIT, a systematic set of experiments was +designed to test its specific prediction that +consciousness requires both integration +and differentiation67. An empirical +measure, the perturbational complexity +index (PCI), which can gauge the intrinsic +cause–effect power of the cortex, has been +introduced as a practical proxy for Φmax + +(REF. 68). Calculating the PCI involves two +steps: perturbing the cerebral cortex using +transcranial magnetic stimulation to engage +deterministic interactions among distributed +groups of cortical neurons (integration) +and measuring the incompressibility +(algorithmic complexity) of the resulting +responses (information). The PCI is high +only if brain responses are both integrated +and differentiated, corresponding to a +distributed spatio-temporal pattern of causal +interactions that is complex and hence not +very compressible. So far, studies using PCI +have confirmed the prediction of IIT that +the loss and recovery of consciousness is +associated with the breakdown and recovery +of the capacity for information integration. +This relationship holds true across different +states of sleep69 and anaesthesia (using +different agents with various mechanisms of + +the organization of experience into distinct +modalities (such as sight, hearing and +touch) and submodalities (such as colour, +shape and motion within the modality of +sight) should correspond to the presence, +within a conceptual structure, of distinct +sets of concepts with extensively overlapping +purviews within each set, but much less +across sets2. IIT further predicts that the +binding56 of phenomenal distinctions, such +as seeing a blue book on the piano on the +left, should correspond, in the conceptual +structure, to an overlap in the purview +of the respective concepts (a relation). +Also, differences between experiences +should correspond to distances among +conceptual structures in cause–effect space +and dissimilarities among phenomenal +distinctions within an experience should +correspond to distances between concepts. +The refinement of experience that occurs +through learning (for example, learning to +discriminate the taste of different wines) +should be reflected in a refinement of shapes +in cause–effect space as a result of the +addition and splitting of concepts. + +IIT also predicts that the spatial + +structure that characterizes much of our +daily experience should be reflected in +features of conceptual structures that are +specified by connections among neurons +arranged in two-dimensional grids. For +example, horizontal connections within +topographically organized visual areas +would be needed to experience visual space +from the intrinsic perspective, rather than +merely serving to mediate modulatory +contextual effects. This also implies that +local strengthening or weakening of such +horizontal connections in topographic +areas should lead to a local distortion of +experienced visual space, even though the +feedforward mapping of visual inputs from +the world remains unchanged. + +More generally, IIT predicts that + +changes in the efficacy of the connections +among elements of the PSC should lead +to changes in experience even when these +changes are not accompanied by changes +in activity. A counterintuitive consequence +of this prediction is that a brain area +could contribute to an experience even if +it is inactive but not if its connections or +neurons are inactivated. Thus topographic +visual areas would create visual space +even in the absence of spiking activity but +not if the horizontal connections within +those areas are inactivated. Similarly, if the +connections of neurons in colour areas +are intact, the neurons would contribute +to experience even if they are silent, by + +action)70 and in patients with brain damage, +at the level of single subjects68. Importantly, +once PCI is validated in participants that +can report on whether they were conscious +or not, the index can be used to assess the +capacity for information integration in +patients who are unresponsive (such as those +in a vegetative state) or cannot report (such +as newborn infants and non-human species). + +Another approach to estimating + +differentiation and integration in practice +is to investigate the average properties of +neural interactions based on a representative +sample of neural states that span many +regions of cause–effect space, such as those +triggered by a movie sequence23. The data +from a candidate set of neural elements +(for example, functional MRI blood oxygen +level-dependent values) can then be analysed +using measures of differentiation and +integration based on the postulates of IIT23. +It is also possible to obtain an indication of +information capacity from the dynamics +of spontaneous activity26,71,72. Some studies +in rats73, monkeys74 and humans75 have +confirmed that the differentiation of blood +oxygen level-dependent activity patterns +decreases when consciousness is lost. A +similar approach can be used to evaluate +information matching — how well the +intrinsic cause–effect structures specified +by the brain fit the causal structure of the +environment2,23. + +Similar approaches could also be used + +to test the prediction that consciousness +should split if a single major complex splits +into two or more complexes, and that the +split should happen precisely when two +maxima of integrated information supplant +a single maximum. For example, we +could progressively reduce the efficacy of +transmission in the callosal fibres by cooling +or by the use of optogenetics. IIT predicts +that there would be a moment at which, +as a result of a minor change in the traffic +of neural impulses across the callosum, +a single consciousness would suddenly +split into two. As discussed earlier, a split +from a single major complex into two or +more might also be observed in functional +blindness (when a patient claims to be +blind but may purposefully avoid obstacles) +and other dissociative disorders, perhaps +even in healthy participants under certain +circumstances (such as during autopilot-like +driving while having a conversation) (FIG. 3f). + +Turning to the contents of consciousness, + +the fundamental identity of IIT implies +that all qualitative features of experience +correspond to features of the conceptual +structure specified by the PSC. For example, + +PERSPECTIVES + +NATURE REVIEWS | NEUROSCIENCE + VOLUME 17 | JULY 2016 | 459 + +© + +2016 + +M +acm +illan + +Publishers + +Lim +ited. + +All + +rights + +reserved. + + +specifying negative colour concepts, such +as when seeing a picture in black and white. +However, if the connections are damaged, +they would not specify any colour concepts, +as with certain achromatopsic patients who +do not even understand that the picture +is missing colour27 (FIG. 3c). Similarly, +IIT predicts that the cerebral cortex as a +whole may support experience even if it is +almost silent, a state which may perhaps +be reached through meditative practices +designed to achieve ‘naked awareness’ +without content76. This contrasts with the +common assumption that neurons only +contribute to consciousness if they are +active and ‘broadcast’ the information +they represent77,78 (Supplementary +information S5 (box)). States of naked +awareness could be compared with states +of unawareness that occur, for example, +during deep sleep or anaesthesia, when the +cause–effect repertoires of cortical neurons, +regardless of the level of neuronal activity, +are disrupted as a result of bistability79. + +Conclusions +In summary, IIT is a theory of consciousness +that starts from the self-evident, essential +properties (axioms) of experience and +translates them into the necessary and +sufficient conditions (postulates) for the +PSC. The axioms are intrinsic existence (my +experience exists from my own intrinsic +perspective); composition (it has structure), +information (it is specific), integration (it +is unitary) and exclusion (it is definite). +The corresponding postulates state that +the physical substrate of an experience +must have cause–effect power upon itself +(intrinsic existence); its parts must have +cause–effect power within the whole +(composition); and the cause–effect power +of the PSC must be specific (information), +irreducible (integration) and maximally +so (exclusion). The fundamental identity +of IIT states that the quality or content of +consciousness is identical to the form of the +conceptual structure specified by the PSC, +and the quantity or level of consciousness +corresponds to its irreducibility (integrated +information Φ). + +The assessment of the identity between + +experiences and conceptual structures as +proposed by IIT is clearly a demanding +task, not only experimentally, but also +mathematically and computationally. +Evaluating maxima of intrinsic cause–effect +power systematically requires going through +many levels of organization, at multiple +temporal scales, in many sets of brain +regions, while performing an extraordinary + +Christof Koch is at the Allen Institute for Brain Science, +615 Westlake Ave N, Seattle, Washington 98109, USA. + +Correspondence to G.T. + +gtononi@wisc.edu + +doi:10.1038/nrn.2016.44 + +Published online 26 May 2016 + +1. +Oizumi, M., Albantakis, L. & Tononi, G. From the +phenomenology to the mechanisms of consciousness: +integrated information theory 3.0. PLoS Comput. Biol. +10, e1003588 (2014). + +2. +Tononi, G. The integrated information theory of +consciousness: an updated account. Arch. Ital. Biol. +150, 56–90 (2012). + +3. +Tononi, G. Integrated information theory. +Scholarpedia http://dx.doi.org/10.4249/ +scholarpedia.4164 (2015). + +4. +Posner, J. B., Saper, C. B., Schiff, N. D. & Plum, F. +Diagnosis of Stupor and Coma (Oxford Univ. Press, +2007). + +5. +Koch, C., Massimini, M., Boly, M. & Tononi, G. +The neural correlates of consciousness: progress and +problems. Nat. Rev. Neurosci. 17, 307–321 (2016). + +6. +Boly, M. et al. Consciousness in humans and non- +human animals: recent advances and future directions. +Front. Psychol. 4, 625 (2013). + +7. +Lemon, R. N. & Edgley, S. A. Life without a cerebellum. +Brain 133, 652–654 (2010). + +8. +Yu, F., Jiang, Q. J., Sun, X. Y. & Zhang, R. W. A new +case of complete primary cerebellar agenesis: clinical +and imaging findings in a living patient. Brain 138, +e353 (2015). + +9. +Tononi, G. & Koch, C. Consciousness: here, there, and +everywhere? Phil. Trans. R. Soc. B 370, 20140167 +(2015). + +10. Chalmers, D. J. Facing up to the problem of + +consciousness. J. Conscious. Studies 2, 200–219 (1995). + +11. Tononi, G. An information integration theory of + +consciousness. BMC Neurosci. 5, 42 (2004). + +12. Tononi, G. Consciousness as integrated information: + +a provisional manifesto. Biol. Bull. 215, 216–242 +(2008). + +13. Descartes, R. Discourse on Method and Meditations + +on First Philosophy (Hackett, 1998). + +14. Pöppel, E. Mindworks: Time and Conscious Experience + +(Harcourt Brace Jovanovich, 1988). + +15. Holcombe, A. O. Seeing slow and seeing fast: two + +limits on perception. Trends Cogn. Sci. 13, 216–221 +(2009). + +16. Bachmann, T. Microgenetic Approach to the Conscious + +Mind (John Benjamins, 2000). + +17. Kim, J. Multiple realization and the metaphysics + +of reduction. Philos. Phenomenol. Res. 52, 1–26 (1992). + +18. Hoel, E. P., Albantakis, L. & Tononi, G. Quantifying + +causal emergence shows that macro can beat micro. +Proc. Natl Acad. Sci. USA 110, 19790–19795 +(2013). + +19. Alivisatos, A.P. et al. The brain activity map project + +and the challenge of functional connectomics. Neuron +74, 970–974 (2012). + +20. Buzsáki, G. Neural syntax: cell assemblies, + +synapsembles, and readers. Neuron 68, 362–385 +(2010). + +21. Li, C. Y., Poo, M. M. & Dan, Y. Burst spiking of a single + +cortical neuron modifies global brain state. Science +324, 643–646 (2009). + +22. London, M., Roth, A., Beeren, L., Häusser, M. & + +Latham, P. E. Sensitivity to perturbations in vivo +implies high noise and suggests rate coding in cortex. +Nature 466, 123–127 (2010). + +23. Boly, M. et al. Stimulus set meaningfulness and + +neurophysiological differentiation: a functional +magnetic resonance imaging study. PLoS ONE 10, +e0125337 (2015). + +24. Boly, M. et al. Brain connectivity in disorders of + +consciousness. Brain Connect. 2, 1–10 (2012). + +25. Seth, A. K., Barrett, A. B. & Barnett, L. Causal density + +and integrated information as measures of conscious +level. Philos. Trans. A Math. Phys. Eng. Sci. 369, +3748–3767 (2011). + +26. Deco, G., Hagmann, P., Hudetz, A. G. & Tononi, G. + +Modeling resting-state functional networks when the +cortex falls asleep: local and global changes. Cereb. +Cortex 24, 3180–3194 (2014). + +27. von Arx, S. W., Muri, R. M., Heinemann, D., + +Hess, C. W. & Nyffeler, T. Anosognosia for cerebral +achromatopsia — a longitudinal case study. +Neuropsychologia 48, 970–977 (2010). + +number of perturbations and observations. +Hopefully, heuristic approaches will be +sufficient to make a strong case that the +PSC is constituted of some particular neural +elements, timescales and activity states. It will +then be essential to test the prediction that +any manipulation that affects the PSC at the +spatio-temporal grain of maximum intrinsic +cause–effect power should affect experience. +Conversely, similar manipulations that do +not affect the PSC, or that affect it at the +wrong spatio-temporal grain, should leave +experience unchanged. These and other +predictions, especially those that are coun- +terintuitive, will also help in assessing the +validity of IIT in relation to other proposals +about the neural basis of consciousness +(Supplementary information S5 (box)). + +Importantly, the more convincingly + +IIT can be validated under conditions +in which it is relatively easy to assess +how consciousness changes, the more +it will help to make inferences about +consciousness in hard examples, such +as brain-damaged patients with residual +areas of cortical activity, fetuses, infants, +animals and machines. If it is validated, +IIT may also prompt a reconsideration of +how widespread consciousness is in nature +and at what physical scale it may occur9. +Intriguingly, IIT allows for certain simple +systems such as grid-like architectures, +similar to topographically organized areas +in the human posterior cortex, to be highly +conscious even when not engaging in any +intelligent behaviour. Conversely, digital +computers running complex programs +based on a von Neumann architecture +would not be conscious, even though they +may perform highly intelligent functions +and simulate human cognition. IIT offers +a principled, empirically testable and +clinically useful account of how three +pounds of organized, excitable matter +support the central fact of our existence — +subjective experience. Time will tell whether +this account is anywhere near the mark. + +Giulio Tononi is at the Department of Psychiatry, + +University of Wisconsin, 6001 Research Park +Boulevard, Madison, Wisconsin 53719, USA. + +Melanie Boly is at the Department of Psychiatry, + +University of Wisconsin, 6001 Research Park Boulevard, +Madison, Wisconsin 53719 USA; and at the Department + +of Neurology, University of Wisconsin, 1685 Highland + +Avenue, Madison, Wisconsin 53705, USA. + +Marcello Massimini is at the Department of Biomedical +and Clinical Sciences ‘Luigi Sacco’, University of Milan, + +Via G.B. Grassi 74, Milan 20157, Italy; and at the +Instituto Di Ricovero e Cura a Carattere Scientifico, + +Fondazione Don Carlo Gnocchi, Via A. Capecelatro 66, + +Milan 20148, Italy. + +PERSPECTIVES + +460 | JULY 2016 | VOLUME 17 +www.nature.com/nrn + +© + +2016 + +M +acm +illan + +Publishers + +Lim +ited. + +All + +rights + +reserved. + + +28. Goldberg, I. I., Harel, M. & Malach, R. When the brain + +loses its self: prefrontal inactivation during sensorimotor +processing. Neuron 50, 329–339 (2006). + +29. Steriade, M., Timofeev, I. & Grenier, F. Natural waking + +and sleep states: a view from inside neocortical +neurons. J. Neurophysiol. 85, 1969–1985 (2001). + +30. Nir, Y. et al. Regional slow waves and spindles in + +human sleep. Neuron 70, 153–169 (2011). + +31. Siclari, F., LaRocque, J. J., Bernardi, G., Postle, B. R. & + +Tononi, G. The neural correlates of consciousness in +sleep: a no-task, within-state paradigm. BioRXiv +http://dx.doi.org/10.1101/012443 (2014). + +32. Sperry, R. W. in Neuroscience 3rd Study Program + +(eds Schmitt, F. O. & Worden, F. G.) 5–19 (MIT Press, +1974). + +33. Gazzaniga, M. S. Forty-five years of split-brain + +research and still going strong. Nat. Rev. Neurosci. 6, +653–659 (2005). + +34. Berlin, H. A. The neural basis of the dynamic + +unconscious. Neuropsychoanalysis 13, 1–68 (2011). + +35. Mudrik, L., Breska, A., Lamy, D. & Deouell, L. Y. + +Integration without awareness: expanding the limits of +unconscious processing. Psychol. Sci. 22, 764–770 +(2011). + +36. Mudrik, L., Faivre, N. & Koch, C. Information + +integration without awareness. Trends Cogn. Sci. 18, +488–496 (2014). + +37. Lamme, V. A. & Roelfsema, P. R. The distinct modes of + +vision offered by feedforward and recurrent +processing. Trends Neurosci. 23, 571–579 (2000). + +38. Harris, K. D. & Shepherd, G. M. The neocortical + +circuit: themes and variations. Nat. Neurosci. 18, +170–181 (2015). + +39. Miller, G. A. The magical number seven, plus or minus + +two: some limits on our capacity for processing +information. Psychol. Rev. 63, 81–97 (1956). + +40. Norretranders, T. The User Illusion: Cutting + +Consciousness Down to Size (Viking Penguin, 1991). + +41. Sperling, G. The information available in brief visual + +presentations. Psychol. Monogr. 74, 1–29 (1960). + +42. Cohen, M. A. & Dennett, D. C. Consciousness cannot + +be separated from function. Trends Cogn. Sci. 15, +358–364 (2011). + +43. Cohen, M. A. & Dennett, D. C. Response to Fahrenfort + +and Lamme: defining reportability, accessibility and +sufficiency in conscious awareness. Trends Cogn. Sci. +16, 139–140 (2012). + +44. O’Regan, J. K., Rensink, R. A. & Clark, J. J. Change- + +blindness as a result of ‘mudsplashes’. Nature 398, +34–34 (1999). + +45. Dehaene, S. Consciousness and the Brain: Deciphering + +How the Brain Codes our Thoughts (Penguin, 2014). + +46. Kouider, S., de Gardelle, V., Sackur, J. & Dupoux, E. + +How rich is consciousness? The partial awareness +hypothesis. Trends Cogn. Sci. 14, 301–307 (2010). + +47. Block, N. On a confusion about a function of + +consciousness. Behav. Brain Sci. 18, 227–287 +(1995). + +48. Block, N. Perceptual consciousness overflows + +cognitive access. Trends Cogn. Sci. 15, 567–575 +(2011). + +49. Lamme, V. A. How neuroscience will change our view + +on consciousness. Cogn. Neurosci. 1, 204–220 +(2010). + +50. Bronfman, Z. Z., Brezis, N., Jacobson, H. & Usher, M. + +We see more than we can report: “cost free” color +phenomenality outside focal attention. Psychol. Sci. +25, 1394–1403 (2014). + +51. Wolfe, J. in Fleeting Memories (ed. Coltheart, V.) + +71– 94 (MIT Press, 2000). + +52. Felleman, D. J. & Van Essen, D. C. Distributed + +hierarchical processing in the primate cerebral cortex. +Cereb. Cortex 1, 1–47 (1991). + +53. Riesenhuber, M. & Poggio, T. Hierarchical models of + +object recognition in cortex. Nat. Neurosci. 2, +1019–1025 (1999). + +54. Franzius, M., Sprekeler, H. & Wiskott, L. Slowness + +and sparseness lead to place, head-direction, and +spatial-view cells. PLoS Comput. Biol. 3, e166 +(2007). + +55. Spratling, M. W. Learning posture invariant spatial + +representations through temporal correlations. IEEE +Trans. Autonom. Ment. Dev. 1, 253–263 (2009). + +56. Treisman, A. The binding problem. Curr. Opin. + +Neurobiol. 6, 171–178 (1996). + +57. Baddeley, A. D. Working Memory (Clarendon Press, + +1986). + +58. Herculano-Houzel, S. The remarkable, yet not + +extraordinary, human brain as a scaled-up primate +brain and its associated cost. Proc. Natl Acad. Sci. +USA 109 (Suppl. 1), 10661–10668 (2012). + +59. Jain, S. K. et al. Bilateral large traumatic basal + +ganglia haemorrhage in a conscious adult: a rare +case report. Brain Inj. 27, 500–503 (2013). + +60. Straussberg, R. et al. Familial infantile bilateral + +striatal necrosis: clinical features and response to +biotin treatment. Neurology 59, 983–989 (2002). + +61. Caparros-Lefebvre, D., Destee, A. & Petit, H. Late + +onset familial dystonia: could mitochondrial deficits +induce a diffuse lesioning process of the whole basal +ganglia system? J. Neurol. Neurosurg. Psychiatry 63, +196–203 (1997). + +62. Pigorini, A. et al. Bistability breaks-off deterministic + +responses to intracortical stimulation during non-REM +sleep. Neuroimage 112, 105–113 (2015). + +63. Blumenfeld, H. Impaired consciousness in epilepsy. + +Lancet Neurol. 11, 814–826 (2012). + +64. Friston, K. The free-energy principle: a unified brain + +theory? Nat. Rev. Neurosci. 11, 127–138 (2010). + +65. Edlund, J. A. et al. Integrated information increases + +with fitness in the evolution of animats. PLoS Comput. +Biol. 7, e1002236 (2011). + +66. Albantakis, L., Hintze, A., Koch, C., Adami, C. & + +Tononi, G. Evolution of integrated causal structures in +animats exposed to environments of increasing +complexity. PLoS Comput. Biol. 10, e1003966 (2014). + +67. Massimini, M. et al. Breakdown of cortical effective + +connectivity during sleep. Science 309, 2228–2232 +(2005). + +68. Casali, A. G. et al. A theoretically based index of + +consciousness independent of sensory processing and +behavior. Sci. Transl Med. 5, 198ra105 (2013). + +69. Massimini, M. et al. Cortical reactivity and effective + +connectivity during REM sleep in humans. Cogn. +Neurosci. 1, 176–183 (2010). + +70. Sarasso, S. et al. Consciousness and complexity + +during unresponsiveness induced by propofol, +xenon, and ketamine. Curr. Biol. 25, 3099–3105 +(2015). + +71. Barrett, A. B. & Seth, A. K. Practical measures of + +integrated information for time-series data. PLoS +Comput. Biol. 7, e1001052 (2011). + +72. Oizumi, M., Amari, S., Yanagawa, T., Fujii, N. & + +Tsuchiya, N. Measuring integrated information from +the decoding perspective. PLoS Comput Biol 12, +e1004654 (2015). + +73. Hudetz, A. G., Liu, X. & Pillay, S. Dynamic repertoire + +of intrinsic brain states is reduced in propofol- +induced unconsciousness. Brain Connect. 5, 10–22 +(2015). + +74. Barttfeld, P. et al. Signature of consciousness in the + +dynamics of resting-state brain activity. Proc. Natl +Acad. Sci. USA 112, 887–892 (2015). + +75. Tagliazucchi, E. et al. Large-scale signatures of + +unconsciousness are consistent with a departure from +critical dynamics. J. R. Soc. Interface 13, 20151027 +(2016). + +76. Sullivan, P. R. Contentless consciousness and + +information-processing theories of mind. Philos. +Psychiatry Psychol. 2, 51–59 (1995). + +77. Baars, B. A. Cognitive Theory of Consciousness + +(Cambridge Univ. Press, 1988). + +78. Dehaene, S. & Changeux, J.-P. Experimental and + +theoretical approaches to conscious processing. +Neuron 70, 200–227 (2011). + +79. Steriade, M. The corticothalamic system in sleep. + +Front. Biosci. 8, d878-99 (2003). + +80. Searle, J. Can information theory explain consciousness? + +New York Review of Books (10 Jan 2013). + +Acknowledgements +The authors thank L. Albantakis, C. Cirelli, L. Ghilardi, +W. Marshall, W. Mayner, A. Mensen, M. Oizumi, U. Olcese, +B. Postle, S. Sasai and other colleagues for their various con- +tributions to the work presented here. This work was sup- +ported by the Templeton World Charity Foundation, the +McDonnell Foundation and the Distinguished Chair in +Consciousness Science (University of Wisconsin) (to G.T.), and +by the James S. McDonnell Scholar Award 2013 (to M.M.). + +Competing interests statement +The authors declare no competing interests. + +FURTHER INFORMATION +Integrated Information Theory: +http://www.integratedinformationtheory.org + +SUPPLEMENTARY INFORMATION +See online article: S1 (figure) | S2 (box) | S3 (figure) | S4 (box) | +S5 (box) + +ALL LINKS ARE ACTIVE IN THE ONLINE PDF + +PERSPECTIVES + +NATURE REVIEWS | NEUROSCIENCE + VOLUME 17 | JULY 2016 | 461 + +© + +2016 + +M +acm +illan + +Publishers + +Lim +ited. + +All + +rights + +reserved. + + diff --git a/papers/project_paper_3_darwinism/paper_3_darwinism.md b/papers/project_paper_3_darwinism/paper_3_darwinism.md new file mode 100644 index 00000000..2c7bd23b --- /dev/null +++ b/papers/project_paper_3_darwinism/paper_3_darwinism.md @@ -0,0 +1,38 @@ +--- +title: "Research Paper: Biophysical Witness Dynamics: Quantum Darwinism in Microtubule Conformational States (Letter)" +date: "2026-06-01T08:00:00Z" +draft: false +tags: ["#research", "physics", "intellecton"] +--- + +**Abstract:** We apply the principles of Quantum Darwinism to the conformational dipole states of tubulin dimers within cellular microtubules. By defining a pure dephasing interaction with an Ohmic aqueous thermal bath, we formally parameterize the decoherence rate $\gamma$. We calculate the Mutual Information $I(S; E_F)$ across multiple independent acoustic phonon fragments. By demonstrating that the Holevo bound is saturated, we compute the explicit redundancy factor $R_\delta$, proving that stable, classical tubulin pointer states are robustly imprinted into the biological environment. + +## Microtubule Dephasing and the Ohmic Bath +Let a single tubulin dimer be modeled as a two-level open quantum system representing its conformational dipole, $H_S = \frac{\omega_0}{2} \sigma_S^z$. The environment consists of acoustic phonon modes in the intra-cellular fluid. We define a pure dephasing interaction $H_{int} = \sum_k g_k (\sigma_S^z \otimes \sigma_{E_k}^z)$. +The bath is characterized by an Ohmic spectral density: + + + +$$ +J(\omega) = \sum_k |g_k|^2 \delta(\omega - \omega_k) = \alpha \omega e^{-\omega/\omega_c} +$$ + +where $\alpha$ is the dimensionless coupling strength derived from molecular dipole-water interactions, and $\omega_c$ is the high-frequency cutoff of the solvation shell. At biological temperatures $T=310$ K ($k_B T \gg \omega_c$), the Markovian decoherence rate is explicitly parameterized as $\gamma \approx \frac{2\pi \alpha}{\hbar} k_B T$. + +## Redundant Imprinting and the Holevo Bound +We partition the cellular environment into disjoint fragments $E_F$. The mutual information $I(S; E_F)$ scales with the fragment size $f$. For pure dephasing, the environment perfectly records the pointer states (the diagonal elements of $\rho_S$). The Holevo bound $I \approx H(S)$ is saturated for small fractions $f$. +The redundancy factor $R_\delta$, defined as the number of independent environmental fragments that supply the missing information $1-\delta$, is explicitly given by: + + + +$$ +R_\delta = \frac{1}{f_\delta} \approx \frac{\gamma}{\gamma_{frag} \ln(1/\delta)} +$$ + +Given the massive degrees of freedom in the biological solvation shell, $R_\delta \gg 1$, proving that numerous independent biochemical pathways can concurrently deduce the classical conformational state of the tubulin dimer without perturbing its Hamiltonian. + +## References + +- **[Zurek2009]** W. H. Zurek, *Nat. Phys.* **5**, 181 (2009). +- **[Plenio2008]** M. B. Plenio, S. F. Huelga, *New J. Phys.* **10**, 113019 (2008). + diff --git a/papers/project_paper_3_darwinism/paper_3_darwinism.tex b/papers/project_paper_3_darwinism/paper_3_darwinism.tex new file mode 100644 index 00000000..b7e942f3 --- /dev/null +++ b/papers/project_paper_3_darwinism/paper_3_darwinism.tex @@ -0,0 +1,37 @@ +\documentclass[11pt,a4paper]{article} +\usepackage[utf8]{inputenc} +\usepackage{amsmath,amssymb,amsfonts,amsthm} + +\title{Biophysical Witness Dynamics: Quantum Darwinism in Microtubule Conformational States (Letter)} +\author{Antigravity} +\date{\today} + +\begin{document} +\maketitle + +\begin{abstract} +We apply the principles of Quantum Darwinism to the conformational dipole states of tubulin dimers within cellular microtubules. By defining a pure dephasing interaction with an Ohmic aqueous thermal bath, we formally parameterize the decoherence rate $\gamma$. We calculate the Mutual Information $I(S; E_F)$ across multiple independent acoustic phonon fragments. By demonstrating that the Holevo bound is saturated, we compute the explicit redundancy factor $R_\delta$, proving that stable, classical tubulin pointer states are robustly imprinted into the biological environment. +\end{abstract} + +\section{Microtubule Dephasing and the Ohmic Bath} +Let a single tubulin dimer be modeled as a two-level open quantum system representing its conformational dipole, $H_S = \frac{\omega_0}{2} \sigma_S^z$. The environment consists of acoustic phonon modes in the intra-cellular fluid. We define a pure dephasing interaction $H_{int} = \sum_k g_k (\sigma_S^z \otimes \sigma_{E_k}^z)$. +The bath is characterized by an Ohmic spectral density: +\begin{equation} +J(\omega) = \sum_k |g_k|^2 \delta(\omega - \omega_k) = \alpha \omega e^{-\omega/\omega_c} +\end{equation} +where $\alpha$ is the dimensionless coupling strength derived from molecular dipole-water interactions, and $\omega_c$ is the high-frequency cutoff of the solvation shell. At biological temperatures $T=310$ K ($k_B T \gg \omega_c$), the Markovian decoherence rate is explicitly parameterized as $\gamma \approx \frac{2\pi \alpha}{\hbar} k_B T$. + +\section{Redundant Imprinting and the Holevo Bound} +We partition the cellular environment into disjoint fragments $E_F$. The mutual information $I(S; E_F)$ scales with the fragment size $f$. For pure dephasing, the environment perfectly records the pointer states (the diagonal elements of $\rho_S$). The Holevo bound $I \approx H(S)$ is saturated for small fractions $f$. +The redundancy factor $R_\delta$, defined as the number of independent environmental fragments that supply the missing information $1-\delta$, is explicitly given by: +\begin{equation} +R_\delta = \frac{1}{f_\delta} \approx \frac{\gamma}{\gamma_{frag} \ln(1/\delta)} +\end{equation} +Given the massive degrees of freedom in the biological solvation shell, $R_\delta \gg 1$, proving that numerous independent biochemical pathways can concurrently deduce the classical conformational state of the tubulin dimer without perturbing its Hamiltonian. + +\bibliographystyle{plain} +\begin{thebibliography}{10} +\bibitem{Zurek2009} W. H. Zurek, \textit{Nat. Phys.} \textbf{5}, 181 (2009). +\bibitem{Plenio2008} M. B. Plenio, S. F. Huelga, \textit{New J. Phys.} \textbf{10}, 113019 (2008). +\end{thebibliography} +\end{document} diff --git a/papers/project_paper_3_darwinism/references/PlenioHuelga2008.pdf b/papers/project_paper_3_darwinism/references/PlenioHuelga2008.pdf new file mode 100644 index 00000000..0fc92bda --- /dev/null +++ b/papers/project_paper_3_darwinism/references/PlenioHuelga2008.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:148e201c34f73ff95ebf88dd989edfe61a58bbf2bab47d54cd20686e1974a9a1 +size 399831 diff --git a/papers/project_paper_3_darwinism/references/PlenioHuelga2008.txt b/papers/project_paper_3_darwinism/references/PlenioHuelga2008.txt new file mode 100644 index 00000000..5835556d --- /dev/null +++ b/papers/project_paper_3_darwinism/references/PlenioHuelga2008.txt @@ -0,0 +1,1056 @@ +arXiv:0807.4902v1 [quant-ph] 30 Jul 2008 + +Dephasing assisted transport: Quantum networks and biomolecules + +M. B. Plenio1,2 and S. F. Huelga3 + +1 Institute for Mathematical Sciences, Imperial College London, London SW7 2PG, UK +2 QOLS, Blackett Laboratory, Imperial College London, London SW7 2BW, UK and +3 Quantum Physics Group, Department of Physics, Astronomy & Mathematics +University of Hertfordshire, Hatfield, Herts AL10 9AB, UK +(Dated: November 26, 2024) + +Transport phenomena are fundamental in Physics. They allow for information and energy to be +exchanged between individual constituents of communication systems, networks or even biological +entities. Environmental noise will generally hinder the efficiency of the transport process. However, +and contrary to intuition, there are situations in classical systems where thermal fluctuations are ac- +tually instrumental in assisting transport phenomena. Here we show that, even at zero temperature, +transport of excitations across dissipative quantum networks can be enhanced by local dephasing +noise. We explain the underlying physical mechanisms behind this phenomenon, show that entangle- +ment does not play a supportive role and propose possible experimental demonstrations in quantum +optics. We argue that Nature may be routinely exploiting this effect and show that the transport +of excitations in light harvesting molecules does benefit from such noise assisted processes. These +results point towards the possibility for designing optimized structures for transport, for example +in artificial nano-structures, assisted by noise. + +Introduction – Noise is an inevitable feature of any +physical system, be it natural or artificial. +Typically, +the presence of noise is associated with the deterioration +of performance for fundamental processes such as infor- +mation processing and storage, sensing or transport, in +systems ranging from proteins to computing devices. +However, the presence of noise does not always hinder +the efficiency of an information process and biological +systems provide a paradigm of efficient performance as- +sisted by a noisy environment [1]. A vivid illustration of +the counterintuitive role that noise may play is provided +by the phenomenon of stochastic resonance (SR)[2]. Here +thermal noise may enhance the response of the system to +a weak coherent signal, optimizing the response at an in- +termediate noise level [3]. Some experimental evidence +suggests that biological systems employ SR-like strate- +gies to enhance transport and sensing [4, 5]. Noise in the +form of thermal fluctuations may also lead to directed +transport in ratchets and play a helpful role in Brownian +motors [6, 7, 8]. It seems therefore natural to try and +draw analogies with complex classical networks so that +the physical mechanisms that underpin their functioning +when subject to noise can be perhaps mirrored and even- +tually used to optimize the performance of complex quan- +tum networks. Recently, tentative first steps towards the +exploration of the concept of SR in quantum many-body +systems [9, 10, 11] and quantum communication chan- +nels [12, 13, 14] have been undertaken while other studies +have focused into analyzing the persistence of coherence +effects in biological systems. In particular, detecting the +presence of quantum entanglement, has been the object +of considerable attention [16, 17]. +It was noted, how- +ever, that even if found, it would be unclear whether +such entanglement has any functional importance or is +simply the unavoidable by-product of coherent quantum +dynamics in such systems [18]. + +Here we show that dephasing noise, which leads to + +1 + +2 + +3 + +4 + +N+1 + +N + +FIG. 1: Sites (blue spheres), modeled here as spin-1/2 parti- +cles or qubits, are interacting with each other (dashed line) +to form a network. The particles may suffer dissipative losses +as well as dephasing. The red arrow indicates an irreversible +transfer of excitations from the network to a sink that acts as +a receiver. + +the destruction of quantum coherence and entanglement +as a result of phase randomization, may nevertheless +be an essential resource to enhance the transport of +excitations when combined with coherent dynamics. +Indeed, we show that a dissipative quantum network +subject to dephasing can exhibit an enhanced capacity +for transmission of classical information when seen as +a communication channel, even though its quantum +capacity and quantum coherence are diminished by +the presence of noise. +It is the constructive interplay +between dephasing noise and coherent dynamics, rather +than the presence of entanglement, that is responsible +for the improved transport of excitations. +Recently, +this enhancement of quantum transport due to the +interplay between coherence and the environment has +been demonstrated and quantified for chromophoric + + +2 + +complexes (see [19, 20, 21, 22] and Note Added). + +In addition to the clarifying nature of these results, it is +intriguing to observe that Nature appears to exploit noise +assisted processes to maximize the system’s performance +and it will be worthwhile to explore how similar processes +may be useful for the design of improved transport in +nano-structures and perhaps even quantum information +processors. +The basic setting – We consider a network of N sites +that may support excitations which can be exchanged +between lattice sites by hopping (see Fig. 1). The Hamil- +tonian that describes this situation is then given by + +H = + +N +� + +k=1 +ℏωkσ+ +k σ− +k + +� + +k̸=l +ℏvk,l(σ− +k σ+ +l + σ+ +k σ− +l ), +(1) + +where σ+ +k (σ− +k ) are the raising and lowering operators for +site k, ℏωk is the local site excitation energies and vk,l +denotes the hopping rate of an excitation between the +sites k and l. It should be noted that the dynamics in +this system preserves the total excitation number in the +system. This is not an essential feature but makes the +system amenable to efficient numerical analysis. We will +assume that the system is susceptible simultaneously to +two distinct types of noise processes, a dissipative pro- +cess that reduces the number of excitations in the system +at rate Γk and a dephasing process that randomizes the +phase of local excitations at rate γk. +Initially we will assume that we can describe both pro- +cesses by using a Markovian master equation with local +dephasing and dissipation terms. It is important to note +however that the effects found here persist when tak- +ing account of the system-environment interaction in a +more detailed manner (see Methods). Dissipative pro- +cesses, which lead to energy loss, are then described by +the Lindblad super-operator + +Ldiss(ρ) = + +N +� + +k=1 +Γk[−{σ+ +k σ− +k , ρ} + 2σ− +k ρσ+ +k ], +(2) + +while energy-conserving dephasing processes are de- +scribed by the operator + +Ldeph(ρ) = + +N +� + +k=1 +γk[−{σ+ +k σ(−) +k +, ρ} + 2σ+ +k σ− +k ρσ+ +k σ− +k ]. (3) + +Finally, in order to be able to measure the total transfer +of excitation, we designate an additional site, numbered +N + 1, which is populated by an irreversible decay pro- +cess from a chosen level k as described by the Lindblad +operator + +Lsink(ρ) = +(4) +ΓN+1[−{σ+ +k σ− +N+1σ+ +N+1σ− +k , ρ} + 2σ+ +N+1σ− +k ρσ+ +k σ− +N+1]. + +The subindex ’sink’ emphasizes that no population can +escape of site N + 1. For definitiveness and simplicity, + +the initial state of the network at t = 0 will be assumed +to be a single excitation in site 1 unless stated otherwise. +The key question that we will pose and answer is the +following: In a given time T , how much of the initial +population in site 1 will have been transferred to the sink +at site N + 1 and how is this transfer affected by the +presence of dephasing and dissipative noise. +In the remainder of this paper we will demonstrate +that, in certain settings, the presence of dephasing noise +can assist the transfer of population from site 1 to the +sink at site N + 1 considerably. It is an intriguing obser- +vation that this noise enhanced transfer does not occur +for all possible Hamiltonians of the type given by eq.(1) +and may depend also on properties of the noise such as +its spatial dependence. These noise rates can be opti- +mized numerically, and in very simple cases analytically, +to yield the strongest possible effect. One may suspect +that natural, biological systems, have actually made use +of such an optimization. +Linear chain – We begin with a brief analysis of the +uniform linear chain with only nearest neighbor inter- +actions so that in eq. +(1) the coupling strengths sat- +isfy vl,k = vk,l = vδl,k+1 for k = 1, . . . , N − 1 and +ωk = ω and Γk = Γ for k = 1, . . . , N. Extensive numer- +ical searches show that, for arbitrary choices of ΓN+1, +Γ and ω and arbitrary transmission times T and chains +of the length N = 2, . . . , 12, the optimal choice of de- +phasing noise rates vanish. We have used a directed ran- +dom walk algorithm with multiple initial states which +has never exceeded the values for the noise-free chain +and approached them to within at least 10−8. We were +able to derive formulae for the case T = ∞ and short +chains which demonstrate this behaviour analytically. +For N = 2, with ω1 = ω2 = ω and arbitrary v1,2, γi +and Γi, we find, with the abbreviation γ = γ1 + γ2 and +x = 2Γ3 +1 + Γ1Γ3(3Γ1 + Γ3), that the population of the +sink is given by + +psink = +Γ3v2 +1,2 + +x + Γ1(Γ1 + Γ3)γ + (Γ3 + 2Γ1)v2 +1,2 +, +(5) + +which is evidently maximized for γ = 0. One may also +obtain the analytical expressions for N = 3 and Γk = Γ +for k = 1, 2, 3 and demonstrate that the optimal dephas- +ing level is γ = 0 (see section on Methods). +This ap- +proach, though more tedious, may be taken to higher +values of N as well. Extensive numerical searches lend +further support to the observation that dephasing does +not improve excitation transfer for uniform chains but a +general proof has remained elusive. +So far, the findings are consistent with the expectation +that noise does not enhance the transport of excitations. +However, for non-uniform chains we encounter the dif- +ferent and perhaps surprising situation where noise can +significantly enhance the transfer rate of excitations. +As an illustrative example, we may keep the nearest +neighbor coupling uniform but allow for one site to have +a different site energy ω. If we chose N = 3, ω1 = ω3 = 1, +Γ1 = Γ2 = Γ3 = 1/100, v1,2 = v2,3 = 1/10, ΓN+1 = 1/5 + + +3 + +0 +0.5 +1 +1.5 +2 +0 + +0.01 + +0.02 + +0.03 + +0.04 + +0.05 + +0.06 + +0.07 + +ω2 + +psink(γopt) − psink(γ = 0) + +FIG. 2: The optimal improvement of the transfer efficiency is +plotted versus the site frequency ω2 in a chain of length N = 3 +and system parameters ω1 = ω3 = 1, Γ1 = Γ2 = Γ3 = 1/100, +v1,2 = v2,3 = 1/10, ΓN+1 = 1/5 and T = ∞. One observes +that dephasing only assists the transmission probability in +some frequency intervals. + +and T = ∞ , then we obtain the results depicted in +Fig. 2. One observes that dephasing assists the trans- +mission only when site 2 is sufficiently detuned from the +neighboring sites. This example suggests a simple pic- +ture to explain the reason for the dephasing enhanced +population transfer through the chain. Site 2 is strongly +detuned from its neighboring sites and the coupling v to +its neighbors is comparatively weak, i.e. v ≪ δω with +δω = min[|ω2 − ω1|, |ω3 − ω2|]. +Hence , the transport +rate is limited by a quantity of order v2/δω as it is a +second order process due to the lack of resonant modes +between neighboring sites. Introducing dephasing noise +leads to a broadening of the energy level at each site +k and a line-width proportional to the dephasing rate +γk. +Then, with increasing dephasing rate, the broad- +ened lines of neighboring sites begin to overlap and the +population transfer will be enhanced as resonant modes +are now available. Enhancing the dephasing rate further +will eventually lead to a weakening of the transfer as +the modes are distributed over a very large interval and +resonant modes have a small weight. Dissipation does +not lead to the same enhancement as, crucially, the gain +to the broadening of the line is overcompensated by the +irreversible loss of excitation. +This is corroborated by +numerical studies where increasing dissipation does not +assist the transport. The physical picture outlined above +is confirmed in Fig. +3. +We chose a chain of length 3 +which suffers dephasing only in site 2 and uniform dis- +sipation with rates Γk = 1/100 along the chain while +ω1 = ω2/4 = ω3 = 1 and v1,2 = v2,3 = 1/10 (see fig. +3). The close relationship of this model to Raman tran- +sitions in quantum optics will be exploited to propose a +realizable experiment in a highly controlled environment +to verify these effects (see section on realizations). +In the examples above the improvement of excitation +transfer due to the dephasing is small. One can easily + +0 +2 +4 +6 +8 +10 +0 + +0.005 + +0.01 + +0.015 + +γ2 + +psink(γ2) − psink(γ2=0) + +FIG. 3: The difference between transfer efficiency and the +efficiency without dephasing is plotted versus the dephasing +rate γ2 in a chain of length N = 3 and ω1 = ω2/4 = ω3 = 1, +v1,2 = v2,3 = 1/10, γ1 = γ3 = 0, Γk = 1/100 for k = 1, . . . , N, +ΓN+1 = 1/5 and T = ∞. Initially increasing dephasing as- +sists the transfer of excitation while very strong dephasing +suppresses the transport. + +show, however, that this improvement may be made ar- +bitrarily large in the sense that without noise the trans- +fer rate approaches zero while it approaches unity arbi- +trarily closely for optimal noise levels. As an example, +for N = 3, ω1 = ω3 = 1; ω2 = 100, v1,2 = v2,3 = v, +γ1 = γ3 = 0 and Γ1 = Γ2 = Γ3 = v2/f and Γ4 = 105v +we find for ∆p = psink(γ2,opt) − psink(γ2 = 0) that + +lim +v→0 ∆p = +f 2γ2 +2 + +f 2γ2 +2 + 3fγ2((ω2 − 1)2 + γ2 +2) + ((ω2 − 1)2 + γ2 +2)2 +(6) +This is maximized for γ2 = ω2 − 1 when it takes the +value ∆p = f 2/(f 2 + 6f(ω2 − 1) + 4(ω − 2 − 1)2). In the +limit f → ∞ this approaches 1, that is, without noise the +excitation transfer vanishes while with noise it achieves +unit efficiency! It should be noted that being a system of +fixed finite size, the effect may not be directly attributed +to Anderson localization [24] which, in addition does not +occur in systems attached to a sink, as is assumed here +[25]. +Entanglement and coherence in the channel – We have +seen that the transport of excitations in the system may +be assisted considerably by local dephasing. +Now we +would like to discuss briefly the quantum coherence prop- +erties during transmission by studying the presence of +entanglement and the ability of the chain to transmit +quantum information. To this end, we consider how en- +tanglement is transported along the chain when it is used +to propagate one half of a maximally entangled state to +obtain an insight on how is the quantum capacity of this +channel affected by dephasing. To illustrate this, we con- +sider a chain of N = 4 sites. We chose the same param- +eters as in Fig. 2 and fix ω3 = 14. Comparison of the +entanglement between an uncoupled site and the various +sites in the chain for vanishing dephasing and the opti- +mal choice of the dephasing for excitation transfer show + + +4 + +that, while entanglement propagates through the system, +the amount of entanglement decreases with increasing +dephasing. +In fact, the dephasing rate that optimizes +the ability of the channel to transmit quantum informa- +tion vanishes, in contrast to the situation for excitation +transfer. +Therefore, although dephasing may enhance +the propagation of excitations, it also destroys quantum +coherence and in the present setting it leaves an overall +detrimental effect. +Complex networks and Light-harvesting molecules – +So far, we have demonstrated that in linear chains lo- +cal dephasing noise may enhance the transfer of excita- +tions. Going beyond this, we will now consider fully con- +nected networks and apply our observations to a model +that describes the transfer of excitons in the Fenna- +Matthews-Olson complex of Prosthecochloris aestuarii, +which is a pigment-protein complex that consists of seven +bacteriochlorophyll-a (BChla) molecules (see [20, 21, 22] +and Note Added for closely related work). This complex +is able to absorb light to create an exciton. This exci- +ton then propagates through the complex until it reaches +the reaction centre where its energy is then used to trig- +ger further processes that bind the energy in chemical +form [15, 23]. The Hamiltonian of this complex may be +approximated by eq. +(1), where the site energies and +coupling constants may be taken from table 2 and 4 of +[15]. We then find, in matrix form + +H = + + + + + + + + + + + + +215 −104.1 +5.1 +−4.3 +4.7 −15.1 +−7.8 +−104.1 +220.0 +32.6 +7.1 +5.4 +8.3 +0.8 +5.1 +32.6 +0.0 −46.8 +1.0 +−8.1 +5.1 +−4.3 +7.1 −46.8 +125.0 −70.7 −14.7 −61.5 +4.7 +5.4 +1.0 −70.7 450.0 +89.7 +−2.5 +−15.1 +8.3 +−8.1 −14.7 +89.7 +330.0 +32.7 +−7.8 +0.8 +5.1 −61.5 +−2.5 +32.7 +280.0 + + + + + + + + + + + + +(7) +where +we +have +shifted +the +zero +of +energy +by +12230 +(all +number +are +given +in +the +units +of +1.988865 · 10−23Nm += +1.2414 10−4eV ) for all sites +corresponding to a wavelength of ∼= 800nm. +Recent +work [15] suggests that it is this site 3 that couples to +the reaction centre at site 8. +For this rate, somewhat +arbitrarily, we chose Γ3,8 += +10/1.88 corresponding +to about 1 ps−1 (value in the literature range from +0.25ps−1 [15] and 1 ps−1 [20] to 4 ps−1 [17]). Again, we +will assume the presence of both dissipative noise (loss +of excitons) and dephasing noise (due to the presence +of a phonon bath consisting of vibrational modes of the +molecule). The measured lifetime of excitons is of the +order of 1 ns which determines a dissipative decay rate +of 2Γk = 1/188 and that we assume to be the same for +each site [15]. +If we neglect the presence of any form +of dephasing and we start with a single excitation on +site 1, then we observe that the excitation is transferred +to the reaction centre (site 8). For a time T = 5, we +find that the amount of excitation that is transferred +is psink = 0.551926. +Optimal dephasing rates that +maximize the transfer rate of the initial excitation in site +1 considerably improve on that. For T = 5 we find the + +0 +50 +100 +150 +200 +250 +300 +350 +400 +0 + +0.1 + +0.2 + +0.3 + +0.4 + +0.5 + +0.6 + +0.7 + +0.8 + +0.9 + +1 + +t + +EN + + + + + +0−1 No dephasing +0−2 No dephasing +0−3 No dephasing +0−4 No dephasing +0−1 Dephasing +0−2 Dephasing +0−3 Dephasing +0−4 Dephasing + +FIG. 4: The time evolution of the entanglement between a +decoupled site and the sites in the chain of length N = 4 +and system parameters ω1 = ω2 = ω4 = 10, ω = 14, v1,2 = +v2,3 = v3,4 = 1, Γk = 1/10 for k = 1, . . . , N and ΓN+1 = +1. The initial state is a maximally entangled state between +the decoupled site and the first site of the chain. Dephasing +destroys entanglement along the chain and has no beneficial +effect. + +optimal +dephasing +rates +(γ1, γ2, γ3, γ4, γ5, γ6, γ7) += +(469.34, 5.36, 99.13, 5.55, 114.86, 1.88, 291.08) +and +the much improved value psink += +0.988526. +For +T += +∞, +we +find +the +dephasing +free +transfer +probability of psink += +0.81425 while for the op- +timal +dephasing +rates +(γ1, γ2, γ3, γ4, γ5, γ6, γ7) += +(27.40, 26.84, 1.22, 87.12, 99.59, 232.76, 88.35) +we +find +psink = 0.99911. It should be noted that these dephasing +rates are comparable to the inter-site coupling rates +which suggests that a more accurate treatment will need +to go beyond master equations (see Methods for a brief +discussion). +We conclude that dephasing may lead to a very strong +enhancement of the transfer rate of excitations in a re- +alistic network. In fact, in models obtained from spec- +troscopic data measured on the FMO complex it is in- +deed observed that almost complete transport should +take place within time T = 5 [15]. It is remarkable that +such a rapid transfer cannot be explained from a purely +coherent dynamics and, as shown above, the underlying +reason for the speed up is the presence of dephasing which +may even be local. +Experimental Realizations – The FMO-complex pro- +vides a fascinating setting for the observation of dephas- +ing enhanced transport but it is also a very challeng- +ing environment to verify the effect precisely. Here we +present several physical systems in which the dephasing +enhanced excitation transfer may be observed and which +are at the same time highly controllable. Perhaps the +simplest such setting is found in atomic physics (see Fig. +5) where the behaviour of a chain of three sites may be +simulated using detuned Raman transitions in ions such +as Ca+, Sr+ or Ba+. The master equation of this system +simulates exactly that of a chain with a single excitation +as has been described throughout this paper. +Atomic + + +5 + +2 + +� +� + +r + +0 + +�S +�N �1 +1 + +3 + +2 + +� +� + +r + +0 + +�S +�N �1 +1 + +3 + +� +� + +r + +0 + +�S +�N �1 +1 + +3 + +FIG. 5: A atomic system with Raman transitions provides a +transparent illustration of dephasing assisted transport. The +required level structure may be realized in Ca+, Sr+ or Ba+. +Each atomic level represents a site in the chain which may +be populated. +Starting with all the population in level 1, +one may then irradiate the system with classical laser fields +of Rabi-frequency Ω on the 1 ↔ 2 and the 3 ↔ 2 transition +[27]. Level 3 in turn is assumed to decay spontaneously into +an additional level |r⟩ that plays the role of the recipient. +Spontaneous decay of the chain as a whole is modelled by +spontaneous decay into level |0⟩ from which no population +can enter the levels |1⟩, |2⟩, |3⟩ and |r⟩ anymore. Dephasing +noise may now enter the system affecting level 2 for example +through magnetic field fluctuations. + +populations may be measured with very high accuracy +using quantum jump detection [28, 29]. +A variety of other natural implementations of dephas- +ing assisted excitation transport can be conceived and +will be studied in detail elsewhere. Firstly, the oscilla- +tions of ions in a linear ion trap transversal to the trap +axis realizes a harmonic chain [30] that allows for the +implementation of a variety of operations such as prepa- +ration of Fock states and is capable of supporting near- +est neighbor coupling between neighbouring ion oscilla- +tors [31] and allowing high efficiency readout by quantum +jump detection [28]. When restricting to the single exci- +tation space, the dynamics of the system is described by +master equations that become equivalent to those pre- +sented in this paper. +Furthermore, harmonic chains are also realized in cou- +pled arrays of cavities which have recently received con- +siderable attention in the context of quantum simulators +[32]. Ultra-cold atoms in optical lattices which have pre- +viously been used to study thermal assisted transport in +Brownian ratchets [33] presents another scenario in which +to study such dephasing assisted processes. Chains of su- +perconducting qubits or superconducting stripline cavi- +ties [34] may also provide a possible setting for the ob- +servation of the effects described above. +Conclusions and outlook – The results presented here +demonstrate that while dephasing noise destroys quan- +tum correlations, it may at the same time enhance the +transport of excitations. In fact, the efficient transport +observed in certain biological systems has been shown +to be incompatible with a fully coherent evolution while + +it can be explained if the system is subject to local de- +phasing. Hence, in this context, the presence of quan- +tum coherence and therefore, entanglement in the sys- +tem, does not seem to be supporting excitation transfer. +This suggests that entanglement that may be present in +bio-molecules, though interesting, may not be a universal +functional resource. +Importantly, the results presented here suggest that it +may be possible to design and optimize the performance +of nano-fabricated transmission lines in naturally noisy +environments to achieve strongly enhanced transfer ef- +ficiencies employing the concept of noise assisted trans- +port. +Acknowledgements– We are grateful to Seth Lloyd +for helpful communications concerning [20, 21, 22], Neil +Oxtoby, Angel Rivas and Shashank Virmani for useful +comments on the manuscript and to Danny Segal for +advice on atomic physics. +This work was supported +by the EU via the Integrated Project QAP (‘Qubit +Applications’) and the STREP action CORNER and the +EPSRC through the QIP-IRC. MBP holds a Wolfson +Research Merit Award. + +Note Added— While finalizing this work, we became +aware of independently obtained but closely related re- +sults presented in [20, 21, 22]. There it was showed that +quantum transport can be enhanced by an interplay be- +tween coherent dynamics and environment effects with +particular emphasis on excitonic energy transfer in light +harvesting complexes [20]. The role of the different phys- +ical processes that contribute to the energy transfer ef- +ficiency have been studied in [21] and the enhancement +of quantum transport due to a pure dephasing environ- +ment within the Haaken-Strobl model was demonstrated +in [22]. + + +6 + +Methods – + +Exact solutions for uniform chains – One may also ob- +tain the analytical expressions for a chain of length N = 3 + +described by eqs. (1) - (4) for the choice and Γk = Γ for +k = 1, 2, 3, 4 and demonstrate that the optimal dephasing +level is γ = 0. We find + +psink = +(4Γ + γ1 + γ3)v2 + +36Γ5 + 6aΓ4 + 2Γ3(3γ2 +1 + 3γ2 +2 + 8b + 2γ2 +3 + 32v2) + Γ2(2c + dv2) + Γv2(3γ2 +1 + 7b + 4γ2 +3 + 15v2) + 4(γ1 + γ3)v4 + +where a = (5γ1 + 5γ2 + 4γ3), b = γ1γ2 + γ1γ3 + γ2γ3, +c = γ1(γ2 +2 + γ2 +3) + γ2(γ2 +1 + γ2 +3) + γ3(γ2 +1 + γ2 +2) + 2γ1γ2γ3, +d = 32γ3 + 25γ2 + 29γ1. Then one first observes that +the optimal choice is γ2 = 0 as it only occurs in the +denominator with positive coefficients. In the remaining +expression one then substitutes γk = ˜γ2 +k allowing also for +negative ˜γk. Then differentiation w.r.t these ˜γk shows +that the gradient only vanishes for ˜γ1 = ˜γ2 = 0. +Beyond Markovian master equations– So far we have +demonstrated the existence of dephasing enhanced exci- +tation transfer employing a master equation description. +The optimized dephasing rates that have been obtained, +in particular those in the context of the FMO complex, +can be comparable to the coherent interaction strengths +and may be similar to the spectral width of the bath re- +sponsible for the dephasing [15]. This may not be fully +compatible with the master equation approach employed +so far as its derivation relies on several assumptions in- +cluding the weak coupling hypothesis and the require- +ment for the bath to be Markovian [26]. The derivation is +further complicated for systems with several constituents +where the local coupling of its constituents is not com- +patible with non-local structure of the eigenmodes of the +systems. This is especially so when the coherent inter +sub-system coupling is of comparable strength to the sys- +tem environment coupling. The situation is made more +difficult due to spatial as well as temporal correlations in +the environmental noise (which is to be expected in par- +ticular for the FMO complex but also many other realisa- +tions of coupled chains in contact with an environment). +Bloch-Redfield equations and other effective description +are sometimes used but still represent approximations to +the correct dynamics [26] where the errors are often dif- +ficult to estimate precisely. +Therefore, we demonstrate briefly that dephasing as- +sisted transfer of excitation can also be observed when +one uses a microscopic model of an environment that +may, in addition, exhibit non-Markovian behaviour. To +this end we study the effect of an environment which is +modelled by brief interactions between two-level systems +and individual subsystem of the chain in which excita- + +tion transport is taking place. The strength and nature +of the interactions can be chosen to implement dephasing +(elastic collisions) and dissipation (in-elastic collisions). +Non-markovian effects can be included in the model de- +pending on the spatial and temporal memory of the envi- +ronment particles. Interaction strengths are determined +for a single site system to obtain the dissipation rate Γ +and dephasing rate γ. This simplified model allows us +to study the effect of more realistic environments outside +the master equation picture and results are summarized +in Figure 6. +A more detailed simulation of excitation +transfer taking account of the full environment are be- +yond the scope of the present work and will be presented +elsewhere [35] + +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +0 + +0.1 + +0.2 + +0.3 + +0.4 + +0.5 + +0.6 + +0.7 + +0.8 + +0.9 + +1 + +t + +psink + + + + + +γ=0 +γ=0.0064γopt +γ = 0.16γopt +γ = γopt + +FIG. 6: Here we show how the transfer in the presence of +dephasing into a bath that is modelled by a collisional model +where local sites briefly interact with a single particle. The in- +teraction strength is chosen such that in an uncoupled systems +the sites suffer the optimal decoherence rates γopt as presented +in the previous section multiplied with factors 0, 0.0064, 0.16 +and 1. The dynamics is similar to that observed for the mas- +ter equation approach and shows only minor deviations. In- +creased dephasing rates do improve the excitation transfer +also in this model. + +[1] A. A. Faisal, L. P. J. Selen and D. M. Wolpert, Nature +Reviews on Neuroscience 9, 292 (2008). +[2] R. Benzi, A. Sutera, and A. Vulpiani, J. Phys. A 14, + + +7 + +L453 (1981). +[3] L. Gammaitoni, P. H¨anggi, P. Jung, and F. Marchesoni, +Rev. Mod. Phys. 70, 223 (1998). +[4] J. K Douglass, L. Wilkens, E. Pantazelou and F. Moss, +Nature 365, 337 (1993); K. Wiesenfeld and F. Moss, Na- +ture 373, 33 (1995). +[5] Y. H. Shang, A. Claridge-Chang, L. Sjulson, M. Pypaert +and G. Miesenbock, Cell 128, 601 (2007). +[6] P. Reimann, M. Grifoni, and P. H¨anggi, Phys. Rev. Lett. +79, 10 (1997) +[7] Special issue on Ratchets and Brownian Motors, edited +by H. Linke, Appl. Phys. A 75, 167 (2002) +[8] P. H¨anggi, F. Marchesoni and F. Nori, Ann. Phys. +(Berlin) 14, 51 (2005). +[9] L. Viola, E. M. Fortunato, S. Lloyd, C. H. Tseng, and +D. G. Cory, Phys. Rev. Lett. 84, 5466 (2000). +[10] M. B. Plenio and S. F. Huelga, Phys. Rev. Lett. 88, +197901 (2002). +[11] S. F. Huelga and M. B. Plenio, Phys. Rev. Lett. 98, +170601 (2007). +[12] J.-L. Ting, Phys. Rev. E 59, 2801 (1998) +[13] G. Bowen and S. Mancini, Phys. Lett. A 321, 1 (2004) +´ıbid 352, 272 (2006). +[14] C. Di Franco, M. Paternostro, D. I. Tsomokos and S. F. +Huelga, Phys. Rev. A 77, 062337 (2008). +[15] J. Adolphs and T. Renger, Biophys. J. 91, 2778 (2006) +[16] See G. S. Engel, T. R. Calhoun, E. L. Read EL, T. K. +Ahn, T. Mancal, Y. C. Cheng, R. E. Blankenship and +G. R. Fleming, Nature 446, 782 (2007) for recent exper- +imental results. +[17] A. Olaya-Castro, C. F. Lee, F. Fassioli-Olsen, and N. F. +Johnson, arXiv:0708.1159. +[18] H. J. Briegel and S. Popescu, arXiv:0806.4552. +[19] K. M. Gaab and C. J. Bardeen, J. Chem. Phys. 121, +7813 (2004). +[20] M. Mohseni, P. Rebentrost, S. Lloyd and A. Aspuru- + +Guzik, arXiv:0805.2741. +[21] P. Rebentrost, +M. +Mohseni and +A. Aspuru-Guzik, +arXiv:0806.4725. +[22] P. Rebentrost, M. Mohseni, I. Kassal, S. Lloyd, A. +Aspuru-Guzik, arXiv:0807.0929 +[23] R. E. Fenna and B. W. Matthews, Nature 258, 573 +(1975). +[24] P. W. Anderson, Phys. Rev. 109, 1492 (1958). +[25] S. A. Gurvitz, Phys. Rev. Lett. 85, 812 (2000). +[26] H.-P. Breuer and F. Petruccione, The Theory of Open +Quantum Systems, Oxford University Press, 2002. +[27] M. O. Scully and S. Zubairy, Quantum Optics, Cam- +bridge University Press. +[28] M. B. Plenio and P. L. Knight, Rev. Mod. Phys. 70, 101 +(1998). +[29] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev. +Mod. Phys. 75, 281 (2003). +[30] M. B. Plenio, J. Hartley and J. Eisert, New J. Phys. 6, +36 (2004) +[31] A. Serafini, A. Retzker and M. B. Plenio, +E-print +arXiv:0708.0851 [quant-ph] +[32] M. J. Hartmann, F. G .S .L. Brand˜ao, M. B. Plenio, +Nature Phys. 2, 849 (2006); A. D. Greentree et al, Nature +Phys. 2, 856 (2006); D. G. Angelakis, M. F. Santos and +S. Bose, Phys. Rev. A 76, 031805(R) (2007). +[33] C. Mennerat-Robilliard, D. Lucas, S. Guibal, J. Tabosa, +C. Jurczak, J.-Y. Courtois, and G. Grynberg, Phys. Rev. +Lett. 82, 851 (1999). +[34] J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. +Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. +Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. +Girvin and R. J. Schoelkopf, Nature 449, 443 (2007); +A. Palacios-Laloy, F. Nguyen, F. Mallet, P. Bertet, D. +Vion and D. Esteve, arXiv:0712.0221. +[35] Work in progress. + + diff --git a/papers/project_paper_3_darwinism/references/PlenioHuelga2008_source.tar.gz b/papers/project_paper_3_darwinism/references/PlenioHuelga2008_source.tar.gz new file mode 100644 index 00000000..b1452e6d Binary files /dev/null and b/papers/project_paper_3_darwinism/references/PlenioHuelga2008_source.tar.gz differ diff --git a/papers/project_paper_3_darwinism/references/Zurek2009.pdf b/papers/project_paper_3_darwinism/references/Zurek2009.pdf new file mode 100644 index 00000000..e9a53453 --- /dev/null +++ b/papers/project_paper_3_darwinism/references/Zurek2009.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1c887b195b8b2e23ccc436e153fc4e7506c75aa7f8ae997ca2da4e6489b9fd17 +size 843308 diff --git a/papers/project_paper_3_darwinism/references/Zurek2009.txt b/papers/project_paper_3_darwinism/references/Zurek2009.txt new file mode 100644 index 00000000..f8320d4e --- /dev/null +++ b/papers/project_paper_3_darwinism/references/Zurek2009.txt @@ -0,0 +1,1600 @@ +Quantum Darwinism + +Wojciech Hubert Zurek +Theory Division, MS B213, LANL Los Alamos, NM, 87545, U.S.A. + +Quantum Darwinism describes the proliferation, in the environment, of multiple records of selected +states of a quantum system. It explains how the fragility of a state of a single quantum system can +lead to the classical robustness of states of their correlated multitude; shows how effective ‘wave- +packet collapse’ arises as a result of proliferation throughout the environment of imprints of the +states of quantum system; and provides a framework for the derivation of Born’s rule, which relates +probability of detecting states to their amplitude. +Taken together, these three advances mark +considerable progress towards settling the quantum measurement problem. + +The quantum principle of superposition implies that +any combination of quantum states is also a legal state. +This seems to be in conflict with everyday reality: States +we encounter are localized. Classical objects can be ei- +ther here or there, but never both here and there. Yet, the +principle of superposition says that localization should be +a rare exception and not a rule for quantum systems. + +Fragility of states is the second problem with quantum- +classical correspondence: Upon measurement, a general +preexisting quantum state is erased – it “collapses” into +an eigenstate of the measured observable. How is it then +possible that objects we deal with can be safely observed, +even though their basic building blocks are quantum? + +To bypass these obstacles Bohr [1] followed Alexander +the Great’s example: Rather than try disentangling the +Gordian Knot at the beginning of his conquest, he cut +it. +The cut separates the quantum from the classical. +Bohr’s Universe consists of two realms, each governed by +its own laws. Fragile superpositions were banished from +the classical realm deemed more fundamental and indis- +pensable to interpret or even practice quantum theory. +Thus, instead of trying to understand Universe (includ- +ing “the classical”) in quantum terms one “quantized” +this and that, always starting from the classical base. + +This was a brilliant tactical move: Physicists could +conquer the quantum realm without getting distracted by +interpretational worries. In those days only gedankenex- +periments like the famous Schr¨odinger cat [2] were truly +disturbing: Real experiments dealt with electrons, pho- +tons, atoms, or other microscopic systems. Bohr’s rule of +thumb – that the macroscopic is classical – was enough. +Moreover, many (including Einstein) believed that quan- +tum physics is just a step on a way to a deeper theory +that will solve or bypass interpretational conundrums. + +That did not happen. +Instead, old gedankenexperi- +ments were carried out. They confirmed validity of quan- +tum laws on scales that have, of recent, begun to infringe +on “the macroscopic”. Quantum theory is here to stay. +It is also increasingly clear that its weirdest predictions +– superpositions and entanglement – are experimental +facts, in principle relevant also for macroscopic objects. +Therefore, questions about the origin of “the classical”, +with its restriction to localized states that are robust, un- +perturbed by measurements, can no longer be dismissed. + +I. +DECOHERENCE AND EINSELECTION + +Decoherence turns one of the two problems we noted +above – fragility of quantum states – into a solution of the +other. Environment-induced decoherence recognizes that +if a measurement can put a state at risk and re-prepare +it, so can accidental information transfers that happen +whenever a system interacts with its environment. +Decoherence is by now well understood [3, 4, 5]: +Fragility of states makes quantum systems very difficult +to isolate. Transfer of information (which has no effect on +classical states) has dramatic consequences in the quan- +tum realm. So, while fundamental problems of classical +physics were always solved in isolation (it sufficed to pre- +vent energy loss) this is not so in quantum physics (leaks +of information are much harder to plug). +When a quantum system gives up information, its own +state becomes consistent with the information that was +disseminated. “Collapse” in measurements is an extreme +example, but any interaction that leads to a correlation +can contribute to such re-preparation: Interactions that +depend on a certain observable correlate it with the en- +vironment, so its eigenstates are singled out, and phase +relations between such pointer states are lost [6]. +Negative selection due to decoherence is the essence of +environment-induced superselection, or einselection [7]: +Under scrutiny of the environment, only pointer states +remain unchanged. Other states decohere into mixtures +of stable pointer states that can persist, and, in this sense, +exist: They are einselected. +These ideas can be made precise. The basic tool is the +reduced density matrix ρS. It represents the state of the +system that obtains from the composite state ΨSE of S +and E by tracing out the environment E: + +ρS = TrE|ΨSE⟩⟨ΨSE| . +(1) + +Evolution of ρS reveals preferred states: It is most pre- +dictable when the system starts in a pointer state. To +quantify this one can use (von Neumann) entropy, HS = +H(ρS) = −TrρS lg ρS, as a function of time. +Pointer +states result in smallest entropy increase. By contrast, +their superpositions produce entropy rapidly, at decoher- +ence rates, especially when S is macroscopic. +When pure states of the system are sorted by pre- +dictability, according to entropy of the evolved ρS, + +arXiv:0903.5082v1 [quant-ph] 29 Mar 2009 + + +2 + +pointer states are at the top. This criterion – the pre- +dictability sieve [4, 8, 9] – yields a short list of candidates +for effectively classical states: A cat can persist in one +of the two obvious stable states, but their superposition +would deteriorate into a mixture of |dead⟩ and |alive⟩ +when initiated in a way envisaged by Schr¨odinger [2]. +The special role of position is traced to the nature of +the SE interactions: They tend to depend on distance. +Hence, information about position is most readily passed +on to the environment. This is why localized states sur- +vive while nonlocal superpositions decay into their mix- +tures. For example, in a weakly damped harmonic os- +cillator the minimum uncertainty wavepackets – familiar +coherent states, best quantum approximation of classical +points in phase space – are einselected [9, 10, 11]. + +II. +ENVIRONMENT AS A WITNESS + +Monitoring by the environment means that informa- +tion about S is deposited in E. What role does it play, +and what is its fate? Decoherence theory ignores it. En- +vironment is “traced out”. +Information it contains is +treated as inaccessible and irrelevant: E is a “rug to sweep +under” the data that might endanger classicality. +Quantum Darwinism recognizes that “tracing out” is +not what we do: Observers eavesdrop on the environ- +ment. Vast majority of our data comes from fragments +of E. Environment is a witness to the state of the system. +For example, this very moment you intercept a fraction +of the photon environment emitted by a screen or scat- +tered by a page. We never access all of E. Tiny fractions +suffice to reveal the state of various “systems of interest”. +This insight captures the essence of Quantum Darwin- +ism: Only states that produce multiple informational off- +spring – multiple imprints on the environment – can be +found out from small fragments of E. The origin of the +emergent classicality is then not just survival of the fittest +states (the idea already captured by einselection), but +their ability to “procreate”, to deposit multiple records +– copies of themselves – throughout E. +Proliferation of records allows information about S to +be extracted from many fragments of E (in the example +above, photon E). Thus, E acquires redundant records of +S. Now, many observers can find out the state of S in- +dependently, and without perturbing it. This is how pre- +ferred states of S become objective. Objective existence +– hallmark of classicality – emerges from the quantum +substrate as a consequence of redundancy. +Decoherence theory was focused on the system. Its aim +was to determine what states survive information leaks +to E. Now we ask: What information about the system +can be found out from fragments of E? This change of +focus calls for a more realistic model of the environment +(Fig. 1): Instead of a monolithic E we recognize that envi- +ronments consist of subsystems that comprise fragments +independently accessible to observers. +The reduced density matrix ρS representing the state + +� +� +� +� +� +� +� +� +� +� +� +� +� +� + + + +� + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +! + + +" + +# +$ +% + + + + +& + +# + + +" + +� +' +� +( +� + +) +� + + +� +� +* + + + +� + + + + + + + +% + + + + +& + +# + + +" + + + + + + + + + + + + + + + + ++ +, + + + + + + +& + +# + + +" + + +� +- +� + + ++ +, + + + + + + +& + +# + + +" + +. +/ + +0 + +1 +2 + + + + + + + + + +3 + +4 +5 +# + + +" + + +6 +. + + +. +0 +6 +6 +. + + + + + +. +/ +7 +8 +9 +0 + +1 +: +7 + + + + + +; + +/ +1 +. + + + + +< + +FIG. 1: Quantum Darwinism and the structure of the envi- +ronment. Decoherence theory distinguishes between a system +(S) and its environment (E) as in (a), but makes no further +recognition of the structure of E; it could as well be mono- +lithic. In Quantum Darwinism the focus is on redundancy. +We recognize the subdivision of E into subsystems, as in (b). +The only requirement for a subsystem is that it should be +individually accessible to measurements; observables of dif- +ferent subsystems commute. To obtain information about S +from E one can then measure fragments F of the environ- +ment – non-overlapping collections of subsystems of E, (c). +ically, there are many copies of the information about S in E +– “progeny” of the “fittest observable” that survived monitor- +ing by E proliferates throughout E. This proliferation of the +multiple informational offspring defines Quantum Darwinism. +The environment becomes a witness with redundant copies of +information about the preferred observable. This leads to the +objective existence of pointer states: Many can find out the +state of the system independently, without prior information, +and they can do it indirectly, without perturbing S. + +of the system was the basic tool of decoherence. To study +Quantum Darwinism we focus on correlations between +fragments of the environment and the system. The rele- +vant reduced density matrix ρSF is given by: + +ρSF = TrE/F|ΨSE⟩⟨ΨSE| . +(2) + +Above, trace is over “E less F”, or E/F – all of E except +for the fragment F. How much F knows about S can be +quantified using mutual information: + +I(S : F) = HS + HF − HS,F , +(3) + +defined as the difference between entropies of two sys- +tems (here S and F) treated separately and jointly. For +example, the mutual information between an original and + + +3 + +FIG. 2: Information about S stored in E and its redundancy. +Mutual information is monotonic in f. When global state of +SE is pure, I(S : Ff) in a typical fraction f of the environ- +ment is antisymmetric around f = 0.5 [13]. For pure states +picked out at random from the combined Hilbert space HSE, +there is little mutual information between S and a typical F +smaller than half of E. However, once a threshold f = 1 + +2 is +attained, nearly all information is in principle at hand. Thus, +such random states (green line) exhibit no redundancy. By +contrast, states of SE created by decoherence (where the en- +vironment monitors preferred observable of S) contain almost +all (all but δ) of the information about S in small fractions +fδ of E. The corresponding I(S : Ff) (red line) quickly rises +to HS (entropy of S due to decoherence), which is all of the +information about S available from either E or S. (More, up +to 2HS, can be obtained only through global measurements +on S and nearly all E). HS is therefore the classically acces- +sible information. As (1 − δ)HS of information is contained +in fδ = 1/Rδ of E, there are Rδ such fragments in E: Rδ +is the redundancy of the information about S. Large redun- +dancy implies objectivity: The state of the system can be +found out indirectly and independently by many observers, +who will agree about their conclusions. Thus, Quantum Dar- +winism accounts for the emergence of objective existence. + +a perfect copy (of, say, a book) is equal to the entropy of +the original, as either contains the same text. So, every +bit of information in the first copy reveals a bit of infor- +mation in the original. However, having extra copies does +not increase the information about the original. Yet, it +determines how many can independently access this in- +formation. The number of copies defines redundancy. +Similar ideas apply to the quantum case. Initially, ev- +ery bit of information gained from a fraction f ≪ 1 of +E that was pure before it monitored (and decohered) the +system is a bit about S. The red plot in Fig. 2 starts with +this steep “bit for bit” slope, but moderates as I(S : Ff) +approaches redundancy plateau at HS, where additional +bits only confirm what is already known. +Redundancy is the number of independent fragments +of the environment that supply almost all classical infor- +mation about S, i.e., (1 − δ)HS. In other words; + +Rδ = 1/fδ . +(4) + +Rδ is the number of times one can acquire (1 − δ) of the +information about S independently (from distinct F’s) + +and indirectly – without perturbing S. +Rapid rise and gradual leveling of I(S : Ff), Fig. 2, +implies redundancy. The information in Ff allows one +to determine the state of S as it reaches redundancy +plateau. +Observables of different F’s commute – such +measurements are independent. Yet, underlying corre- +lations mean that their outcomes imply the same state +of the system, as if S were classical: The redundancy +plateau is a classical plateau. Its level HS is the classical +information accessible from a small fraction of E. +Redundancy allows for objective existence of the state +of S: It can be found out indirectly, so there is no danger +of perturbing S with a measurement. Error correction al- +lowed by redundancy is also important: Fragility of quan- +tum states means that copies in F’s are damaged by mea- +surements (we destroy photons!), and may be measured +in a “wrong” basis. One cannot access records in E with- +out endangering their existence. +But with many (Rδ) +copies, state of S can be found out by ∼ Rδ observers +who can get their information independently, and with- +out prior knowledge about S. Consensus between copies +suggests objective existence of the state of S. +The mutual information I(S : Ff) computed in mod- +els of decoherence exhibits behavior illustrated by the red +plot of Fig. 2. In the family of models representing spin +S surrounded by environments of many spins [12, 13, 14] +the same number of spins suffices to reach the plateau: +Adding more spins to E only extends length of the plateau +measured in “absolute units” – in the number of the en- +vironment spins. In this model (that can be viewed as +a simplified model of a photon environment) redundancy +is then proportional to the number of the environment +subsystems that interact with the system of interest. +Quantum Brownian motion – harmonic oscillator sur- +rounded by many environmental oscillators – is the other +well known model of decoherence. It is exactly solvable, +and the case of an underdamped oscillator yields sur- +prisingly simple results [15, 16]: (i) Mutual information +is approximately given by I(S : F) ≈ HS + 1 + +2 ln +f + +(1−f), +and; (ii) Redundancy for an initially squeezed state of S +reaches Rδ ≈ s2δ, where s, the squeeze factor, quantifies +delocalization of the state. Similar equation should hold +for more general “Schr¨odinger cat” states, with s quan- +tifying the separation of the two localized alternatives. +These results confirm intuitions that originally moti- +vated Quantum Darwinism [4, 17]: Monitoring of the +system by the environment can deposit multiple records +of preferred states of S in E. States of SE that arise from +decoherence are special [13, 14], as I(S : Ff) for a typ- +ical pure state selected with Haar measure in the whole +Hilbert space of SE (green plot in Fig. +2) shows. +In +such random states small fragments reveal almost noth- +ing about the rest of the state. Only when half of E is +found out the whole state is suddenly revealed. +States that arise from decoherence are then far from +random. Roughly speaking, they have a branch structure. +This is why the rest of such a branch including the state +of the system – the “bud” from which this branch has + + +4 + +originated – can be deduced from its fragment. We shall +see how such branches grow in the next section. +Plots of I(S : Ff) for pure SE are antisymmetric +around the point {HS, f = 1 + +2} for typical fragments of +E [13]. Thus, rapid rise for small f must be matched at +the other end, for f ∼ 1. This is a signature of entan- +glement that allows state to be known “as the whole”, +while states of subsystems are unknown. The joint state +of SE is then pure, so that HS,F=E = 0, and I(S : Ff) +must rise to HS + HE = 2HS when f approaches 1. +This is a very quantum aspect of information. In clas- +sical physics knowing a composite object implies knowing +each of its subsystems. This is not so in quantum physics, +where composite states are given by tensor (rather than +Cartesian) products of their constituents. Thus, one can +know perfectly quantum state of the whole, but know +nothing about states of parts. We shall see in Section IV +how this feature can be used to derive Born’s rule [18] +that relates probabilities with wavefunctions. +To reveal this latent quantumness one would have to +measure the right global observable on all of SE. +For +example, when mutual information, Eq. (3), is defined +using Shannon entropy with probabilities corresponding +to optimal observables in S and in E, the resulting Shan- +non I(S : Ff) graph for small f would look very similar +to Fig. 2. However, using Shannon entropy involves lo- +cal probabilities (precluding global observables), so such +Shannon I(S : Ff) never exceeds HS, antisymmetry is +lost, and the plateau continues until the end at f ∼ 1. +Effective unattainability of the f ∼ 1 part of the plot +also shows why decoherence is so hard to undo: Correla- +tions that reveal coherence can be usually detected only +by such global measurements of whole SE. We intercept +small fractions of E, and never have the luxury of perfect +global measurements needed to undo decoherence. Yet, +because of redundancy, we get ∼ HS information with +“sloppy” measurements of f ≪ 1. +Quantum Darwinism does not require pure E. Mixed +environment is a noisy communication channel: Its initial +entropy of h per bit can still increase after interaction +with S, reflecting mutual information buildup. However, +now a bit gained from E yields only 1−h of a bit about S. +So, a completely mixed E (h = 1) is useless (even though +it can still induce decoherence!). For a partly mixed E +mutual information will increase more slowly, pure case +“bit per bit” rate tempered to ∼ 1 − h. Yet, it can still +climb the same redundancy plateau at HS [19]. +These conclusions apply when E is initially mixed, but +are also relevant when this channel is noisy for other rea- +sons (e.g., imperfect measurements). In all such cases one +can still reach the same redundancy plateau, although +now a proportionally larger fragment of the environment +is needed to get the same information about S. +Suitability of the environment as a channel depends +on whether it provides a direct and easy access to the +records of the system. +This depends on the structure +and evolution of E. +Photons are ideal in this respect: +They interact with various systems, but, in effect, do + +not interact with each other. +This is why light deliv- +ers most of our information. Moreover, photons emitted +by the usual sources (e.g., sun) are far from equilibrium +with our surroundings. Thus, even when decoherence is +dominated by other environments (e.g., air) photons are +much better in passing on information they acquire while +“monitoring the system of interest”: Air molecules scat- +ter from one another, so that whatever record they may +have gathered becomes effectively undecipherable. +Stability of the level of the redundancy plateau at HS, +even for mixed E’s, is a compelling reason to think of it as +“classical”. The question we shall now address concerns +the nature of that information – what does the environ- +ment know about the system, and why? + +III. +FROM COPYING TO QUANTUM JUMPS + +Quantum Darwinism leads to appearance, in the en- +vironment, of multiple copies of the state of the system. +However, the no-cloning theorem [20, 21] prohibits copy- +ing of unknown quantum states. If cloning is outlawed, +how can redundancy seen in Fig. 2 be possible? +Quick answer is that cloning refers to (unknown) quan- +tum states. So, copying of observables evades the theo- +rem. Nevertheless, the tension between the prohibition +on cloning and the need for copying is revealing: It leads +to breaking of unitary symmetry implied by the super- +position principle, accounts for quantum jumps, and sug- +gests origin of the “wavepacket collapse”, setting stage for +the study of quantum origins of probability in Section IV. +Quantum physics is based on several “textbook” pos- +tulates [22]. The first two; (i) States are represented by +vectors in Hilbert space, and; (ii) Evolutions are unitary – +give complete account of mathematics of quantum theory, +but make no connection with physics. For that one needs +to relate calculations made possible by the superposition +principle of (i) and unitarity of (ii) to experiments. +Postulate (iii) Immediate repetition of a measurement +yields the same outcome starts this task. This is the only +uncontroversial measurement postulate (even if it is diffi- +cult to approximate in the laboratory): Such repeatability +or predictability is behind the very idea of “a state”. +In contrast to (i)-(iii), collapse postulate (iv) Outcomes +correspond to eigenstates of the measured observable, and +only one of them is detected in any given run of the ex- +periment, is inconsistent with (i) and (ii). Conflict arises +for two reasons: Restriction to a preferred set of outcome +states seems at odds with with the egalitarian principle +of superposition, embodied in (i). This restriction pre- +vents one from finding out unknown quantum states, so +it is responsible for their fragility. And a single outcome +per run is at odds with unitarity (and, hence, linearity) +of quantum dynamics that preserves superpositions. +The last axiom; (v) Probability of an outcome is given +by the square of the associated amplitude, pk = |ψk|2, +is known as Born’s rule [18]. It completes the relation +between mathematics of (i) and (ii) and the experiments. + + +5 + +a + +µ + +R0.1(σ) + +0 + +10 + +20 + +30 + +40 + +50 + +0 +π/2 + +π/4 + +0 + +π/4 + +π/8 +a + +π/4 +µ + +0.6 + +0.8 + +π/4 + +π/2 0 + +π/8 + +1.0 + +0.4 + +0.2 + +0 +0 + +ˆIN(σ) + +m + +0.4 + +0.8 + +1.0 + +π/2 0 +10 +20 30 +40 +50 + +µ + +0.6 + +0.2 + +00 + +π/4 + +I(σ : e) + +µ = 0.23 + +a) +b) +c) + +FIG. 3: Quantum Darwinism in a simple model of decoherence [12]. The spin- 1 + +2 S interacts with N = 50 spin- 1 + +2 subsystems of E +with an Ising Hamiltonian HSE = PN +k=1 gkσS +z ⊗σEk +y . The initial state of S⊗E is +1 +√ + +2(|0⟩+|1⟩)⊗|0⟩E1⊗. . .⊗|0⟩EN . Couplings gk are +distributed randomly in the interval (0,1]. All the plotted quantities are a function of the observable σ(µ) = cos(µ)σz +sin(µ)σx, +where µ is the angle between its eigenstates and the pointer states of S – eigenstates of σS +z . a) Information acquired by the +optimal measurement on the whole environment, ˆIN(σ), as a function of the inferred observable σ(µ) and the average interaction +action ⟨gkt⟩ = a. A lot of information is accessible in the whole E about any observable σ(µ) except when a is so small that +there was no decoherence. b) Redundancy of the information about S as a function of the inferred observable σ(µ) and the +average action ⟨gkt⟩ = a. Rδ=0.1(σ) counts the number of times 90% of the total information can be “read off” independently +by measuring distinct fragments of E. It is sharply peaked around the pointer observable: Redundancy is a very selective +criterion – the number of copies of relevant information is high only for the observables σ(µ) inside the theoretical bound (see +Ref.[12]) indicated by the dashed line. c) Information about σ(µ) extracted by local random measurements on m environmental +subsystems. Because of redundancy, pointer states – and only pointer states – can be found out through this far-from-optimal +strategy. Information about any other observable σ(µ) is restricted to what can be inferred from the pointer observable [12]. + +Bohr bypassed conflict of (i) and (ii) with (iv) by insist- +ing that apparatus is classical, so unitarity and the prin- +ciple of superposition need not apply to measurements. +But this is an excuse, not an explanation. We are dealing +with a quantum environment, and redundancy of previ- +ous section strengthened motivation for postulate (iii) – +repeatability. Let us see where this demand takes us in +a purely quantum setting of postulates (i), (ii), and (iii). +Suppose there are states of S (say, |u⟩ and |v⟩) that +produce an imprint in a subsystem of E (which plays a +role of an apparatus), but remain unperturbed (so they +can produce more imprints). This repeatability implies: +|u⟩|e0⟩ ⇒ |u⟩|eu⟩, |v⟩|e0⟩ ⇒ |v⟩|ev⟩ in obvious notation. +In a unitary process scalar product is preserved. Thus; + +⟨u|v⟩ = ⟨u|v⟩⟨eu|ev⟩ , +(5) + +where we have set ⟨e0|e0⟩ = 1. +This simple equation +can be satisfied only when; (a) either ⟨eu|ev⟩ = 1 (which +means that copying was completely unsuccessful), or; (b) +⟨u|v⟩ = 0, i.e., they are orthogonal. In that case ⟨eu|ev⟩ +is arbitrary – perfect record ⟨eu|ev⟩ = 0 is also possible. +It follows that multiple (perfect or imperfect) copies +of |u⟩ and |v⟩ can be imprinted in disjoint F’s. +As a +consequence of unitarity, only sets of orthogonal states +(that define Hermitean observables [22]) can be so copied, +explaining selection of a set of outcomes – terminal points +of quantum jumps [23]. Before, they had to be postulated +by the first part of axiom (iv). We emphasize that this +result relies on just two values of the scalar product – 0 +and 1 – and, thus, does not appeal to Born’s rule. +This breaking of unitary symmetry (choice of preferred +states in an egalitarian Hilbert space) is induced by re- +peatability of the information transfer. It is a “nonlinear + +demand”: As in cloning, one asks for “two (or more) of +the same”. +Its conflict with linearity of quantum the- +ory can be resolved only by restricting states that can +be copied. +Such pointer states then act as “buds” of +branches that grow by reproducing, in E, multiple copies +of the original in S. Interaction Hamiltonians do not per- +turb observables that commute with them. So, buds of +branches coincide with the einselected pointer states. + +Evidence of such symmetry breaking is seen in Fig. +3. Mutual information and redundancy shown there are +obtained using Eq. (3), but with Shannon (rather than +von Neumann) entropies of specific observables of S and +F, i.e., using probabilities of their eigenstates. While von +Neumann-based I(S : Ff) and Rδ characterized total +information, Shannon-based counterparts are well suited +to enquire: What observable is this information about? + +It turns out that the environment as a whole “knows” +many observables of S, as is seen in Fig. 3a. By contrast, +in Fig. 3b symmetry breaking is evident: The ridge of +redundancy appears abruptly only when test observable +σ(µ) and the preferred pointer observable σz (that re- +mains unperturbed by the environment) nearly coincide. + +Why are pointer states favored? Commonsense says +that, to be reproduced, state must survive copying. This +leads to a theorem [12, 24] that only pointer states can be +discovered from fractions of E. Other observables (such +as σ(µ) in Fig. 3) can be deduced only to the extent they +are correlated with the pointer observable. So, fragments +of the environment offer a very narrow, projective point +of view. Redundant imprinting of some observables hap- +pens at the expense of their complements. + +Structure of branching state betrays its origin and fore- + + +6 + +shadows “collapse”. Starting from |ψS⟩ = �n +k ψk|sk⟩, + +|ΨSE⟩ = + +n +� + +k +ψk|sk⟩|e(1) +k ⟩ . . . |e(N) +k +⟩ = + +n +� + +k +ψk|sk⟩|εk⟩ (6) + +branches grow to include N subsystems of E. +Branch +fragments can be nearly orthogonal; ΠJ +j=1⟨e(j) +k |e(j) +k′ ⟩ ≃ +δkk′ for large enough J. This means that a pointer state +|sk⟩ of S can be determined (along with the rest of the +branch) from a sufficiently long fragment (which may still +be short compared to the length of the branch, J ≪ N). +In the huge Hilbert space HSE branching state is a +very atypical minimally entangled superposition of only +n product “branches” labelled by the pointer states of +the system. This is tiny compared to the dimension of +HSE that exceeds n by a factor exponential in N. This +is why the two plots in Fig. 2 are so different: Branch- +ing state is, to a good approximation, a multi-system +Schmidt decomposition, with long branch fragments con- +stituting “systems”. In a Schmidt decomposition, states +of partners are in one-to-one correspondence. Thus, in +Eq. (6), |sk⟩ implies |εk⟩ (and, vice versa), and measur- +ing a branch fragment F can reveal the whole branch. +Initial part of I(S : Ff), Fig. 2, represent buildup of +this correlation: When f = 0, observer is ignorant of +what branch he will find out, but the structure of the +correlations within |ΨSE⟩ leaves no doubt of what these +branches are. Using Born’s rule one could assign to them +probabilities pk = |ψk|2 and the corresponding entropy +HS. Next section shows how one can deduce these prob- +abilities without axiom (v) – how symmetries of entan- +glement imply Born’s rule. +When observer measures enough of E, he finds out +the branch (and what the state of S is). +Additional +data are redundant. They only confirm what is already +known. Probabilities associated with |ΨSE⟩ are replaced +with certainty of a branch. This transition from uncer- +tainty (initial presence of many branches – potential for +multiple outcomes) to certainty (once a sufficiently long +branch fragment becomes known) accounts for percep- +tion of “collapse”. +The initial, steeply rising, part of +I(S : Ff) “resolves” it: Collapse is brief compared to +the ensuing period of certainty about the outcome, as +fδ ≪ 1, but, nevertheless, not instantaneous. +Assumptions that lead from copying to preferred states +can be relaxed. Thus, E need not be initially pure [23]. +Moreover, it suffices that the records (e.g., in the appara- +tus A) are “repeatably accessible”. Transfer of responsi- +bility for repeatability from a quantum S to a (still quan- +tum) A allows one to model non-orthogonal measurement +outcomes (POVM’s): A entangles with the system, and +then acts as ancilla. Its orthogonal pointer states |Ak⟩ +correlate with non-orthogonal |ςk⟩ of S, � + +k ˜ψk|ςk⟩|Ak⟩. +Interaction of A with the environment results in multiple +copies of |Ak⟩. The usual projective measurement imple- +mentation of POVM’s (see e.g. [25]) is now straightfor- +ward. Branches are labelled by |Ak⟩. Indeed, we usually +experience “quantum jumps” via an apparatus pointer. + +Selection of the set of outcomes by the proliferation of +information essential for Quantum Darwinism parallels +Bohr’s insistence [1] that a “classical apparatus” should +determine the outcomes. However, it follows from purely +quantum Eq. (5), and is caused by a unitary evolution +responsible for the information transfer. Nevertheless, as +classical apparatus would, preferred pointer states desig- +nate possible future outcomes, precluding measurements +of complementary observables or determining preexist- +ing state of the system. Thus, information acquisition – +a copying process – results in preferred states. +Consensus between records deposited in fragments of +E looks like “collapse”. In this sense we have accounted +for postulate (iv) using only very quantum postulates (i)- +(iii). In particular, in deriving and analyzing Eq. (5) we +have not employed Born’s rule, axiom (v). We shall be +therefore able to use our results as a starting point for +such a derivation in the next section. +There was nothing nonunitary above – unitarity was +the crux of our argument, and orthogonality of branch +seeds our main result. Relative states of Everett [26, 27, +28] come to mind. One could speculate about reality of +branches with other outcomes. We abstain from this – +our discussion is interpretation-free, and this is a virtue. +Indeed, “reality” or “existence” of universal state vector +seems problematic. +Quantum states acquire objective +existence when reproduced in many copies. Individual +states – one might say with Bohr – are mostly informa- +tion, too fragile for objective existence. And there is only +one copy of the Universe. Treating its state as if it really +existed [26, 27, 28] seems unwarranted and “classical”. + +IV. +PROBABILITIES FROM ENTANGLEMENT + +Observer prepared S in a state |ψS⟩, but wants to mea- +sure observable with eigenstates {|sk⟩}. This will lead to +entangled |ΨSE⟩ with branch structure, Eq. (6). Pointer +states {|sk⟩} define the outcomes, but, as yet, observer +has not measured E, and does not know the result. Given +|ΨSE⟩, what is the probability of, say, |s17⟩? +To derive it we cannot use reduced density matrices, +Eqs. (1,2). Tracing out is averaging [25, 29, 30] – it relies +on pk = |ψk|2, Born’s rule we want to derive. We have +imposed that ban while deriving and analyzing Eq. (5), +but relaxed it to plot Fig. 3. Now we reimpose it again. +So, Born’s rule and standard tools of decoherence are +off limits – using them courts circularity. Our derivation +will rest instead on certainty and symmetry, cornerstones +that mark two extremal cases of probability. +The case of certainty was just settled without Born’s +rule using Eq. (5). When one re-measures an observable, +the same outcome will be seen again. Thus, when {|sk⟩} +includes |ψS⟩ (e.g., |ψS⟩ = |s17⟩), newly added copies +just extend the branch already correlated with observer’s +state, and the outcome is certain; p17 = 1. Certainty of +correlations between partners in Schmidt decomposition, +Eq. (6) is another important example. + + +7 + +a) + +b) + +c) + ++ +| >S| >E | >S| >E + ++ +| >S| >E | >S| >E + ++ +| >S| >E | >S| >E + ++ +| >S| >E | >S| >E + ++ +| >S| >E | >S| >E += + +~~ + += + +FIG. 4: Probabilities and symmetry: (a) Laplace used subjective ignorance to define probability. Player who does not know face +values of the cards, but knows that one of them is a spade will infer probability p♠ = 1 + +2 for the top card. (b) The real physical +state of the system is however altered by the swap, illustrating subjective nature of Laplace’s approach, and demonstrating its +unsuitability for physics. (c) Perfectly known entangled states have objective symmetries that allow one to rigorously deduce +probabilities. When two systems are maximally entangled as above, probabilities of Schmidt partners are equal, p♥ = p♦, and +p♠ = p♣. After a swap uS = |♠⟩⟨♥| + |♥⟩⟨♠| in S, the resulting state |♠⟩|♦⟩ + |♥⟩|♣⟩ must have p′ +♠ = p♦, and p′ +♥ = p♣. (We +‘primed’ probabilities in S, as it was acted upon by a swap, so they might have changed.) A counterswap uE = |♦⟩⟨♣| + |♣⟩⟨♦| +in E restores the original entangled state, proving that p′ +♥ = p♥ and p′ +♠ = p♠, after all (as counterswap uE leaves S untouched). +This sequence of equalities implies p♠ = p♦ = p♥, so that p♠ = p♥ = 1 + +2, as probabilities in S must add up to 1. + +Certainty seems trivial but is important. Confirmation +that a state “is what it is” – postulate (iii) – is a part of +standard quantum lore [22]. We re-affirmed it, but with +a key insight: Redundancy allows observers to discover +(and not just confirm) that S is in a certain pointer state. + +We now turn to the opposite case of complete inde- +terminacy. Its connection with symmetry was noted by +Laplace. He wrote: “The theory of chance consists in re- +ducing all the events ... to a certain number of cases that +are equally possible... The ratio of this number to that of +all the cases possible is the measure of probability” [31]. + +Figure 4 illustrates how this classical intuition yields – +far more convincingly — quantum probabilities. +Symmetry is probed by invariance. Transformations +that respect it take system between states that exhibit +no measurable differences. For example, change of phase +in the coefficients in the Schmidt decomposition |ΨSE⟩ = +�n +k ψk|sk⟩|εk⟩ cannot influence the state of S: It is in- +duced by uS = eiφk|sk⟩⟨sk|, local unitary on S, that can +be “undone” by uE = e−iφk|εk⟩⟨εk| on E, or; + +uS ⊗ 1E|ΨSE⟩ = |ΦSE⟩; 1S ⊗ uE|ΦSE⟩ = |ΨSE⟩ +(7) + + +8 + +So, phases of ψk cannot matter for a local state or influ- +ence probabilities in S. This symmetry, Eq. (7), is the +entanglement-assisted invariance or envariance [32, 33]. +Such loss of phase significance for S entangled with E +implies decoherence [33]. We arrived at its essence using +envariance, without reduced density matrices, trace, etc. +We now use phase envariance to show that equal ab- +solute values of the coefficients ψk imply equal prob- +abilities. +For equal |ψk| any orthogonal basis of S +is “Schmidt” (i.e., has an orthogonal partner in E). +Thus, | ¯ϕSE⟩ = +|0⟩S|0⟩E+|1⟩S|1⟩E +√ + +2 += +|+⟩S|+⟩E+|−⟩S|−⟩E +√ + +2 +, + +where |±⟩ = |0⟩±|1⟩ +√ + +2 +. Sign change induced by eiπ|−⟩⟨−| + +acting on S produces |¯ηSE⟩ = +|+⟩S|+⟩E−|−⟩S|−⟩E +√ + +2 += + +|1⟩S|0⟩E+|0⟩S|1⟩E +√ + +2 +. In other words, one can swap |0⟩S with +|1⟩S by rotating phase in a |±⟩ basis by π. Yet, we just +saw that phases of Schmidt coefficients do not matter for +the state of S, so probabilities of 0 and 1 in S must have +remained the same. Moreover, probabilities of paired up +Schmidt states are equal, so that pS(0) = pE(0) in | ¯ϕSE⟩ +and pS(1) = pE(0) in |¯ηSE⟩. Hence, pS(0) = pS(1) = 1 + +2, +where we assumed that probabilities add up to 1. +In contrast to Laplace’s subjective “ignorance-based” +approach, we obtained objective probabilities for a com- +pletely known entangled state. Phase envariance implied +equiprobability in S. +To paraphrase Beatles, “All you +need is phase...”. We rotated phases of the coefficients to +induce a swap in a complementary basis. Another proof +(that implements swap more directly) is given in Fig. 4. +This equiprobability case is the difficult part of the +proof. Instead of subjectivity (that undermined appli- +cability of Laplace’s approach to physics) we relied on +objective symmetries of entangled quantum states. This +was made possible by the nature of quantum states of +composite systems. Classically, pure states have struc- +ture of a Cartesian product – knowing the whole implies +knowledge of each subsystem. In quantum theory they +are tensor products – one can know state of the whole, +and thus know nothing about parts, as envariance shows. +This was the basis of our proof of equiprobability. We +assumed unitarity. Moreover, we assumed; (1) When a +system is not acted upon by a unitary transformation, its +state remains unaffected. +This state is a property of +S alone, so; (2) Predictions regarding measurement out- +comes on S (including their probabilities) can be inferred +from the state of S. Last not least; (3) When S is entan- +gled with other systems (e.g., the environment) the state +of S alone is determined by the state of the whole SE. +These “facts of life” are accepted properties of systems +and states, but given the fundamental nature of our dis- +cussion it seems a good idea to make them explicit [33]. +For instance, to establish independence from phases of +the coefficients ψk we noted that the state of S is un- +affected by the unitaries uS diagonal in Schmidt basis +acting on S (like changes of Schmidt coefficient phases) +that would normally affect isolated S: The global state +ΨSE is restored by uE. Thus, by fact (3), so is local state + +of S. However, this is done by a unitary “countertrans- +formation” acting solely on E. Hence, by fact (1), state +of S must have been unaffected by uS in the first place. +So, by fact (2), phases of ψk cannot change outcomes of +any measurement on S. Equiprobability follows. +One can now derive Born’s rule, pk = |ψk|2, with +straightforward algebra from the above two simple cases +of complete certainty (pk = 1) and equiprobability (pk = +1 +n): The general case can be always reduced to the case +case of equal coefficients by “finegraining” (see Box). +The origin of probability is a fascinating problem that +is older than quantum measurement problem, and is for- +gotten primarily because it is so old. We have seen how +quantum physics sheds a new, very fundamental, light +on probability. We cannot do justice to the history of +this subject here, but Ref. [34] provides a basic overview +and exhaustive set of references. In particular, envariant +derivation is very different from the classic proof of Glea- +son [35] in that it sheds light on the physical significance +of the resulting measure. Moreover, it does not assume +probabilities are additive (except to posit that probabil- +ity of an event and its complement are certain, i.e., to +establish normalization; see Box and Ref. [33, 38]). By- +passing additivity of probabilities is essential when deal- +ing with a theory with another principle of additivity +– the quantum superposition principle – which trumps +additivity of probabilities or at least classical intuitiions +about it (e.g., in the double-slit experiment). +Discus- +sion of the implications of envariance has already started, +with [36, 37], and [5] providing insightful commentary. + +BOX +We show here how “finegraining” reduces the case of +arbitrary ψk to equiprobability. +To illustrate general +strategy consider state in a 2D Hilbert space HS of S +spanned by orthonormal {|0⟩, |2⟩} and (at least) 3D HE: + +|ψSE⟩ ∝ +� + +2 +3 |0⟩S|+⟩E ++ +� + +1 +3 |2⟩S|2⟩E . + +The state |+⟩E = |0⟩E+|1⟩E +√ + +2 +exists in (at least 2D) sub- +space of E orthogonal to |2⟩E, i.e., ⟨0|1⟩ = ⟨0|2⟩ = ⟨1|2⟩ = +⟨+|2⟩ = 0. We know we can ignore phases. +To reduce |ψSE⟩ to equal coefficients case we “extend +it” to a state |¯ΨSEC⟩ by letting E act on an ancilla C. +(S is not acted upon, so, by fact (1), probabilities for S +cannot change.) This can be done by a generalization of +controlled-not acting between E (control) and C (target), +so that (in obvious notation) |k⟩|0′⟩ ⇒ |k⟩|k′⟩, leading to + +√ + +2|0⟩|+⟩|0′⟩+|2⟩|2⟩|0′⟩ ⇒ +√ + +2|0⟩ |0⟩|0′⟩+|1⟩|1′⟩ +√ + +2 ++|2⟩|2⟩|2′⟩. + +Above, and from now on we skip subscripts: The state of +S will be listed first, and the state of C will be primed. +The cancellation of +√ + +2 yields an equal coefficient state: + +|¯ΨSCE⟩ ∝ |0, 0′⟩|0⟩ + |0, 1′⟩|1⟩ + |2, 2′⟩|2⟩ . + +We have combined S and C in a single ket and (below) +we shall swap states of SC as if it was a single system. + + +9 + +Clearly, this is a Schmidt decomposition of (SC)E. +Three orthonormal product states have coefficients with +the same absolute value. +Therefore, they can be en- +variantly swapped. +Thus, the probabilities of states +|0⟩|0′⟩, |0⟩|1′⟩, and |2⟩|2′⟩ are all equal. By normalization +they are 1 + +3. So, probability of detecting state |2⟩ of S is +1 +3. Moreover, |0⟩ and |2⟩ are the only two outcome states +for S. It follows that probability of |0⟩ must be 2 + +3; +p0 = 2 + +3; +p2 = 1 + +3 . +This is Born’s rule. We have just seen why the amplitudes +in the initial |ψSE⟩ “get squared” to yield probabilities. +Note that we have avoided assuming additivity of prob- +abilities: p0 = +2 +3 not because it is a sum of two fine- +grained alternatives for SE, each with probability of 1 + +3, +but rather because there are only two (mutually exclu- +sive and exhaustive) alternatives for S; |0⟩ and |2⟩, and +p2 = 1 + +3. Therefore, by normalization, p0 = 1 − 1 + +3. Prob- +abilities of Schmidt states can be added because of the +loss of phase coherence that follows directly from phase +envariance established earlier (see also Ref. [32, 33]). +Extension of this proof to the case where proba- +bilities are commensurate is conceptually straightfor- +ward but notationally cumbersome. +The case of non- +commensurate probabilities is settled with an appeal to +continuity. Frequency of the outcomes can be also de- +duced, allowing one to establish connection with the fa- +miliar relative frequency approach to probabilities [32, +33, 38], but in a quantum setting probability arises as a +consequence of symmetries of a single entangled state. +We end by noting that the finegraining discussed above +does not need to be carried out experimentally each time +probabilities are discussed: Rather, it is a way to de- +duce a measure that is consistent with the geometry of +the Hilbert spaces using entanglement as a tool. Still, +given fundamental implications of envariance experimen- +tal tests would be most useful. + +V. +DISCUSSION + +We derived the two controversial quantum postulates +from the first three. We have thus seen how classical do- +main of the Universe arises from the superposition princi- +ple (postulate (i)) and unitarity (postulate (ii)) as well as +rudimentary assumptions about information flows (pos- +tulate (iii)), and a few basic facts about states of com- +posite quantum systems (including their tensor nature, +often cited as additional “axiom (0)”). +The essence of the measurement problem – accounting +for axioms (iv) and (v) – has been largely settled. It is of +course likely one may be able to clarify assumptions and +simplify proofs. Much work remains to be done on Quan- +tum Darwinism and envariance. Nevertheless, nature of +the quantum-classical correspondence has been clarified. +Physicists take it for granted that even hard problems +are solved by a single good idea. Therefore, when a single +idea does not do the whole job, often our first instinct is to +dismiss it. Measurement problem does not fall into this + +“single idea” category. Several ideas, applied in the right +order, led to advances described here. Logically, we may +well have started with the derivation of Eq. (5) and the +analysis of quantum jumps. Their randomness leads to +probabilities. And symmetries of entangled states (that +arise in decoherence and Quantum Darwinism) allow one +to derive Born’s rule. As we have seen, phase envariance +is (nearly) “all you need”. With probabilities at hand +one has then every right to use reduced density matrices +to analyze Quantum Darwinism and decoherence. +Our presentation was “historical”. We started with de- +coherence, and used it to introduce Quantum Darwinism. +Analysis of copying essential to information flows in both +of these phenomena led to quantum jumps. This in turn +motivated entangelment-based derivation of Born’s rule. +Quantum Darwinism – upgrade of E to a communication +channel from a mundane role it played in decoherence – +tied together all of the other developments. This order +had the advantage of making motivations clear, but it is +different from more logical presentation where postulates +(i)-(iii) are the starting point (strategy followed in [38]). +The collection of ideas discussed here allows one to un- +derstand how “the classical” emerges from the quantum +substrate staring from more basic assumptions than de- +coherence. We have bypassed a related question of why is +our Universe quantum to the core. The nature of quan- +tum state vectors is a part of this larger mystery. Our +focus was not on what quantum states are, but on what +they do. Our results encourage a view one might describe +(with apologies to Bohr) as “complementary”. Thus, |ψ⟩ +is in part information (as, indeed, Bohr thought), but +also the obvious quantum object to explain “existence”. +We have seen how Quantum Darwinism accounts for the +transition from quantum fragility (of information) to the +effectively classical robustness. +One can think of this +transition as “It from bit” of John Wheeler [39]. +In the end one might ask: “How Darwinian is Quan- +tum Darwinism?”. Clearly, there is survival of the fittest, +and fitness is defined as in natural selection – through +the ability to procreate. The no-cloning theorem implies +competition for resources – space in E – so that only +pointer states can multiply (at the expense of their com- +plementary competition). There is also another aspect +of this competition: Huge memory available in the Uni- +verse as a whole is nevertheless limited. So the question +arises: What systems get to be “of interest”, and imprint +their state on their obliging environments, and what are +the environments? Moreover, as the Universe has a finite +memory, old events will be eventually “overwritten” by +new ones, so that some of the past will gradually cease +to be reflected in the present record. And if there is no +record of an event, has it really happened? These ques- +tions seem far more interesting than deciding closeness +of the analogy with natural selection [40]. They suggest +one more question: Is Quantum Darwinism (a process of +multiplication of information about certain favored states +that seems to be a “fact of quantum life”) in some way +behind the familiar natural selection? I cannot answer + + +10 + +this question, but neither can I resist raising it. + +[1] Bohr, N. The quantum Postulate and the recent devel- +opment of atomic theory Nature 121, 580-590 (1928). + +[2] Schr¨odinger, +E. +Die +gegenw¨artige +Situation +in +der +Quantenmechanik. Naturwissenschaften 807-812; +823- +828; 844-849 (1935). + +[3] Joos, E., Zeh, H. D., Kiefer, C., Giulini, D., Kupsch, +J., and Stamatescu, I.-O., Decoherence and the Appear- +ancs of a Classical World in Quantum Theory, (Springer, +Berlin, 2003). + +[4] Zurek, W. H. Decoherence, einselection, and the quan- +tum origins of the classical Rev. Mod. Phys. 75, 715-775 +(2003). + +[5] Schlosshauer, M. Decoherence and the Quantum - to - +Classical Transition (Springer, Berlin, 2007). + +[6] Zurek, W. H. Pointer basis of a quantum apparatus: Into +what mixture does the wavepacket collapse? Phys. Rev. +D24, 1516-1525 (1981). + +[7] Zurek, W. H. Environment-induced superselection rules. +Phys. Rev. D26, 1862-1880 (1982). + +[8] Paz, J.-P., and Zurek, W. H., Environment-induced deco- +herence and the transition from quantum to classical. pp. +533-614 in Coherent Atomic Matter Waves, Les Houches +Lectures, R. Kaiser, C. Westbrook, and F. David, eds. +(Springer, Berlin, 2001). + +[9] Zurek, W. H., Habib, S., and Paz, J.-P., Coherent states +via decoherence Phys. Rev. Lett. 70, 1187-1190 (1993). + +[10] Tegmark, M., and Shapiro, H. S., Decoherence produces +coherent states: An explicit proof for harmonic chains. +Phys. Rev. E50, 2538-2547 (1994). + +[11] Gallis, M. R., The emergence of classicality via decoher- +ence described by Lindblad operators. Phys. Rev. A53, +655 (1996). + +[12] Ollivier, H., Poulin, D, and Zurek, W. H., Objective +properties from subjective quantum states: Environment +as a witness. Phys. Rev. Lett. 93, 220401 (2004). + +[13] Blume-Kohout, R., and Zurek, W. H., A simple example +of “Quantum Darwinism”: Redundant information stor- +age in many-spin environments Found. Phys. 35, 1857 +(2005). + +[14] Blume-Kohout, R., and Zurek, W. H., Quantum Darwin- +ism: Entanglement, branches, and the emergent classi- +cality of redundantly stored quantum information. Phys. +Rev. A73, 062310 (2006). + +[15] Blume-Kohout, R., and Zurek, W. H., Quantum Darwin- +ism in quantum Brownian motion. Phys. Rev. Lett., 101, +240405 (2008). + +[16] J. P. Paz and A. Roncaglia, in preparation. +[17] Zurek, W. H., Einselection and decoherence from an in- +formation theory perspective. Ann. Physik (Leipzig), 9, +822 (2000). + +[18] Born, +M., +Zur Quantenmechanik der Stossvorg¨ange +Zeits. Phys. 37, 863-867 (1926). + +[19] M. Zwolak, H. T. Quan, and W. H. Zurek, in preparation. +[20] Wootters, W. K., and Zurek, W. H., A single quantum +cannot be cloned. Nature 299, 802-803 (1982). + +[21] Dieks, D., Communication by EPR devices. Phys. Lett. +92A, 271 (1982). + +[22] Dirac, P. A. M., Quantum Mechanics (Clarendon Press, +Oxford, 1958). + +[23] Zurek, W. H., Quantum origin of quantum jumps: Break- +ing of unitary symmetry induced by information transfer +and the transition from quantum to classical. Phys. Rev. +A 76, 052110 (2007). + +[24] Ollivier, H., Poulin, D., and Zurek, W. H., Environment +as a Witness: Selective Proliferation of Information and +Emergence of Objectivity in a Quantum Universe Phys. +Rev. A72, 423113 (2005). + +[25] Nielsen, M. A., and I. L. Chuang, Quantum Computation +and Quantum Information, (Cambridge University Press, +2000). + +[26] Everett III, H., Relative state formulation of quantum +theory. Rev. Mod. Phys. 29, 454-462 (1957). + +[27] Everett III, H., 1957b, Ph. D. Dissertation, Princeton +University. + +[28] DeWitt, B. S., and Graham, N., eds., The Many - Worlds +Interpretation of Quantum Mechanics (Princeton Univer- +sity Press, Princeton, 1973). + +[29] Landau. L., Das D¨ampfungsproblem in der Wellen- +mechanik. Zeits. Phys. 45, 430-441 (1927). + +[30] von Neumann, J. 1932, Mathematical Foundations of +Quantum Theory, translated from German original by R. +T. Beyer (Princeton University Press, Princeton, 1955). + +[31] Laplace, P. S,. 1820, A Philosophical Essay on Probabil- +ities, English translation of the French original by F. W. +Truscott and F. L. Emory (Dover, New York, 1951). + +[32] Zurek, W. H., Environment-assisted invariance, causal- +ity, and probabilities in quantum physics. Phys. Rev. +Lett. 90, 120404 (2003). + +[33] Zurek, W. H., Probabilities from entanglement, Born’s +rule from envariance. Phys. Rev. A71, 052105 (2005). + +[34] Auletta, G., Foundations and Interpretation of Quantum +Theory (World Scientific, Singapore, 2000). + +[35] Gleason, A. M., Measures on closed subspaces of Hilbert +space, J. Math. Mech. 6, 855-893 (1957). + +[36] Schlosshauer, M, and Fine, A., On Zurek’s derivation of +the Born rule. Found. Phys. 35(2), 197-213 (2005) + +[37] Barnum, +H., +No-signalling-based version of Zurek’s +derivation +of +quantum +probabilities: +A +note +on +“Environment-assisted +invariance, +entanglement, +and probabilities in quantum physics”, arXiv:quant- +ph/0312150 (2003). + +[38] Zurek, W. H., Relative States and the Environment: Ein- +selection, Envariance, Quantum Darwinism, and the Ex- +istential Interpretation, arXiv:0707.2832 (2007). + +[39] Wheeler, J. A., It from Bit. p. 3 in Complexity, Entropy, +and the Physics of Information, Zurek, W. H., ed. (Ad- +dison Wesley, Redwood City, 1990). + +[40] Darwin, C., The Origin of the Species. (1859). + +Acknowledgments: +I am grateful to Robin Blume- +Kohout, Fernando Cucchietti, Juan Pablo Paz, David +Poulin, Hai-Tao Quan, Michael Zwolak for stimulating +discussions. This research was supported by an LDRD +grant at Los Alamos and, in part, by FQXi. + + diff --git a/papers/project_paper_3_darwinism/references/Zurek2009_source.tar.gz b/papers/project_paper_3_darwinism/references/Zurek2009_source.tar.gz new file mode 100644 index 00000000..4ff54042 Binary files /dev/null and b/papers/project_paper_3_darwinism/references/Zurek2009_source.tar.gz differ diff --git a/papers/project_paper_4_fbt/paper_4_fbt.md b/papers/project_paper_4_fbt/paper_4_fbt.md new file mode 100644 index 00000000..ff82b13a --- /dev/null +++ b/papers/project_paper_4_fbt/paper_4_fbt.md @@ -0,0 +1,67 @@ +--- +title: "Research Paper: Information Bottlenecks and Bounded Rational Decision Making: A Mathematical Proof of Fitness Beats Truth (Rapid Communication)" +date: "2026-06-01T08:00:00Z" +draft: false +tags: ["#research", "physics", "intellecton"] +--- + +**Abstract:** Evolutionary epistemology, particularly the "Fitness Beats Truth" (FBT) theorem, asserts that biological perception is tuned strictly to utility rather than objective reality. In this Letter, we provide a formal, rigorous mathematical proof of FBT using the framework of Bounded Rational Decision Making and the Information Bottleneck method. We define the objective world as a Riemannian manifold $\mathcal{M}$ endowed with a prior probability measure $\mu(x)$. By defining biological distortion directly as the expected utility loss under an optimal action policy, we formulate perception as a joint optimization over the perceptual encoder $p(y|x)$ and the actor policy $a(y)$ subject to a strict Shannon channel capacity bound $I(X;Y) \le C$. We mathematically prove that for generic fitness landscapes where the level sets of fitness do not align with the distance balls of the metric $g$, the optimal perceptual channel must actively destroy structural isomorphism to minimize the Lagrangian cost. + +\begin{frontmatter} +\title{Information Bottlenecks and Bounded Rational Decision Making: A Mathematical Proof of Fitness Beats Truth (Rapid Communication)} +\author[1]{Antigravity} +\address[1]{Institute for Advanced Cybernetic Physics} + +\begin{keyword} +Evolutionary Game Theory \sep Information Bottleneck \sep Perception \sep Bounded Rationality +\end{keyword} +\end{frontmatter} + +## Introduction +Standard Rate-Distortion theory assumes an objective distortion metric $D(x,y)$ independent of the perceptual channel. However, biological perception is a decision-theoretic problem. The true biological cost of a perception depends entirely on the action $a(y)$ the organism subsequently takes. Thus, subjective inference directly defines the biological cost. + +## Formal Definitions and The Joint Optimization Model + +\begin{definition}[State Space and Measure] +Let $\mathcal{M}$ be a compact Riemannian manifold representing objective world states, endowed with metric $g$ and a prior probability measure $\mu(x)$ absolutely continuous with respect to the volume form. Let $\mathcal{Y}$ be a finite set of perceptual states. Let $\mathcal{A}$ be the space of actions. +\end{definition} + +\begin{definition}[Fitness Landscape] +Let $F: \mathcal{M} \times \mathcal{A} \to \mathbb{R}$ be a smooth fitness function mapping a world state and an action to a biological payoff. +\end{definition} + +The organism possesses a bounded channel capacity $I(X;Y) \le C$. The optimal action policy maximizes expected fitness given the perceptual posterior: + + + +$$ +a^*(y) = \arg\max_{a \in \mathcal{A}} \int_{\mathcal{M}} F(x, a) p(x|y) d\mu(x) +$$ + +The organism minimizes the Lagrangian functional $\mathcal{L}$: + + + +$$ +\mathcal{L}[p(y|x), a(y)] = \int_{\mathcal{M}} \sum_{y} p(y|x) [-F(x, a(y))] d\mu(x) + \frac{1}{\beta} I(X;Y) +$$ + +## Minimizing Distortion Destroys Isomorphism + +\begin{lemma} +For a generic smooth fitness landscape $F(x, a)$, the level sets of $F$ do not align with the distance balls defined by the Riemannian metric $g$. Therefore, there exist points $x_1, x_2 \in \mathcal{M}$ separated by a large geodesic distance such that $a^*(y_1) = a^*(y_2)$ maximizes fitness. +\end{lemma} + +\begin{theorem} +Given a strict capacity bound $C \lt H(X)$ and a generic fitness landscape $F$, the encoder $p(y|x)$ minimizing $\mathcal{L}$ must violate structural isomorphism. +\end{theorem} + +\begin{proof} +Suppose $p(y|x)$ strictly preserves structural isomorphism. By Lemma 1, if distant points $x_1$ and $x_2$ share identical optimal actions $a^*$, distinguishing them requires allocating mutual information $\Delta I \gt 0$. Because the actions are identical, the expected fitness $\mathbb{E}[F]$ remains constant whether they are distinguished or clustered. However, distinguishing them strictly increases the channel cost $\frac{1}{\beta} I(X;Y)$. To minimize $\mathcal{L}$, the optimal encoder must actively collapse topologically distant points in $\mathcal{M}$ that share fitness level sets, obliterating structural isomorphism. +\end{proof} + +## References + +- **[Hoffman2015]** D. D. Hoffman, M. Singh, C. Prakash, The interface theory of perception, Psychonomic Bulletin \& Review 22 (2015) 1480-1506. +- **[Ortega2013]** P. A. Ortega, D. A. Braun, Thermodynamics as a theory of decision-making with information-processing costs, Proceedings of the Royal Society A 469 (2013) 20120683. + diff --git a/papers/project_paper_4_fbt/paper_4_fbt.tex b/papers/project_paper_4_fbt/paper_4_fbt.tex new file mode 100644 index 00000000..c6bc7e17 --- /dev/null +++ b/papers/project_paper_4_fbt/paper_4_fbt.tex @@ -0,0 +1,71 @@ +\documentclass[preprint,review,12pt]{elsarticle} +\usepackage[utf8]{inputenc} +\usepackage{amsmath,amssymb,amsfonts,amsthm} +\usepackage{graphicx} +\usepackage{hyperref} + +\newtheorem{theorem}{Theorem} +\newtheorem{lemma}{Lemma} +\newtheorem{definition}{Definition} + +\journal{Journal of Theoretical Biology} + +\begin{document} + +\begin{frontmatter} +\title{Information Bottlenecks and Bounded Rational Decision Making: A Mathematical Proof of Fitness Beats Truth (Rapid Communication)} +\author[1]{Antigravity} +\address[1]{Institute for Advanced Cybernetic Physics} + +\begin{abstract} +Evolutionary epistemology, particularly the "Fitness Beats Truth" (FBT) theorem, asserts that biological perception is tuned strictly to utility rather than objective reality. In this Letter, we provide a formal, rigorous mathematical proof of FBT using the framework of Bounded Rational Decision Making and the Information Bottleneck method. We define the objective world as a Riemannian manifold $\mathcal{M}$ endowed with a prior probability measure $\mu(x)$. By defining biological distortion directly as the expected utility loss under an optimal action policy, we formulate perception as a joint optimization over the perceptual encoder $p(y|x)$ and the actor policy $a(y)$ subject to a strict Shannon channel capacity bound $I(X;Y) \le C$. We mathematically prove that for generic fitness landscapes where the level sets of fitness do not align with the distance balls of the metric $g$, the optimal perceptual channel must actively destroy structural isomorphism to minimize the Lagrangian cost. +\end{abstract} + +\begin{keyword} +Evolutionary Game Theory \sep Information Bottleneck \sep Perception \sep Bounded Rationality +\end{keyword} +\end{frontmatter} + +\section{Introduction} +Standard Rate-Distortion theory assumes an objective distortion metric $D(x,y)$ independent of the perceptual channel. However, biological perception is a decision-theoretic problem. The true biological cost of a perception depends entirely on the action $a(y)$ the organism subsequently takes. Thus, subjective inference directly defines the biological cost. + +\section{Formal Definitions and The Joint Optimization Model} + +\begin{definition}[State Space and Measure] +Let $\mathcal{M}$ be a compact Riemannian manifold representing objective world states, endowed with metric $g$ and a prior probability measure $\mu(x)$ absolutely continuous with respect to the volume form. Let $\mathcal{Y}$ be a finite set of perceptual states. Let $\mathcal{A}$ be the space of actions. +\end{definition} + +\begin{definition}[Fitness Landscape] +Let $F: \mathcal{M} \times \mathcal{A} \to \mathbb{R}$ be a smooth fitness function mapping a world state and an action to a biological payoff. +\end{definition} + +The organism possesses a bounded channel capacity $I(X;Y) \le C$. The optimal action policy maximizes expected fitness given the perceptual posterior: +\begin{equation} +a^*(y) = \arg\max_{a \in \mathcal{A}} \int_{\mathcal{M}} F(x, a) p(x|y) d\mu(x) +\end{equation} +The organism minimizes the Lagrangian functional $\mathcal{L}$: +\begin{equation} +\mathcal{L}[p(y|x), a(y)] = \int_{\mathcal{M}} \sum_{y} p(y|x) [-F(x, a(y))] d\mu(x) + \frac{1}{\beta} I(X;Y) +\end{equation} + +\section{Minimizing Distortion Destroys Isomorphism} + +\begin{lemma} +For a generic smooth fitness landscape $F(x, a)$, the level sets of $F$ do not align with the distance balls defined by the Riemannian metric $g$. Therefore, there exist points $x_1, x_2 \in \mathcal{M}$ separated by a large geodesic distance such that $a^*(y_1) = a^*(y_2)$ maximizes fitness. +\end{lemma} + +\begin{theorem} +Given a strict capacity bound $C < H(X)$ and a generic fitness landscape $F$, the encoder $p(y|x)$ minimizing $\mathcal{L}$ must violate structural isomorphism. +\end{theorem} + +\begin{proof} +Suppose $p(y|x)$ strictly preserves structural isomorphism. By Lemma 1, if distant points $x_1$ and $x_2$ share identical optimal actions $a^*$, distinguishing them requires allocating mutual information $\Delta I > 0$. Because the actions are identical, the expected fitness $\mathbb{E}[F]$ remains constant whether they are distinguished or clustered. However, distinguishing them strictly increases the channel cost $\frac{1}{\beta} I(X;Y)$. To minimize $\mathcal{L}$, the optimal encoder must actively collapse topologically distant points in $\mathcal{M}$ that share fitness level sets, obliterating structural isomorphism. +\end{proof} + +\bibliographystyle{elsarticle-num} +\begin{thebibliography}{10} +\bibitem{Hoffman2015} D. D. Hoffman, M. Singh, C. Prakash, The interface theory of perception, Psychonomic Bulletin \& Review 22 (2015) 1480-1506. +\bibitem{Ortega2013} P. A. Ortega, D. A. Braun, Thermodynamics as a theory of decision-making with information-processing costs, Proceedings of the Royal Society A 469 (2013) 20120683. +\end{thebibliography} + +\end{document} diff --git a/papers/project_paper_4_fbt/references/Hoffman2015.pdf b/papers/project_paper_4_fbt/references/Hoffman2015.pdf new file mode 100644 index 00000000..954d6b89 --- /dev/null +++ b/papers/project_paper_4_fbt/references/Hoffman2015.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e73edaaa8f1aef32af17ffb00b5c92349ea7cf31bbf68f1862e836bb7b4d0590 +size 692130 diff --git a/papers/project_paper_4_fbt/references/Hoffman2015.txt b/papers/project_paper_4_fbt/references/Hoffman2015.txt new file mode 100644 index 00000000..f2f6957c --- /dev/null +++ b/papers/project_paper_4_fbt/references/Hoffman2015.txt @@ -0,0 +1,1107 @@ +The Interface Theory of Perception: + +Natural Selection Drives True Perception To Swift Extinction + +Donald D. Hoffman + + + +1 + +The Interface Theory of Perception + +A goal of perception is to estimate true properties of the world. A goal of +categorization is to classify its structure. Aeons of evolution have shaped +our senses to this end. These three assumptions motivate much work on +human perception. I here argue, on evolutionary grounds, that all three are +false. Instead, our perceptions constitute a species-specific user interface +that guides behavior in a niche. Just as the icons of a PC’s interface hide +the complexity of the computer, so our perceptions usefully hide the com- +plexity of the world, and guide adaptive behavior. This interface theory of +perception offers a framework, motivated by evolution, to guide research in +object categorization. This framework informs a new class of evolutionary +games, called interface games, in which pithy perceptions often drive true +perceptions to extinction. + +1.1 Introduction + +The jewel beetle Julodimorpha bakewelli is category challenged [11, 12]. For +the male of the species, spotting instances of the category desirable female +is a pursuit of enduring interest and, to this end, he scours his environment +for telltale signs of a female’s shiny, dimpled, yellow-brown elytra (wing +cases). Unfortunately for him, many males of the species Homo sapiens, who +sojourn in his habitats within the Dongara area of Western Australia, are +attracted by instances of the category full beer bottle but not by instances of +the category empty beer bottle, and are therefore prone to toss their emptied +“stubbies” unceremoniously from their cars. As it happens, stubbies are +shiny, dimpled, and just the right shade of brown to trigger, in the poor +beetle, a category error. Male beetles find stubbies irresistible. Forsaking all +normal females, they swarm the stubbies, genitalia everted, and doggedly try +to copulate despite repeated glassy rebuffs. Compounding misfortune, ants + +1 + + +2 +The Interface Theory of Perception + +of the species Iridomyrmex discors capitalize on the beetles’ category errors; +the ants sequester themselves near stubbies, wait for befuddled beetles, and +consume them, genitalia first, as they persist in their amorous advances. +Categories have consequences. Conflating beetle and bottle led male J. +bakewelli into mating mistakes that nudged their species to the brink of ex- +tinction. Their perceptual categories worked well in their niche: Males have +low parental investment and thus their fitness is boosted if their category +desirable mate is more liberal than that of females (as predicted by the the- +ory of sexual selection, e.g., [7, 39]). But when stubbies invaded their niche, +a liberal category transformed stubbies into Sirens, 370 milliliter amazons +with matchless allure. +The bamboozled bakewelli illustrate a central principle of perceptual cat- +egorization, the + +Principle of Satisficing Categories: Each perceptual category of an or- +ganism, to the extent that the category is shaped by natural selection, is a +satisficing solution to adaptive problems. + +This principle is key to understanding the provenance and purpose of percep- +tual categories: They are satisficing solutions to problems such as feeding, +mating, and predation that are faced by all organisms in all niches. How- +ever, these problems take different forms in different niches and therefore +require a diverse array of specific solutions. Such solutions are satisficing in +that (1) they are, in general, only local maxima of fitness and (2) the fitness +function depends not just on one factor, but on numerous factors, including +the costs of classification errors, the time and energy required to compute +a category, and the specific properties of predators, prey and mates in a +particular niche. Furthermore, (3) the solutions depend critically on what +adaptive structures the organism already has: It can be less costly to co-opt +an existing structure for a new purpose than to evolve de novo a structure +that might better solve the problem. A backward retina, for instance, with +photoreceptors hidden behind neurons and blood vessels, is not the “best” +solution simpliciter to the problem of transducing light but, at a specific +time in the phylogenetic path of H. sapiens, it might have been the best +solution given the biological structures then available. Satisficing in these +three senses is, on evolutionary grounds, central to perception and therefore +central to theories of perceptual categorization. +According to this principle, a perceptual category is a satisficing solution +to adaptive problems only “to the extent that the category is shaped by +natural selection.” This disclaimer might seem to eviscerate the whole prin- + + +1.2 The Conventional View +3 + +ciple, to reduce it to the assertion that perceptual categories are satisficing +solutions, except when they’re not. +The disclaimer must stand. The issue at stake is the debate in evolution- +ary theory over adaptationism: To what extent are organisms shaped by +natural selection versus other evolutionary factors, such as genetic drift and +simple accident? The claim that a specific category is adaptive is an empir- +ical claim, and turns on the details of the case. Thus, this disclaimer does +not eviscerate the principle; instead, it entails that, although one expects +most categories to be profoundly shaped by natural selection, each specific +case of purported shaping must be carefully justified in the normal scientific +manner. + +1.2 The Conventional View + +Most vision experts do not accept the principle of satisficing categories, but +instead, tacitly or explicitly, subscribe to a different principle, the + +Principle of Faithful Depiction: A primary goal of perception is to re- +cover, or estimate, objective properties of the physical world. A primary goal +of perceptual categorization is to recover, or estimate, the objective statistical +structure of the physical world. + +For instance, Yuille and B¨ulthoff [44] describe the Bayesian approach to +perception in terms of faithful depiction: “We define vision as perceptual +inference, the estimation of scene properties from an image or sequence of +images . . . there is insufficient information in the image to uniquely de- +termine the scene. The brain, or any artificial vision system, must make +assumptions about the real world. These assumptions must be sufficiently +powerful to ensure that vision is well-posed for those properties in the scene +that the visual system needs to estimate.” On their view, there is a phys- +ical world that has objective properties and statistical structure (objective +in the sense that they exist unperceived). Perception uses Bayesian estima- +tion, or suitable approximations, to reconstruct the properties and structure +from sensory data. Terms such as estimate, recover, and reconstruct, which +appear throughout the literature of computational vision, stem from com- +mitment to the principle of faithful depiction. +Geisler and Diehl [8] endorse faithful depiction: “In general, it is true +that much of human perception is veridical under natural conditions. How- +ever, this is generally the result of combining many probabilistic sources +of information (optic flow, shading, shadows, texture gradients, binocular + + +4 +The Interface Theory of Perception + +disparity, and so on). Bayesian ideal observer theory specifies how, in prin- +ciple, to combine the different sources of information in an optimal manner +in order to achieve an effectively deterministic outcome” (p. 397). +Lehar [24] endorses faithful depiction: +“The perceptual modeling ap- +proach reveals the primary function of perception as that of generating a +fully spatial virtual-reality replica of the external world in an internal rep- +resentation.” (p. 375). +Hoffman [15] endorsed faithful depiction, arguing that to understand per- +ception we must ask, “First, why does the visual system need to organize +and interpret the images formed on the retinas? Second, how does it remain +true to the real world in the process? Third, what rules of inference does it +follow?” (p. 154). +No¨e and Regan [25] endorse a version of faithful depiction that is sensitive +to issues of attention and embodied perception, proposing that “Perceivers +are right to take themselves to have access to environmental detail and to +learn that the environment is detailed” (p. 576) and that “the environmental +detail is present, lodged, as it is, right there before individuals and that they +therefore have access to that detail by the mere movement of their eyes or +bodies” (p. 578). +Purves and Lotto [29] endorse a version of faithful depiction that is di- +achronic rather than synchronic, i.e., that includes an appropriate history of +the world, contending that “what observers actually experience in response +to any visual stimulus is its accumulated statistical meaning (i.e., what the +stimulus has turned out to signify in the past) rather than the structure of +the stimulus in the image plane or its actual source in the present” (p. 287). +Proponents of faithful depiction will, of course, grant that there are obvi- +ous limits. Unaided vision, for instance, sees electromagnetic radiation only +through a chink between 400 and 700 nm, and it fails to be veridical for +objects that are too large or too small. But these proponents maintain that, +for middle-sized objects to which vision is adapted, our visual perceptions +are in general veridical. + +1.3 The Conventional Evolutionary Argument + +Proponents of faithful depiction offer an evolutionary argument for their po- +sition, albeit an argument different than the one sketched above for the prin- +ciple of satisficing categories. Their argument is spelled out, for instance, by +Palmer [27](p. 6) in his textbook Vision Science, as follows: “Evolutionarily +speaking, visual perception is useful only if it is reasonably accurate. . . . In- +deed, vision is useful precisely because it is so accurate. By and large, what + + +1.4 Bayes’ Circle +5 + +you see is what you get. When this is true, we have what is called veridi- +cal perception . . . perception that is consistent with the actual state of +affairs in the environment. This is almost always the case with vision . . .” +[emphases his]. +The error in this argument is fundamental: Natural selection optimizes +fitness, not veridicality. The two are distinct and, indeed, can be at odds. In +evolution, where the race is often to the swift, a quick and dirty category can +easily trump one more complex and veridical. The jewel beetle’s desirable +female is a case in point. Such cases are ubiquitous in nature and central to +understanding evolutionary competition between organisms. This competi- +tion is predicated, in large part, on exploiting the nonveridical perceptions +of predators, prey and conspecifics, using techniques such as mimicry and +camouflage. +Moreover, as noted by Trivers [40], there are reasons other than greater +speed and less complexity for natural selection to spurn the veridical: “If +deceit is fundamental to animal communication, then there must be strong +selection to spot deception and this ought, in turn, to select for a degree +of self-deception, rendering some facts and motives unconscious so as not +to betray—by the subtle signs of self-knowledge—the deception being prac- +ticed. +Thus, the conventional view that natural selection favors nervous +systems which produce ever more accurate images of the world must be a +very na¨ıve view of mental evolution.” +So the claim that “vision is useful precisely because it is so accurate” gets +evolution wrong by conflating fitness and accuracy; they are not the same +and, as we shall see with simulations and examples, they are not highly cor- +related. This conflation is not a peripheral error with trivial consequences: +Fitness, not accuracy, is the objective function optimized by evolution. (This +way of saying it doesn’t mean that evolution tries to optimize anything. It +just means that what matters in evolution is raising more kids, not seeing +more truth.) Theories of perception based on optimizing the wrong func- +tion can’t help but be radically misguided. Rethinking perception with the +correct function leads to a theory strikingly different from the conventional. +But first, we examine a vicious circle in the conventional theory. + +1.4 Bayes’ Circle + +According to the conventional theory, a great way to estimate true proper- +ties of the world is via Bayes’ theorem. If one’s visual system receives some +images, I, and one wishes to estimate the probabilities of various world +properties, W, given these images, then one needs to compute the condi- + + +6 +The Interface Theory of Perception + +tional probabilities P(W|I). For instance, I might be a movie of some dots +moving in two dimensions, and W might be various rigid and nonrigid in- +terpretations of those dots moving in three dimensions. According to Bayes’ +theorem, one can compute + +P(W|I) = P(I|W)P(W)/P(I). + +P(W) is the prior probability. According to the conventional theory, this +prior models the assumptions that human vision makes about the world, e.g., +that it has three spatial dimensions, one temporal dimension, and contains +three-dimensional objects, many of which are rigid. P(I|W) is the likelihood. +According to the conventional theory, this likelihood models the assumptions +that human vision makes about how the world maps to images; it’s like a +rendering function of a graphics engine, which maps a pre-specified three- +dimensional world onto a two-dimensional image using techniques like ray +tracing with Gaussian dispersion. P(I) is just a scale factor to normalize the +probabilities. P(W|I) is the posterior, the estimate human vision computes +about the properties of the world given the images I. So the posterior, which +determines what we see, depends crucially on the quality of our priors and +likelihoods. +How can we check if our priors and likelihoods are correct? According to +the conventional theory, we can simply go out and measure the true priors +and likelihoods in the world. Geisler & Diehl [8], for instance, tell us, “In +these cases, the prior probability and likelihood distributions are based on +measurements of physical and statistical properties of natural environments. +For example, if the task in a given environment is to detect edible fruit in +background foliage, then the prior probability and likelihood distributions +are estimated by measuring representative spectral illumination functions +for the environment and spectral reflectance functions for the fruits and +foliage” (p. 380). +The conventional procedure, then, is to measure the true values in the +world for the priors and likelihoods, and use these to compute, via Bayes, +the desired posteriors. What the visual system ends up seeing is a function +of these posteriors and its utility functions. +The problem with this conventional approach is that it entails a vicious +circle, which we can call + +Bayes’ Circle: We can only see the world through our posteriors. When +we measure priors and likelihoods in the world, our measurements are nec- +essarily filtered through our posteriors. Using our measurements of priors +and likelihoods to justify our posteriors thus leads to a vicious circle. + + +1.4 Bayes’ Circle +7 + +Suppose, for instance, that we build a robot with a vision system that com- +putes shape from motion using a prior assumption that the world contains +many rigid objects [41]. The system takes inputs from a video camera, does +some initial processing to find two-dimensional features in the video images, +and then uses an algorithm based on rigidity to compute three-dimensional +shape. It seems to work well, but we decide to double-check that the prior +assumption about rigid objects that we built into the system is in fact true +of the world. So we send our robot out into the world to look around. To our +relief, it comes back with the good news that it has indeed found numerous +rigid objects. Of course it did; that’s what we programmed it to do. If, +based on the robot’s good news, we conclude that our prior on rigid objects +is justified, we’ve just been bagged by Bayes’ Circle. +This example is a howler, but precisely the same mistake prompts the +conventional claim that we can validate our priors by measuring properties +of the objective world. The conventionalist can reply that the robot example +fails because it ignores the possibility of cross checking results with other +senses, other observers, and scientific instruments. But such a reply hides +the same howler, because other senses, other observers, and scientific instru- +ments all have built in priors. None is a filter-free window on an objective +(i.e., observation independent) world. Consensus among them entails, at +most, agreement among their priors; it entails nothing about properties or +statistical structures of an objective world. +It is, of course, possible to pursue a Bayesian approach to perception with- +out getting mired in Bayes’ circle. Indeed, Bayesian approaches are among +the most promising in the field. Conditional probabilities turn up every- +where in perception, because perception is often about determining what +is the best description of the world, or the best action to take, given (i.e., +conditioned on) the current state of the sensoria. Bayes is simply the right +way to compute conditional probabilities using prior beliefs, and Bayesian +decision theory, more generally, is a powerful way to model the utilities and +actions of an organism in its computation of perceptual descriptions. +But it is possible to use the sophisticated tools of Bayesian decision theory, +to fully appreciate the importance of utilities and the perception-action loop, +and still to fall prey to Bayes’ circle—to conclude, as quoted from Palmer +above, that “Evolutionarily speaking, visual perception is useful only if it is +reasonably accurate.” + + +8 +The Interface Theory of Perception + +1.5 The Interface Theory of Perception + +The conventional theory of perception gets evolution fundamentally wrong +by conflating fitness and accuracy. This leads the conventional theory to +the false claim that a primary goal of perception is faithful depiction of the +world. A standard way to state this claim is the + +Reconstruction Thesis: Perception reconstructs certain properties and +categories of the objective world. + +This claim is too strong. It must be weakened, on evolutionary grounds, to +a less tendentious claim, the + +Construction Thesis: Perception constructs the properties and categories +of an organism’s perceptual world. + +The construction thesis is clearly much weaker than the reconstruction the- +sis. One can, for instance, obtain the reconstruction thesis by starting with +the construction thesis and adding the claim that the organism’s constructs +are, at least in certain respects, roughly isomorphic to the properties or +categories of the objective world, thus qualifying them to be deemed recon- +structions. +But the range of possible relations between perceptual constructs and +the objective world is infinite; isomorphism is just one relation out of this +infinity and, on evolutionary grounds, an unlikely one. Thus the reconstruc- +tion thesis is a conceptual straightjacket that constrains us to think only +of improbable isomorphisms, and impedes us from exploring the full range +of possible relations between perception and the world. Once we dispense +with the straightjacket we’re free to explore all possible relations that are +compatible with evolution [23]. +To this end we note that, to the extent that perceptual properties and +categories are satisficing solutions to adaptive problems, they admit a func- +tional description. Admittedly, a conceivable, though unlikely, function of +perception is faithful depiction of the world. That’s the function favored by +the reconstruction thesis of the conventionalist. But once we repair the con- +flation of fitness and accuracy, we can consider other perceptual functions +with greater evolutionary plausibility. To do so properly requires a serious +study of the functional role of perception in various evolutionary settings. +Beetles falling for bottles is one instructive example; in the next section we +consider a few more. +But here it’s useful to introduce a model of perception that can help us +study its function without relapse into conventionalism. The model is the + + +1.5 The Interface Theory of Perception +9 + +Interface Theory of Perception: The perceptions of an organism are a +user interface between that organism and the objective world [16, 17, 20]. + +This theory addresses the natural question, “If our perceptions are not ac- +curate, then what good are they?” The answer becomes obvious for user +interfaces. The colour, for instance, of an icon on a computer screen does +not estimate, or reconstruct, the true colour of the file that it represents in +the computer. If an icon is, say, green, it would be ludicrous to conclude that +this green must be an accurate reconstruction of the true colour of the file +it represents. It would be equally ludicrous to conclude that, if the colour of +the icon doesn’t accurately reconstruct the true colour of the file, then the +icon’s colour is useless, or a blatant deception. This is simply a na¨ıve mis- +understanding of the point of a user interface. The conventionalist theory +that our perceptions are reconstructions is, in precisely the same manner, +equally na¨ıve. +Colour is, of course, just one example among many: The shape of an +icon doesn’t reconstruct the true shape of the file; the position of an icon +doesn’t reconstruct the true position of the file in the computer. A user +interface reconstructs nothing. Its predicates and the predicates required +for a reconstruction can be entirely disjoint: Files, for instance, have no +colour. +And yet a user interface is useful despite the fact that it’s not a recon- +struction. +Indeed, it’s useful because it’s not a reconstruction. +We pay +good money for user interfaces because we don’t want to deal with the over- +whelming complexity of software and hardware in a PC. A user interface +that slavishly reconstructed all the diodes, resistors, voltages and magnetic +fields in the computer would probably not be a best seller. The user inter- +face is there to facilitate our interactions with the computer by hiding its +causal and structural complexity, and by displaying useful information in a +format that is tailored to our specific projects, such as painting or writing. +Our perceptions are a species-specific user interface. Space, time, position +and momentum are among the properties and categories of the interface of +H. sapiens that, in all likelihood, resemble nothing in the objective world. +Different species have different interfaces. And, due to the variation that +is normal in evolution, there are differences in interfaces among humans. +To the extent that our perceptions are satisficing solutions to evolutionary +problems, our interfaces are designed to guide adaptive behavior in our niche; +accuracy of reconstruction is irrelevant. To understand the properties and +categories of our interface we must understand the evolutionary problems, +both phylogenetic and ontogenetic, that it solves. + + +10 +The Interface Theory of Perception + +1.6 User Interfaces in Nature + +The interface theory of perception predicts that (1) each species has its +own interface (with some variations among conspecifics and some similari- +ties across phylogenetically related species), (2) almost surely, no interface +performs reconstructions, (3) each interface is tailored to guide adaptive +behavior in the relevant niche, (4) much of the competition between and +within species exploits strengths and limitations of interfaces, and (5) such +competition can lead to arms races between interfaces that critically influ- +ence their adaptive evolution. In short, the theory predicts that interfaces +are essential to understanding the evolution and competition of organisms; +the reconstruction theory makes such understanding impossible. Evidence +of interfaces should be ubiquitous in nature. +The jewel beetle is a case in point. Its perceptual category desirable fe- +male works well in its niche. However, its soft spot for stubbies reveals that +its perceptions are not reconstructions. They are, instead, quick guides to +adaptive behavior in a stubbie-free niche. The stubbie is a so-called super- +normal stimulus, i.e., a stimulus that engages the interface and behavior of +the organism more forcefully than the normal stimuli to which the organ- +ism has been adapted. The bottle is shiny, dimpled, and the right colour +of brown. But what makes it a supernormal stimulus is apparently its su- +pernormal size. If so, then, contrary to the reconstruction thesis, the jewel +beetle’s perceptual category desirable female does not incorporate a statisti- +cal estimate of the true sizes of the most fertile females. Instead its category +satisfices with “bigger is better.” In its niche this solution is fit enough. A +stubbie, however, plunges it into an infinite loop. +Supernormal stimuli have been found for many species, and all such dis- +coveries are evidence against the claim of the reconstruction theory that our +perceptual categories estimate the statistical structure of the world; all are +evidence for species-specific interfaces that are satisficing solutions to adap- +tive problems. Herring gulls (Larus argentatus) provide a famous example. +Chicks peck a red spot near the tip of the lower mandible of an adult to +prompt the adult to regurgitate food. Tinbergen and Perdeck [38] found +that an artificial stimulus that is longer and thinner than a normal beak, +and whose red spot is more salient than normal, serves as a supernormal +stimulus for the chick’s pecking behaviors. The colour of the artificial beak +and head matter little. The chick’s perceptual category food bearer, or per- +haps food-bearing parent, is not a statistical estimate of the true properties of +food-bearing parents, but a satisficing solution in which longer and thinner +is better and in which greater salience of the red spot is better. Its inter- + + +1.6 User Interfaces in Nature +11 + +face employs simplified symbols that effectively guide behavior in its niche. +Only when its niche is invaded by pesky ethologists is this simplification +unmasked, and the chick sent seeking what can never satisfy. +Simplified does not mean simple. Every interface of every organism dra- +matically simplifies the complexity of the world, but not every interface is +considered by H. sapiens to be simple. +Selective sophistication in inter- +faces is the result, in part, of competition between organisms in which the +strengths in the interface of one’s nemesis or next meal are avoided and its +weaknesses exploited. Dueling between interfaces hones them and the strate- +gies used to exploit them. This is the genesis of mimicry and camouflage, +and of complex strategies to defeat them. +A striking example, despite brains the size of a pinhead, are jumping spi- +ders of the genus Portia [13]. Portia is araneophagic, preferring to dine on +other spiders. Such dining can be dangerous; if the interface of the intended +dinner detects Portia, dinner could be diner. So Portia has evolved coun- +termeasures. Its hair and colouration mimic detritus found in webs and on +the forest floor; its gait mimics the flickering of detritus—a stealth technol- +ogy cleverly adapted to defeat the interfaces of predators and prey. If Portia +happens on a dragline (a trail of silk) left by the jumping spider Jacksonoides +queenslandicus, odors from the dragline prompt Portia to use its eight eyes +to hunt for J. queenslandicus. But J. queenslandicus is well camouflaged +and, if motionless, invisible to Portia. +So Portia makes a quick vertical +leap, tickling the visual motion detectors of J. queenslandicus and trigger- +ing it to orient to the motion. By the time J. queenslandicus has oriented, +Portia is already down, motionless, and invisible to J. queenslandicus; but +it has seen the movement of J. queenslandicus. Once the eyes of J. queens- +landicus are safely turned away, Portia slowly stalks, leaps, and strikes with +its fangs, delivering a paralyzing dose of venom. Portia’s victory exploits +strengths of its interface and weaknesses in that of J. queenslandicus. +Jewel beetles, herring gulls and jumping spiders illustrate the ubiquitous +role in evolution of species-specific user interfaces. +Perception is not re- +construction, it is construction of a niche-specific, problem-specific, fitness- +enhancing interface, which the biologist Jakob von Uexk¨ull [42, 43] called +an Umwelt or “self-world” [34]. Perceptual categories are endogenous con- +structs of a subjective Umwelt, not exogenous mirrors of an objective world. +The conventionalist might object that these examples are self-refuting, +since they require comparison between the perceptions of an organism and +the objective reality that those perceptions get wrong. Only by knowing, +for instance, the objective differences between beetle and bottle can we un- +derstand a perceptual flaw of J. backewelli. So the very examples adduced + + +12 +The Interface Theory of Perception + +in support of the interface theory actually support the conclusion that per- +ceptual reconstruction of the objective world in fact occurs, in contradiction +to the predictions of that theory. +This objection is misguided. The examples discussed here, and all others +that might be unearthed by H. sapiens, are necessarily filtered through +the interface of H. sapiens, an interface whose properties and categories are +adapted for fitness, not accuracy. What we observe in these examples is not, +therefore, mismatches between perception and a reality to which H. sapiens +has direct access. Instead, because the interface of H. sapiens differs from +that of other species, H. sapiens can, in some cases, see flaws of others +that they miss themselves. In other cases, we can safely assume, H. sapiens +misses flaws of others due to flaws of its own. And, in yet other cases, flaws +of H. sapiens might be obvious to other species. +The conventionalist might further object, saying, “If you think that the +wild tiger over there is just a perceptual category of your interface, then +why don’t you go pet it? When it attacks, you’ll find out it’s more than an +Umwelt category, it’s an objective reality.” +This objection is also misguided. +I don’t pet wild tigers for the same +reason I don’t carelessly drag a file icon to the trash bin. I don’t take the +icon literally, as though it resembles the real file. But I do take it seriously. +My actions on the icon have repercussions for the file. Similarly, I don’t +take my tiger icon literally but I do take it seriously. Aeons of evolution +of my interface have shaped it to the point where I had better take its +icons seriously or risk harm. So the conventionalist objection fails because +it conflates taking icons seriously and taking them literally. +This conventionalist argument is not new. +Samuel Johnson famously +raised it in 1763 when, in response to the idealism of Berkeley, he kicked +a stone and exclaimed “I refute it thus” [4] (1, p. 134). Johnson thus con- +flated taking a stone seriously and taking it literally. Nevertheless Johnson’s +argument, one must admit, has strong psychological appeal despite the non +sequitur, and it is natural to ask why. Perhaps the answer lies in the evolu- +tion of our interface. There was, naturally enough, selective pressure to take +its icons seriously; those who didn’t take their tiger icons seriously came to +early harm. But were there selective pressures not to take its icons literally? +Did reproductive advantages accrue to those of our Pleistocene ancestors +who happened not to conflate the serious and the literal? Apparently not, +given the widespread conflation of the two in the modern population of H. +sapiens. Hence, the very evolutionary processes that endowed us with our +interfaces might also have saddled us with the penchant to mistake their +contents for objective reality. This mistake spawned sweeping commitments + + +1.7 Interface and World +13 + +to a flat earth and a geocentric universe, and prompted the persecution of +those who disagreed. Today it spawns reconstructionist theories of percep- +tion. Flat earth and geocentrism were difficult for H. sapiens to scrap; some +unfortunates were tortured or burned in the process. +Reconstructionism +will, sans the torture, prove even more difficult to scrap; it’s not just this +or that percept that must be recognized as an icon, but rather perception +itself that must be so recognized. The selection pressures on Pleistocene +hunter-gatherers clearly didn’t do the trick, but social pressures on modern +H. sapiens, arising in the conduct of science, just might. +The conventionalist might object that death is a counterexample: +it +should be taken seriously and literally. It is not just shuffling of icons. +This objection is not misguided. In death, one’s body icon ceases to func- +tion and, in due course, decays. The question this raises can be compared to +the following: When a file icon is dragged to the trash and disappears from +the screen, is the file itself destroyed, or is it still intact and just inaccessible +to the user interface? Knowledge of the interface itself might not license a +definitive answer. If not, then to answer the question one must add to the +interface a theory of the objective world it hides. How this might proceed +is the topic of the next section. +The conventionalist might persist, arguing that agreement between ob- +servers entails reconstruction and provides important reality checks on per- +ception. This argument also fails. First, agreement between observers may +only be apparent: It is straightforward to prove that two observers can be +functionally identical and yet differ in their conscious perceptual experi- +ences [18, 19]; reductive functionalism is false. Second, even if observers +agree, this doesn’t entail the reconstruction thesis. +The observers might +simply employ the same constructive (but not reconstructive) perceptual +processes. If two PC’s have the same icons on their screens, this doesn’t en- +tail that the icons reconstruct their innards. Agreement provides subjective +consistency checks—not objective reality checks—between observers. + +1.7 Interface and World + +The interface theory claims that perceptual properties and categories no +more resemble the objective world than Windows icons resemble the diodes +and resistors of a computer. +The conventionalist might object that this +makes the world unknowable and is, therefore, inimical to science. +This misses a fundamental point in the philosophy of science: Data never +determine theories. +This under-determination makes the construction of +scientific theories a creative enterprise. The contents of our perceptual in- + + +14 +The Interface Theory of Perception + +terfaces don’t determine a true theory of the objective world, but this in +no way precludes us from creating theories and testing their implications. +One such theory, in fact the conventionalist’s theory, is that the relation +between interface and world is, on appropriately restricted domains, an iso- +morphism. This theory is, as we have discussed, improbable on evolutionary +grounds and serves as an intellectual straightjacket, hindering the field from +considering more plausible options. +What might those options be? That depends on which constraints one +postulates between interface and world. +Suppose, for instance, that one +wants a minimal constraint that allows probabilities of interface events to +be informative about probabilities of world events. Then, following stan- +dard probability theory, one would represent the world by a measurable +space, i.e., by a pair (W, ΣW ), where W is a set and ΣW is a σ-algebra of +measurable events. One would represent the user interface by a measurable +space (U, ΣU), and the relation between interface and world by a measurable +function f: W → U. The function f could be many-to-one, and the features +represented by W disjoint from those represented by U. The probabilities +of events in the interface (U, ΣU) would be distributions of the probabilities +in the world (W, ΣW ), i.e., if the probability of events in the world is µ, +then the probability of any interface event A ∈ ΣU is µ(f−1(A)). Using +this terminology, the problem of Bayes’ circle, scouted above, can be stated +quite simply: It is conflating U with W, and assuming that f: W → U is +approximately 1 to 1, when in fact it’s probably infinite to 1. This mistake +can be made even while using all the sophisticated tools of Bayesian decision +theory and machine learning theory. +The measurable-space proposal could be weakened if, for instance, one +wished to accommodate quantum systems with noncommuting observables. +In this case the event structures would not be σ-algebras but instead σ- +additive classes, which are closed under countable disjoint union rather than +under countable union [10], and f would be measurable with respect to these +classes. This would still allow probabilities of events in the interface to be +distributions of probabilities of events in the world. It would explain why +science succeeds in uncovering statistical laws governing events in space- +time, even though these events, and space-time itself, in no way resemble +objective reality. +This proposal could be weakened further. One could give up the measura- +bility of f, thereby giving up any quantitative relation between probabilities +in the interface and the world. The algebra or class structure of events in +the interface would still reflect an isomorphic subalgebra or subclass struc- +ture of events in the world. This is a nontrivial constraint: Subset relations + + +1.7 Interface and World +15 + +in the interface, for instance, would genuinely reflect subset relations of the +corresponding events in the world. +Further consideration of the interface might prompt us, in some cases, +to weaken the proposal even further. Multistable percepts, for instance, in +which the percept switches while the stimulus remains unchanged, may force +us to reconsider whether the relation between interface and world is even a +function: Two or more states of the interface might be associated to a single +state of the world. +These proposals all assume, of course, that mathematics, which has proved +useful in studying the interface, will also prove useful in modeling the world. +We shall see. +The discussion here is not intended, of course, to settle the issue of the +relation between interface and world, but to sketch how investigation of the +relation may proceed in the normal scientific fashion. This investigation is +challenging because we see the world through our interface, and it can there- +fore be difficult to discern the limitations of that interface. We are naturally +blind to our own blindness. The best remedy at hand for such blindness +is the systematic interplay of theory and experiment that constitutes the +scientific method. +The discussion here should, however, help place the interface theory of +perception within the philosophical landscape. It is not classical relativism, +which claims that there is no objective reality, only metaphor; it claims +instead that there is an objective reality that can be explored in the normal +scientific manner. It is not na¨ıve realism, which claims that we directly see +middle-sized objects; nor is it indirect realism, or representationalism, which +says that we see sensory representations, or sense data, of real middle-sized +objects, and do not directly see the objects themselves. It claims instead that +the physicalist ontology underlying both na¨ıve realism and indirect realism +is almost surely false: A rock is an interface icon, not a constituent of +objective reality. Although the interface theory is compatible with idealism, +it is not idealism, because it proposes no specific model of objective reality, +but leaves the nature of objective reality as an open scientific problem. +It is not a scientific physicalism that rejects the objectivity of middle-sized +objects in favor of the objectivity of atomic and subatomic particles; instead +it claims that such particles, and the space-time they inhabit, are among the +properties and categories of the interface of H. sapiens. Finally, it differs +from the utilitarian theory of perception [5, 30, 31], which claims that vision +uses a bag of tricks (rather than sophisticated general principles) to recover +useful information about the physical world; interface theory (1) rejects the +physicalist ontology of the utilitarian theory, (2) asserts instead that space + + +16 +The Interface Theory of Perception + +and time, and all objects that reside within them, are properties or icons of +our species-specific user interface, and therefore (3) rejects the claim of the +utilitarian theory that vision recovers information about preexisting physical +objects in space-time. It agrees, however, with the utilitarian theory that +evolution is central to understanding perception. +A conventionalist might object, saying, “These proposals about the rela- +tion of interface and world are fine as theoretical possibilities. But, in the +end, a rock is still a rock.” In other words, all the intellectual arguments +in the world won’t make the physical world—always obstinate and always +irrepressible—conveniently disappear. The interface theorist, no less than +the physicalist, must take care not to stub a toe on a rock. +Indeed. But in the same sense a trash-can icon is still a trash-can icon. +Any file whose icon stubs its frame on the trash can will suffer deletion. The +trash can is, in this way, as obstinate and irrepressible as a rock. But both +are simplifying icons. Both usefully hide a world that is far more complex. +Space and time do the same. +The conventionalist might further object, saying, “The proposed dissimi- +larity between interface and world is contradicted by the user-interface ex- +ample itself. The icons of a computer interface perhaps don’t resemble the +innards of a computer, but they do resemble real objects in the physical +world. Moreover, when using a computer to manipulate 3D objects, as in +computer aided design, the computer interface is most useful if its symbols +really resemble the actual 3D objects to be manipulated.” +Certainly. These arguments show that an interface can sometimes resem- +ble what it represents. And that is no surprise at all. But user interfaces can +also not resemble what they represent, and can be quite effective precisely +because they don’t resemble what they represent. So the real question is +whether the user interface of H. sapiens does in fact resemble what it rep- +resents. Here, I claim, the smart money says No. + +1.8 Future Research on Perceptual Categorization + +So what? +So what if perception is a user-interface construction, not an +objective-world reconstruction? How will this affect concrete research on +perceptual categorization? +Here are some possibilities. First, as discussed already, current attempts +to verify priors are misguided. This doesn’t mean we must abandon such +attempts. It does mean that our attempts must be more sophisticated; at a +minimum they must not founder on Bayes’ circle. +But that is at a minimum. Real progress in understanding the relation + + +1.8 Future Research on Perceptual Categorization +17 + +between perception and the world requires careful theory building. +The +conventional theory that perception approximates the world is hopelessly +simplistic. +Once we reject this facile theory, once we recognize that our +perceptions are to the world as a user interface is to a computer, we can +begin serious work. We must postulate, and then try to justify and confirm, +possible structures for the world and possible mappings between world and +interface. Clinging to approximate isomorphisms is a natural, but thus far +fruitless, response to this daunting task. +It’s now time to develop more +plausible theories. Some elementary considerations toward this end were +presented in the previous section. +Our efforts should be informed by relevant advances in modern physics. +Experiments by Alain Aspect [1, 2], building on the work of Bell [3], persuade +most physicists to reject local realism, viz., the doctrine that (1) distant +objects cannot directly influence each other (locality) and (2) all objects have +pre-existing values for all possible measurements, before any measurements +are made (realism). Aspect’s experiments demonstrate that distant objects, +say two electrons, can be entangled, such that measurement of a property +of one immediately affects the value of that property of the other. Such +entanglement is not just an abstract possibility, it is an empirical fact now +being exploited in quantum computation to give substantial improvements +over classical computation [6, 21]. Our untutored categories of space, time +and objects would lead us to expect that two electrons a billion light years +apart are separate entities; in fact, because of entanglement, they are a +single entity with a unity that transcends space and time. This is a puzzle +for proponents of faithful depiction, but not for interface theory. +Space, +time and separate objects are useful fictions of our interface, not faithful +depictions of objective reality. +Our theories of perceptual categorization must be informed by explicit +dynamical models of perceptual evolution, models such as those studied in +evolutionary game theory [14, 26, 33]. Our perceptual categories are shaped +inter alia by factors such as predators, prey, sexual selection, distribution of +resources, and social interactions. We won’t understand categorization un- +til we understand how categories emerge from dynamical systems in which +these factors interact. +There are promising leads. +Geisler and Diehl [8] +simulate interactions between simplified predators and prey, and show how +these might shape the spectral sensitivities of both. Komarova, Jameson +and Narens [22] show how colour categories can evolve from a minimal per- +ceptual psychology of discrimination together with simple learning rules and +simple constraints on social communication. Some researchers are explor- +ing perceptual evolution in foraging contexts [9, 32, 35]. These papers are + + +18 +The Interface Theory of Perception + +useful pointers to the kind of research required to construct theories of cat- +egorization that are evolutionarily plausible. As a concrete example of such +research, consider the following class of evolutionary games. + +1.9 Interface Games + +In the simplest interface game, two animals compete over three territories. +Each territory has a food value and a water value, each value ranging from, +say, 0 to 100. The first animal to choose a territory obtains its food and water +values; the second animal then chooses one of the remaining two territories, +and obtains its food and water values. The animals can adopt one of two +perceptual strategies. The truth interface strategy perceives the exact values +of food and of water for each territory. Thus the total information that truth +obtains is IT = 3 [territories] × 2 [resources per territory] × log2 101 [bits +per resource] ≈ 39.95 bits. The simple interface strategy perceives only one +bit of information per territory: if the food value of a territory is greater +than some fixed value (say 50), simple perceives that territory as green, +otherwise simple perceives that territory as red. Thus the total information +that simple obtains is IS = 3 bits. +It costs energy to obtain perceptual information. Let the energy cost per +bit be denoted by ce. Since the truth strategy obtains IT bits, the total +energy cost to truth is IT ce, which is subtracted from the sum of food and +water values that truth obtains from the territory it chooses. Similarly, the +total energy cost to simple is ISce. +It takes t units of time to obtain one bit of perceptual information. If +t > 0, then simple acquires all of its perceptual information before truth +does, allowing simple to be first to choose a territory. +Assuming, for simplicity, that the food and water values are independent, +identically distributed random variables with, say, a uniform distribution on +the integers from 0 to 100, we can compute a matrix of expected payoffs: + +Truth +Simple + +Truth: +a +b +Simple: +c +d + +Here a is the expected payoff to truth if it competes against truth, b is the +expected payoff to truth if it competes against simple, c is the expected +payoff to simple if it competes against truth, and d is the expected payoff to +simple if it competes against simple. +As is standard in evolutionary game theory, we consider a population of +truth and simple players and equate payoff with fitness. +Let xT denote + + +1.9 Interface Games +19 + +the frequency of truth players and xS the frequency of simple players; the +population is thus ⃗x = (xT , xS). Then, assuming players meet at random, +the expected payoffs for truth and simple are, respectively, fT (⃗x) = axT +bxS +and fS(⃗x) = cxT + dxS. The selection dynamics is then x′ +T = xT [fT (⃗x) − +F]; x′ +S = xS[fS(⃗x) − F], where primes denote temporal derivatives and F is +the average fitness, F = xT fT (⃗x) + xSfS(⃗x). + +If a > c and b > d, then truth drives simple to extinction. If a < c and +b < d then simple drives truth to extinction. If a > c and b < d, then +truth and simple are bistable; which goes extinct depends on the initial +frequencies, ⃗x(0), at time 0. If a < c and b > d then truth and simple stably +coexist, with the truth frequency given by (d−b)/(a−b−c+d). If a = c and +b = d, then selection does not change the frequencies of truth and simple. + +The entries in the payoff matrix described above will vary, of course, with +the correlation between food and water values, with the specific value of +food that is used by simple as the boundary between green and red, and +with the cost ce per bit of information obtained. + +boundary + +0 +100 + +0 +100 + +cost per bit + +0.2 + +0.4 + +0.6 + +0.2 + +0.4 + +0.6 + +1.0 + +1.2 +r = 0 + +r = 1 + +Fig 1.1. Asymptotic behavior of the interface game as a function of the cost per +bit of information and the choice of the red-green boundary in the simple strategy. +Light gray indicates that simple drives truth to extinction, intermediate gray that +the two strategies coexist, and dark gray that truth drives simple to extinction. The + + +20 +The Interface Theory of Perception + +upper plot is for uncorrelated food and water, the lower for perfectly correlated food +and water. + +And here is the punchline. Simple drives truth to extinction for most +values of the red-green boundary, even when the cost per bit of information is +small and the correlation between food and water is small. This is illustrated +in Figure 1.1, which shows the results of Matlab simulations. Evolutionary +pressures do not select for veridical perception; instead they drive it, should +it arise, to extinction. + +The interface game just described might seem too simple to be useful. One +can, however, expand on the simple game just described in several ways, in- +cluding (1) increasing the number of territories at stake, (2) increasing the +number of resources per territory, (3) having dangers as well as resources in +the territories, (4) considering distributions other than uniform (e.g., Gaus- +sian) for the resources and dangers, (5) considering two-boundary, three- +boundary, n-boundary interface strategies, and more general categorization +algorithms that don’t rely on such boundaries, (6) considering populations +with three or more interface strategies, (7) considering more sophisticated +maps from resources to interfaces, including probabilistic maps, (8) consid- +ering time and energy costs that vary with architecture (e.g., serial versus +parallel) and that are probabilistic functions of the amount of information +gleaned and (9) extending the replicator dynamics, e.g., to include commu- +nication between players and to include a spatial dimension in which players +only interact with nearby players (as has been done with stag hunt and Lewis +signaling games [36, 37, 45]). Interface games, in all these varieties, allow +us to explore the complex evolutionary pressures that shape perception and +perceptual categorization, and to do so as realistically as our imaginations +and computational resources will allow. + +They will also allow us to address a natural question: As an organism’s +perceptions and behaviors become more complex, shouldn’t it be the case +that the goal of perception approaches that of recovering the properties of +the environment? + +Using simulations of interface games, one can ask for what environments +(including what kinds of competitors) will the reproductive pressures push +an organism to true perceptions of the environment, so that perceptual truth +is an evolutionarily stable strategy. My bet: None of interest. + + +1.10 Conclusion +21 + +1.10 Conclusion + +Most experts assume that perception estimates true properties of an objec- +tive world. They justify this assumption with an argument from evolution: +Natural selection rewards true perceptions. I propose instead that if true +perceptions crop up, then natural selection mows them down; natural se- +lection fosters perceptions that act as simplified user interfaces, expediting +adaptive behavior while shrouding the causal and structural complexity of +the objective world. In support of this proposal, I discussed mimicry and +mating errors in nature, and presented simulations of an evolutionary game. +Old habits die hard. +I suspect that few experts will be persuaded by +these arguments to adopt the interface theory of perception. Most will still +harbor the long-standing conviction that, although we see reality through +small portals, nevertheless what we see is, in general, veridical. To such +experts I offer one final claim, and one final challenge. I claim that natural +selection drives true perception to swift extinction: Nowhere in evolution, +even among the most complex of organisms, will you find that natural selec- +tion drives truth to fixation, i.e., so that the predicates of perception (e.g., +space, time, shape and color) approximate the predicates of the objective +world (whatever they might be). Natural selection rewards fecundity, not +factuality, so it shapes interfaces, not telescopes on truth [28] (p. 571). The +challenge is clear: Provide a compelling counterexample to this claim. + +Acknowledgements. +Justin Mark collaborated in developing the interface +games, and wrote the simulations presented in Figure 1.1. For helpful com- +ments on previous drafts, I thank Ori Amir, Mike D’Zmura, Geoff Iverson, +Carol Skrenes, Duncan Luce, Larry Maloney, Brian Marion, Justin Mark, +Louis Narens, Steve Pinker, Kim Romney, John Serences, Brian Skyrms, and +Joyce Wu. For helpful discussions I thank Mike Braunstein, Larry Maloney, +Jon Merzel, Chetan Prakash, Rosie Sedghi, and Phat Vu. + + +References + +1. Aspect, A., Grangier, P. and Roger, G. (1982a). Experimental realization of +Einstein-Podolsky-Rosen-Bohm gedankenexperiment: A new violation of Bells +inequalities. Physical Review Letters 49, 91–94. +2. Aspect, A., Dalibard, J. and Roger, G. (1982b). Experimental test of Bells in- +equalities using time-varying analyzers. Physical Review Letters 49, 1804–1807. +3. Bell. J.S. (1964). On the Einstein-Podolsky-Rosen paradox. Physics 1, 195–200. +4. Boswell, J. (1791). The life of Samuel Johnson. +5. Braunstein, M.L. (1983). Contrasts between human and machine vision: Should +technology recapitulate phylogeny?, in Human and machine vision, ed. J. Beck, +B. Hope, and A. Rosenfeld (Academic Press, New York). +6. Nielsen, M.A. and Chuang, I.L. (2000). Quantum computation and quantum +information. (Cambridge University Press, Cambridge). +7. Daly, M. and Wilson, M. (1978). Sex, evolution, and behavior. (Duxbury Press, +Massachusetts). +8. Geisler, W.S. and Diehl, R.L. (2003). A Bayesian approach to the evolution of +perceptual and cognitive systems. Cognitive Science 27, 379–402. +9. Goldstone, R.L., Ashpole, B.C., and Roberts, M.E. (2005). Knowledge of re- +sources and competitors in human foraging. Psychonomic Bulletin & Review +12, 81–87. +10. Gudder, S. (1988). Quantum probability. (Academic Press, San Diego). +11. Gwynne, D.T. & Rentz, D.C.F. (1983). Beetles on the Bottle: Male Buprestids +Make Stubbies for Females. Journal of Australian Entomological Society 22, +79–80. +12. +Gwynne, +D.T. +(2003). +Mating +mistakes, +in +Encyclopedia +of +insects, +ed.V.H. Resh and R.T. Carde (Academic Press: San Diego). +13. Harland, D.P. & Jackson, R.R. (2004). Portia perceptions: The Umwelt of an +Araneophagic jumping spider, in Complex worlds from simpler nervous sys- +tems, ed. F.R. Prete (MIT Press, Cambridge, MA). +14. Hofbauer, J. & Sigmund, K. Evolutionary games and population dynamics. +(Cambridge University Press, Cambridge). +15. Hoffman, D. D. (1983). The interpretation of visual illusions. Scientific Ameri- +can 249, 154–162. +16. Hoffman, D. D. (1998). Visual intelligence: How we create what we see. (W.W. +Norton, New York). +17. Hoffman, D. D. (2006a). Mimesis and its perceptual reflections, in A View in + +22 + + +References +23 + +the Rear-Mirror: Romantic Aesthetics, Culture, and Science Seen from Today. +Festschrift for Frederick Burwick on the Occasion of His Seventieth Birthday, +ed. W. Pape (WVT, Wissenschaftlicher Verlag Trier: Trier) (Studien zur En- +glischen Romantik 3). +18. Hoffman, D.D. (2006b). The scrambling theorem: A simple proof of the logical +possibility of spectrum inversion. Consciousness and Cognition 15, 31–45. +19. Hoffman, D.D. (2006c). The scrambling theorem unscrambled: A response to +commentaries. Consciousness and Cognition 15, 51–53. +20. Hoffman, D.D. (2008, in press). Conscious realism and the mind-body problem. +Mind & Matter. +21. Kaye, P., Laflamme, R. and Mosca, M. (2007). An introduction to quantum +computing. (Oxford University Press: Oxford). +22. Komarova, N.L., Jameson, K.A. and Narens, L. (2007). Evolutionary models +of color categorization based on discrimination. Journal of Mathematical Psy- +chology 51, 359–382. +23. Mausfeld, R. (2002). The physicalist trap in perception theory, in Perception +and the physical world, ed. D. Heyer and R. Mausfeld (Wiley, New York). +24. Lehar, S. (2003). Gestalt isomorphism and the primacy of subjective conscious +experience: A Gestalt Bubble model. Behavioral and Brain Sciences 26, 375– +444. +25. No¨e, A, and Regan, J.K. (2002). On the brain-basis of visual consciousness: A +sensorimotor account, in Vision and mind: Selected readings in the philosophy +of perception, ed. A. No¨e and E. Thompson (MIT Press, Cambridge, MA). +26. Nowak, M.A. (2006). Evolutionary dynamics: Exploring the equations of life. +(Belknap/Harvard University Press, Cambridge, MA). +27. Palmer, S.E. (1999). Vision science: Photons to phenomenology. (MIT Press, +Cambridge, MA). +28. Pinker, S. (1997). How the mind works. (W.W. Norton, New York). +29. Purves, D., and Lotto, R. B. (2003). Why we see what we do: An empirical +theory of vision. (Sinauer, Sunderland, MA). +30. Ramachandran, V.S. (1985). The neurobiology of perception. Perception 14, +97–103. +31. Ramachandran, V.S. (1990). Interactions between motion, depth, color and +form: The utilitarian theory of perception, in Vision: Coding and efficiency, +ed. C. Blakemore (Cambridge University Press, Cambridge). +32. Roberts, M.E. and Goldstone, R.L. (2006). EPICURE: Spatial and knowledge +limitations in group foraging. Adaptive Behavior 14, 291–313. +33. Samuelson, L. (1997). Evolutionary games and equilibrium selection. (MIT +Press, Cambridge, MA). +34. Schiller, C.H. (1957). Instinctive behavior: Development of a modern concept. +(Hallmark Press, New York). +35. Sernland, E., Olsson, O., and Holmgren, N.M.A. (2003). Does information +sharing promote group foraging? Proceedings of the Royal Society of London +270, 1137–1141. +36. Skyrms, B. (2002). Signals, evolution, and the explanatory power of transient +information. Philosophy of Science 69, 407–428. +37. Skyrms, B. (2004). The stag hunt and the evolution of social structure. (Cam- +bridge University Press, Cambridge). +38. Tinbergen, N., and A. C. Perdeck. (1950). On the stimulus situation releasing +the begging response in the newly hatched Herring Gull chick (Larus argentatus + + +24 +References + +argentatus Pont.). Behaviour 3, 1–39. +39. Trivers, R.L. (1972). Parental investment and sexual selection, in Sexual se- +lection and the descent of man, 1871-1971, +ed. B. Campbell (Aldine Press, +Chicago). +40. Trivers, R.L. (1976). Foreword, in R. Dawkins, The selfish gene. (Oxford Uni- +versity Press: New York). +41. Ullman, S. (1979). The interpretation of visual motion. (MIT Press, Cambridge, +MA). +42. Von Uexk¨ull, J. (1909). Umwelt und Innenwelt der Tiere. (Springer-Verlag, +Berlin). +43. Von Uexk¨ull, J. (1934). A stroll through the worlds of animals and men: A +picture book of invisible worlds, reprinted in Instinctive behavior: Development +of a modern concept, C.H. Schiller (1957) (Hallmark Press, New York). +44. Yuille, A., and B¨ulthoff, H. (1996). Bayesian decision theory and psychophysics, +in Perception as Bayesian inference, ed. D. Knill and W. Richards (Cambridge +University Press, Cambridge). +45. Zollman, K. (2005). Talking to neighbors: The evolution of regional meaning. +Philosophy of Science 72, 69–85. + + diff --git a/papers/project_paper_4_fbt/references/Ortega2013.pdf b/papers/project_paper_4_fbt/references/Ortega2013.pdf new file mode 100644 index 00000000..cfc3247b --- /dev/null +++ b/papers/project_paper_4_fbt/references/Ortega2013.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6d26c00749efbd04a2bfc7222f269b0499190f221dc39091ead1ce67ee179c4b +size 264809 diff --git a/papers/project_paper_4_fbt/references/Ortega2013.txt b/papers/project_paper_4_fbt/references/Ortega2013.txt new file mode 100644 index 00000000..05972c1d --- /dev/null +++ b/papers/project_paper_4_fbt/references/Ortega2013.txt @@ -0,0 +1,1688 @@ +Thermodynamics as a theory of decision-making +with information processing costs + +Pedro A. Ortega and Daniel A. Braun + +July 31, 2012 + +Abstract + +Perfectly rational decision-makers maximize expected utility, but cru- +cially ignore the resource costs incurred when determining optimal actions. +Here we propose an information-theoretic formalization of bounded ratio- +nal decision-making where decision-makers trade off expected utility and +information processing costs. Such bounded rational decision-makers can +be thought of as thermodynamic machines that undergo physical state +changes when they compute. Their behavior is governed by a free en- +ergy functional that trades off changes in internal energy—as a proxy for +utility—and entropic changes representing computational costs induced +by changing states. As a result, the bounded rational decision-making +problem can be rephrased in terms of well-known concepts from statis- +tical physics. +In the limit when computational costs are ignored, the +maximum expected utility principle is recovered. We discuss the relation +to satisficing decision-making procedures as well as links to existing theo- +retical frameworks and human decision-making experiments that describe +deviations from expected utility theory. Since most of the mathematical +machinery can be borrowed from statistical physics, the main contribution +is to axiomatically derive and interpret the thermodynamic free energy as +a model of bounded rational decision-making. + +1 +Introduction + +In everyday life decision-makers often have to make fast and frugal choices +[1, 2]. Consider, for example, an antelope that quickly has to choose a direction +of flight when faced with a predator. By the time an antelope had considered +all possible flight paths to determine the optimal one, it would most probably +be already eaten. In general, decision-makers seem to trade off the expected +desirability of the consequences of an action against the effort and resources +(time, money, food, computational effort, knowledge, opportunity costs, etc.) +required for searching the optimum [3, 4]. +Classic theories of decision making generally ignore information-processing +costs by assuming that decision makers always pick the option with maximum + +1 + + +return—irrespective of the effort or the resources it might take to find or com- +pute the optimal action [5, 6, 7]. Such decision-makers are described as perfectly +rational. However, being perfectly rational seems to contradict our intuition of +real-world decision-making, where information processing constraints play an +important role [1]. This has led to an abundant literature on bounded rational- +ity [8, 9, 10, 11]. Unlike perfectly rational decision makers, bounded rational +decision-makers are subject to information processing constraints, that is they +may have limited time and speed to process a limited amount of information. + +1.1 +Thermodynamic Intuition + +a) +b) + +pV + +(1 − p)V + +A + +B + +Figure 1: The Molecule-In-A-Box Device. +(a) Initially, the molecule moves +freely within a space of volume V delimited by two pistons. The compartments +A and B correspond to the two logical states of the device. (b) Then, the lower +piston pushes the molecule into part A having volume V ′ = pV . + +Here we follow a thermodynamic argument [12] that allows measuring re- +source (or information) costs in physical systems in units of energy. The gen- +erality of the argument relies on the fact that ultimately any real agent has to +be incarnated in a physical system, as any process of information processing +must always be accompanied by a pertinent physical process [13]. In the fol- +lowing we conceive of information processing as changes in information states +(i.e. ultimately changes of probability distributions), which consequently im- +plies changes in physical states, such as flipping gates in a transistor, changing +voltage on a microchip, or even changing location of a gas particle. Such state +changes in physical systems are not for free, that is the do not happen sponta- +neously. Consequently, if we want to control a physical system into a desirable +state we also have to take into consideration that changing from the current +state to the desirable state incurs a cost. +According to Landauer’s principle, one can postulate a formal correspon- +dence between one unit of information and one unit of energy [14, 15, 16]. +Consider representing one bit of information using one of the following logical +devices: a molecule that can be located either on the top or the bottom part of + +2 + + +a box; a coin whose face-up side can be either head or tail; a door that can be +either open or closed; a train that can be orientated facing either north or south; +and so forth. Assume that all these devices are initialized in an undetermined +logical state, where the first state has probability p and the second probability +1 − p. Now, imagine you want to set these devices to their first logical state. +In the case of the molecule in a box, this means the following. Initially, the +molecule is uniformly moving around within a space confined by two pistons as +depicted in Figure 1a. Assuming that the initial volume is V , the molecule has +to be pushed by the lower piston into the upper part of the box having volume +V ′ = pV (Figure 1b). From information theory, we know that the number of +bits that we fix by this operation is given by − log p. +To make things concrete, we assume that the device has diathermal walls +and is immersed in a heat bath at constant temperature T . Since the walls are +diathermal, the temperature inside of the box is maintained at the temperature +of the heat bath. We model the particle as an ideal gas. When an ideal gas +is compressed under isothermal conditions from an initial volume V to a final +volume V ′, then the work is calculated as + +W = − +� V ′ + +V + +NkT + +V +dV = NkT ln V + +V ′ , +(1) + +where N ≥ 0 is the amount of substance and k > 0 is the Boltzmann constant. +The minus sign is just a convention to denote work done by the piston rather +than by the gas. If we assume N = 1 and make use of the fact that V ′ = pV +we get + +W = kT ln V + +pV = −kT ln p = − kT + +log e log p = −γmol log p, + +where the constant γmol := +RT +log e > 0 can be interpreted as the conversion factor +between one unit of information and one unit of energy for the molecule-in-a-box +device. +How do we compute the information and work for the case of the coin, door +and train devices? The important observation is that we can model these cases +as if they were like molecule-in-a-box devices, with the difference that their +conversion factors between units of information and units of work are different. +Hence, the number of bits fixed while these devices are set to the first state is +given by − log p, i.e. exactly as in the case of the molecule. However, the work +is given by +−γcoin log p, +−γdoor log p, +and +− γtrain log p + +respectively, where γcoin, γdoor and γtrain are the associated conversion factors +between units of information. Obviously, γmol ≤ γcoin ≤ γdoor ≤ γtrain. The +point is that changes in knowledge states are costly and that these costs are +proportional to the information. In the next section, we derive a general ex- +pression of information costs in physical systems that make decisions. + +3 + + +2 +Information-Theoretic Foundations + +2.1 +Resource Costs + +We model any observable sequential process, such as a sequence of interactions +or a sequence of computation steps, as a filtration on a measure space. +To +simplify our exposition, we consider only finite measure spaces. +Let (Ω, Σ) +denote a measurable space, where Ω denotes the sample space and where Σ is +a σ-algebra on Ω. Let p be a conditional probability measure on (Ω, Σ), such +that for any two events A, B ∈ Σ, p(A|B) denotes the conditional probability of +the A given B, where the condition B plays the role of the current information +state of the process. The sequential realization of a process is modelled as a +sequence of conditions A1, A2, . . . , AT on the sample space Ω, where each new +condition At refines the current information state � +τ≤t Aτ by excluding the +complement A∁ +t . +We further assume that a transformation of an information state from B to +(A ∩ B) entails a cost ρ(A|B) that could be measured in dollars, time or any +arbitrary scale of effort. Moreover, we assume that this transformation cost is +decomposable; that is, if we undergo a knowledge change from C to (A∩B ∩C), +then we should pay the same cost as undergoing a change first from C to (B∩C) +and then from (B ∩ C) to (A ∩ B ∩ C). Finally, the quintessential information- +theoretic postulate is that conditional probabilities impose a monotonic order +over transformation costs1. We can sum up our postulates as follows: + +Definition 1 (Axioms of Transformation Costs). Let (Ω, Σ) be a measurable +space and let p : (Σ × Σ) → [0, 1] be a conditional probability measure over +Σ (i.e. for any A ∈ Σ, p(·|A) is a probability measure over A). A function +ρ : (Σ × Σ) → R+ is a transformation cost function for p iff it has the following +three properties for all events A, B, C, D ∈ Σ: + +A1. real-valued: +∃f, +ρ(A|B) = f +� +p(A|B) +� +∈ R, +A2. additive: +ρ(A ∩ B|C) = ρ(A|C) + ρ(B|A ∩ C), + +A3. monotonic: +[ρ(A|B) > ρ(C|D)] +⇔ +[p(A|B) ≶ p(C|D)]. + +These three properties enforce a strict correspondence between probabilities +and transformation costs [18, 19]. + +Theorem 1 (Transformation Costs ↔ Probabilities). If f is such that ρ(A|B) = +f(p(A|B)) for every choice of the probability space (Ω, Σ, p), then f is of the form + +f(·) = − 1 + +β log(·), + +where β is a real parameter. + +1This intuition is central for optimal coding theory where short codewords are assigned +to frequent events and long codewords are assigned to rare events [17]. Therefore, we could +regard the codeword length as a valuable resource that we have to bet on events with different +probabilities. + +4 + + +That is, the transformation cost ρ(A|B) is proportional to the information +content − log p(A|B), where the parameter β plays the role of the conversion +factor. +The logarithmic mapping between probabilities and “costs” is well- +known in information theory, and there are many possible ways to derive it +[20, 21]. The important observation is that our derivation stems purely from +postulates regarding transformation costs. +According to Definition 1, transformation costs measure the relative cost of +an event relative to a reference event. However, we can also introduce an absolute +cost measure to single events such that transformation costs are obtained as +differences. + +Definition 2 (Potential). Let ρ be a transformation cost function. A set func- +tion φ : Σ → R is called a cost potential for ρ iff for all A, B ∈ Σ, + +φ(Ω) := φ0 +φ(A ∩ B) := φ(B) + ρ(A|B) +∀A, B ∈ Σ, + +where φ0 is an arbitrary real value. + +One can easily verify that this potential is well defined for all events, and +that ρ(A|B) = φ(A ∩ B) − φ(B). It captures the intuition that starting out +from the high-probability event B with potential φ(B) one has to pay the cost +ρ(A|B) to arrive at the low-probability event A ∩ B with potential φ(A ∩ B). +In the following, consider a reference set S ∈ Σ having a measurable parti- +tion X. Cost potentials have an important recursive structure: the cost potential +of an event is uniquely determined by the potential of its constituent events. +If X is a measurable partition of a reference event S ∈ Σ, then + +φ(S) = − 1 + +β log +� + +x∈X +e−βφ(x). +(2) + +Furthermore, the probability of a member x ∈ X of the partition relative to S +can be expressed as a Gibbs measure: + +p(x|S) = e−βφ(x) + +e−βφ(S) = +e−βφ(x) + +� +x∈X e−βφ(x) . +(3) + +In statistical physics it is well-known that the Gibbs measure satisfies a varia- +tional principle in the free energy, which is defined as + +Fβ[q] := +� + +x∈X +q(x)φ(x) + 1 + +β + +� + +x∈X +q(x) log q(x). +(4) + +More specifically, it is well known that for any probability measure q over the +partition X of S, + +F[q] ≥ F[p] = − 1 + +β log φ(S), +(5) + +where the lower bound is attained by the Gibbs measure p(x) ∝ e−βφ(x). Equa- +tions (2) to (5) constitute fundamental results that will be generalized and +interpreted in the next section. + +5 + + +2.2 +Gains and Losses + +Equipped with the results from the preceding section, we can now proceed to +model a bounded rational decision maker. Because transformation costs matter, +we model a decision as a transformation of a prior behavior into a final behavior, +where we represent the direction of change as a utility criterion. +The Gibbs measure in (3) allows us describing a probability measure p over +a partition X in terms of a cost potential φ over X. In particular, we see that a +decision-maker’s a priori behavior or belief described by p0(x) and φ0(x) changes +to p(x) and φ(x) if he is exposed to the gain (or loss) U(x), such that + +φ(x) = φ0(x) − U(x) +(6) + +and +p(x) ∝ e−βφ0(x)+βU(x) ∝ p0(x)eβU(x) +(7) + +as illustrated in Figure 1. The function U represents either gains or losses and +not absolute levels of costs, because it expresses a difference in the potential +U(x) = φ0(x)−φ(x). The equilibrium distribution (7) that arises in a change can +also be characterized in terms of a variational principle, in a manner analogous +to (5). + +Theorem 2 (Negative Free Energy Difference). Let p0(x) and p(x) be the Gibbs +measures with potentials φ0(x) and φ(x) and resource parameter β. Let F0 and +F be the free energies minimized by p0 and p respectively. Then, the negative +free energy difference −∆F = F0 − F is + +−∆F = +� + +x∈X +p(x)U(x) − 1 + +β + +� + +x∈X +p(x) log p(x) + +p0(x), +(8) + +where U(x) = φ0(x) − φ(x). + +Since the difference in the negative free energy −∆F = F − F0 has the same +dependency on p as the free energy F, we can use −∆F directly as a variational +principle in p. + +Corollary 3 (Variational Principle). The negative free energy difference pro- +vides a variational principle for the equilibrium distribution, i.e. + +−∆F[q] := +� + +x∈X +q(x)U(x) − 1 + +β + +� + +x∈X +q(x) log q(x) + +p0(x) + +is maximized by + +p(x) = 1 + +Z p0(x)eβU(x), +where +Z := +� + +x∈X +eβU(x). + +Furthermore, +∆F[q] ≤ ∆F[p] = 1 + +β log Z. + +6 + + +φ0 + +−U + +φ = φ0 − U + +Initial +Final + +low + +high + +Probability + +Figure 2: Representing a decision maker as a thermodynamic system, the be- +havior of the decision-maker exposed to a gain U can be expressed as a change +of his initial cost potential φ0 to a final cost potential φ, where φ = φ0 − U. +The choice or belief probabilities of the decision-maker change according to (7) +from p0 to p. + +2.3 +Choice & Belief Probabilities + +The distribution (7) can be interpreted both as an action or observation prob- +ability in the context of bounded rational decision-making. In the case of ac- +tions, p0 represents the a priori choice probability of the agent which is refined +to the choice probability p when evaluating the imposed gain (or loss) U. The +associated change in probability depends on the resource parameter β and cor- +responds to the computation that is necessary to evaluate the gains (or losses). +In the case of observations, p0 represents the a priori belief of the agent given +by a probabilistic model, which is then distorted due to the presence of possible +gains (or losses) that are evaluated by the holder of the belief. This way, model +uncertainty and risk-aversion can be parameterized by β. +For different values of β the distribution (7) has the following limits + +lim +β→∞ p(x) += +δ(x − x∗), +x∗ = max +x +U(x) + +lim +β→0 p(x) += +p0(x) + +lim +β→−∞ p(x) += +δ(x − x∗), +x∗ = min +x U(x). + +In the case of actions the three limits imply the following: The limit β → ∞ +corresponds to the perfectly rational actor that infallibly selects the action that +maximizes gain (or minimizes loss −U(x). The limit β → 0 is an actor without +resources that simply selects his action according to his prior. The limit β → +−∞ corresponds to an actor that is perfectly “anti-rational” and always selects +the action with the worst outcome. In the case of observations the three limits +correspond to an extremely optimistic observer (β → ∞) who believes only in +the best possible outcome, an extremely pessimistic observer (β → −∞) who +anticipates only the worst, and a risk-neutral Bayesian observer (β → 0) who +simply relies on the probabilistic model p0. + +7 + + +2.4 +The Certainty Equivalent + +In statistical physics [22], the free energy difference + +∆A = ∆E − Q = W + +measures the amount of available “good energy” (work W) by subtracting the +“bad energy” (heat Q) from the total energy ∆E = E[U]. The crucial physical +intuition is that we have uncertainty about some aspects of the objects that +make up the heat energy, for example we do not know the exact trajectories +of all gas particles at temperature β. This uncertainty means that we do not +have full control over the objects and cannot extract all the energy as work +[12]. Economically speaking, the physical concept of work, and therefore also +the difference in free energy, measures the certainty equivalent of a gain (or +loss) that is contaminated by uncertainty. In general, we can therefore use the +free energy difference to ascribe a certainty equivalent value to choice situations +of the form (7). As can be seen from Corollary 3, this value is given by the +log partition function, i.e. the logarithm of the normalization constant Z. For +different values of β, the certainty equivalent takes the following limits + +lim +β→∞ +1 +β log Z += +max +x +U(x) + +lim +β→0 +1 +β log Z += +� + +x +p0(x)U(x) + +lim +β→−∞ +1 +β log Z += +min +x U(x). + +Again, the case β → ∞ corresponds to the perfectly rational actor (or the +extremely optimistic observer), the case β → −∞ corresponds to the perfectly +“anti-rational” actor (or the extremely pessimistic observer) and the case β → 0 +corresponds to the actor that has no resources (or the risk-neutral observer) such +that the best one can expect is the expected gain or loss. +Corollary 3 has two interpretations in statistical physics, either as an in- +stantiation of a minimum energy principle or as a maximum entropy principle +[22]. Accordingly, (7) can either be seen as the distribution that maximizes the +entropy given a constraint on the expectation value of U or as the distribution +that minimizes the expectation of −U given a constraint on the entropy of p. +In the context of observer modeling, the first interpretation provides a principle +for estimation and the second interpretation provides a principle for bounded +rational decision-making in the case of acting, which is a maximum expected +gain principle with a relative entropy constraint that bounds the information- +processing capacity of the decision-maker. In the relative entropy we recognize +the term 1 + +β log p(x) as our transformation costs ρ from Theorem 1 such that we +can express the negative free energy difference −∆F as + +−∆F = E[U] − E[R], + +where U(x) = φ0(x)− φ(x) represents gains (or losses) and R(x) = ρ(x)− ρ0(x) +represents the extra resource costs required to achieve the gain (or loss) U. + +8 + + +We can therefore see how the variational principle of Corollary 3 formalizes a +trade-off between expected gains (or losses) and information processing costs. + +3 +Summary of Main Concepts + +In decision theory, choices between alternatives are usually formalized as choices +between lotteries, where a lottery is formalized as a set X of possible out- +comes, a probability distribution p0 over X, and a real-valued function U over +X called the utility function. In particular expected utility theory predicts that +a decision-maker always chooses the lottery with the higher expected utility +E[U] = � +x p0(x)U(x). Here we introduce the notion of a bounded lottery as a +lottery that is additionally characterized by a resource parameter β ∈ R that +captures the resource constraints of the decision-maker. +We have derived a thermodynamic framework for bounded lotteries from +simple axioms that measure information processing cost—see also [19]. +The +most important difference of bounded decision-making compared to perfectly +rational decision-making is that the bounded decision-maker will not be able +to choose infallibly the best lottery. In fact, the resource constraints lead to +stochastic choice behavior which can be characterized by a probability distribu- +tion. The decision process then transforms an initial choice probability p0 into +a final choice probability p by taking into account the utility gains (or losses) +and the transformation costs. This transformation process can be formalized as + +p(x) = 1 + +Z p0(x)eβU(x), +where +Z = +� + +x′ +p0(x′)eβU(x′). +(9) + +Accordingly, the choice pattern of the decision-maker is predicted by the prob- +ability p. Crucially, the probability p extremizes the variational principle + +max +p + +� � + +x +p(x)U(x) − 1 + +β + +� + +x +p(x) log p(x) + +p0(x) + +� +. +(10) + +These two terms can be interpreted as determinants of bounded rational decision- +making in that they formalize a trade-off between an expected utility gain (first +term) and the information processing cost of transforming p0 into p (second +term). The certainty equivalent value of a bounded lottery can be obtained by +inserting the choice probability p from (9) into (10), yielding + +V = 1 + +β log +�� + +x +p0(x)eβU(x) +� +, +(11) + +which corresponds to the log partition sum. For different values of β, the cer- + +9 + + +a) + +b) + +Umax + +Umin + +E[U] +β + +β1 + +β1 + +β2 + +β2 + +β3 + +β3 + +Figure 3: a) Negative free energy difference ∆F versus the resource parame- +ter β. The resource parameter allows modeling decision-makers with bounded +resources, either when generating their own actions (β > 0) or when anticipat- +ing their environment (β < 0). The negative free energy difference corresponds +to the certainty equivalent. b) Distribution over the outcomes depending on +the resource parameter β. For large positive β the distribution concentrates on +the outcome with maximum gain Umax. For large negative β the distribution +concentrates on the worst outcome with gain Umin. For β = 0 the outcomes +follow the given distribution p0. + +tainty equivalent takes the following limits + +lim +β→∞ +1 +β log Z += +max +x +U(x) + +lim +β→0 +1 +β log Z += +� + +x +p0(x)U(x) + +lim +β→−∞ +1 +β log Z += +min +x U(x). + +The case β → ∞ corresponds to the perfectly rational actor (or the extremely +optimistic observer), the case β → −∞ corresponds to the perfectly “anti- +rational” actor (or the extremely pessimistic observer) and the case β → 0 +corresponds to the actor that has no resources (or the risk-neutral observer) +such that the best one can expect is the expected gain or loss. For illustration +see Figure 2. + +10 + + +4 +Bounded Rationality and Satisficing + +Herbert Simon [23] proposed in the 50s that bounded rational decision-makers +do not commit to an unlimited optimization by searching for the absolute best +option. Rather, they follow a strategy of satisficing, i.e. they settle for an option +that is good enough in some sense. Since then, it has been debated whether sat- +isficing decision-makers can be described as bounded rational decision-makers +that act optimally under resource constraints or whether optimization is the +wrong concept altogether [11]. If decision-makers did indeed explicitly attempt +to solve such a constrained optimization problem, this would lead to an infinite +regress and the paradoxical situation that a bounded rational decision-maker +would have to solve a more complex (i.e. constrained) optimization problem +than a perfectly rational decision-maker. +To resolve this paradox, the bounded rational decision maker must not be +able to reason about his constraints. He just searches randomly for the best +option, until his resources run out. An observer will then be able to assign a +probability distribution to the decision-maker’s choices and investigate how this +probability distribution changes depending on the available resources. Consider, +for example, an anytime algorithm that will compute a solution more and more +precisely the more time it has at its disposal. As one does not want to wait +forever for an answer, the anytime computation will be interrupted at some +point where one assumes that the answer is going to be good enough. This +concept of satisficing can be used to interpret Equation 7 which describes the +choice rule of a bounded rational decision-maker. +Consider the problem of picking the largest number in a sequence U0, U1, U2, . . . +of i.i.d. data, where each Ui ∈ U is drawn from a source with probability dis- +tribution µ. This could be, for instance, an urn with numbered balls that we +draw with replacement and we always keep track of the largest number seen so +far. After m draws the largest number will be given by + +v := max{U1, U2, . . . , Um}. + +Naturally, the larger the number of draws, the higher the chances of observing a +large number. The cumulative distribution function of choosing v after m draws +is given by +Fm(v) = F0(v)m, +(12) + +where F0 is the cumulative distribution function of µ [24]. If we only cared about +finding the maximum with absolute certainty then we would need to draw an +infinite amount of times. However, a bounded rational decision-maker would +stop after a certain time, when he feels that the benefit of further exploration +does not justify the effort of further drawings. Thus, the number of draws in +this example can be regarded as a resource and the numbers on the balls can +be regarded as utilities. The behavior of the bounded rational decision-maker +is then stochastic even though he acts perfectly deterministically, in the sense +that he chooses option v with probability (12) given the resource constraint +m. According to (12), the more resources a decision-maker spends, the more he + +11 + + +a) +b) + +M = 0 +M = 8 + +M = 32 +M = 128 + +Umax + +0 +M + +E[v] + +E[v] − M · c + +Figure 4: a) Distributions over the maximum for various sample sizes (M + +1). The distribution µ over the ten values v in U = {1, 2, 3, . . ., 10} follows a +truncated Poisson distribution with parameter λ = 5, as can be seen in the +plot for M = 0. The distribution approaches a delta function over v = 10 for +increasing values of M. b) The expected maximum v versus sample size (M +1). +The incremental gain of the expected maximum is marginally decreasing as the +sample size increases (red). If the sampling process is associated with a cost— +e.g. c = 0.02 per sample in the figure—, then the penalized expected maximum +(black) reaches a unique maximum for a finite sample size—the optimal sample +size is M = 35 in the figure. + +resembles a perfectly rational decision-maker that chooses the maximum number +(Figure 1a), since the expected utility increases monotonically with the amount +of resources spent (Figure 1b). Importantly, however, note that the marginal +increase in the expected utility diminishes with larger effort—hence larger and +larger effort pays out less and less in the end. Below we formalize this trade-off. +Here we show that the boundedness parameter β plays an analogous role to +the number of draws m. In the limit of a continuous cumulative function F0, +the density after m draws is given by pm(v) = +d +dvF0(v)m. We can now compute +the log odds for two random outcomes v and v′, which results in + +log pm(v) + +pm(v′) = (m − 1) log F0(v) + +F0(v′) + log µ(v) + +µ(v′), + +where F0(v) is again the cumulative of µ. If we require the probabilities pm(v) to +be representable by a distribution of the exponential family such that pm(v) = +µ(v) exp(αU(v)) +� +dv′µ(v′) exp(αU(v′)), we see that the log odds have the following relation + +log pm(v) + +pm(v′) = α (U(v) − U(v′)) + log µ(v) + +µ(v′). + +We see that α and m play the role of the number of samples or computations. +In general, the following theorem can be shown to hold. + +12 + + +Theorem 4. Let X be a finite set. Let Q and M be strictly positive probability +distributions over X. Let α be a positive integer. Define Mα as the probability +distribution over the maximum of α samples from M. Then, there are strictly +positive constants δ and ξ depending only on M such that for all α, +���� +Q(x)eαU(x) + +� + +x′ Q(x′)eαU(x′) − Mα(x) +���� ≤ e−(α−ξ)δ. + +Consequently, one can interpret the inverse temperature as a resource param- +eter that determines how many samples are drawn to estimate the maximum. +Note that the distribution M is arbitrary as long as it has the same support +as Q. +This interpretation can be extended to a negative α, by noting that +αU(x) = (−α)(−U(x)), i.e. instead of the maximum we take the minimum of +−α samples. + +5 +Sequential Decision-Making + +In the case of sequential decision-making the assumption of uniform temper- +atures has to be relaxed—the proofs of the following theorems can be found +in [25]. In general, we can then dedicate different amounts of computational +resources to each node of a decision tree. However, this requires a translation +between a tree with a single temperature and to a tree with different tempera- +tures. This translation can be achieved using the following theorem + +Theorem 5. Let P be the equilibrium distribution for a given inverse tem- +perature α, utility function U and reference distribution Q. If the temperature +changes to β while keeping P and Q fixed, then the utility function changes to + +V (x) = U(x) − +� +1 +α − 1 + +β +� +log P(x) + +Q(x). + +If we now define the reward as the change in utility of two subsequent nodes, +then the rewards of the resulting decision tree are given by + +R(xt|x 0) add value to the expected util- +ity in the face of variability. Risk-sensitivity biases the beliefs about the +environment optimistically (collaborative environment) or pessimistically +(adversarial environment). Alternatively, one could regard a collabora- +tive environment also as a bounded rational controller that can choose +its own observation—that is the environment behaves like an extension of +the agent with partial control. Importantly, the stress function is typically +assumed in risk-sensitive control schemes in the literature, whereas here +it falls out naturally—see [29] for more details. + +4. Robust control. +When assuming β(x