Living Reviews in Relativity manuscript No.
(will be inserted by the editor)

The causal set approach to quantum gravity

Sumati Surya

Received: date / Accepted: date

Abstract The causal set theory (CST) approach to quantum gravity postu-
lates that at the most fundamental level, spacetime is discrete, with the space-
time continuum replaced by locally finite posets or “causal sets”. The partial
order on a causal set represents a proto-causality relation while local finite-
ness encodes an intrinsic discreteness. In the continuum approximation the
former corresponds to the spacetime causality relation and the latter to a fun-
damental spacetime atomicity, so that finite volume regions in the continuum
contain only a finite number of causal set elements. CST is deeply rooted in
the Lorentzian character of spacetime, where a primary role is played by the
causal structure poset. Importantly, the assumption of a fundamental discrete-
ness in CST does not violate local Lorentz invariance in the continuum ap-
proximation. On the other hand, the combination of discreteness and Lorentz
invariance gives rise to a characteristic non-locality which distinguishes CST
from most other approaches to quantum gravity.
In this review we give a broad, semi-pedagogical introduction to CST,
highlighting key results as well as some of the key open questions. This review
is intended both for the beginner student in quantum gravity as well as more
seasoned researchers in the field.

Keywords Causal set theory · Quantum gravity

Contents

1
Overview
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2

2
A historical perspective
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7

S. Surya
Raman Research Institute,
CV Raman Ave, Sadashivanagar,
Bangalore 560080,
India
E-mail: ssurya@rri.res.in

arXiv:1903.11544v2  [gr-qc]  28 Aug 2019


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Sumati Surya

3
The causal set hypothesis
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11

3.1
The Hauptvermutung or fundamental conjecture of CST . . . . . . . . . . . .
18

3.2
Discreteness without Lorentz breaking . . . . . . . . . . . . . . . . . . . . . .
21

3.3
Forks in the road: What makes CST so “different”? . . . . . . . . . . . . . . .
23

4
Kinematics or geometric reconstruction
. . . . . . . . . . . . . . . . . . . . . . . .
25

4.1
Spacetime dimension estimators . . . . . . . . . . . . . . . . . . . . . . . . . .
27

4.2
Topological invariants
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30

4.3
Geodesic distance: timelike, spacelike and spatial . . . . . . . . . . . . . . . .
31

4.4
The d’Alembertian for a scalar field
. . . . . . . . . . . . . . . . . . . . . . .
34

4.5
The Ricci scalar and the Benincasa–Dowker action . . . . . . . . . . . . . . .
36

4.6
Boundary terms for the causal set action . . . . . . . . . . . . . . . . . . . . .
39

4.7
Localisation in a causal set
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
42

4.8
Kinematical entropy
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44

4.9
Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45

5
Matter on a continuum-like causal set
. . . . . . . . . . . . . . . . . . . . . . . . .
45

5.1
Causal set Green functions for a free scalar field
. . . . . . . . . . . . . . . .
45

5.2
The Sorkin–Johnston (SJ) vacuum . . . . . . . . . . . . . . . . . . . . . . . .
49

5.3
Entanglement entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50

5.4
Spectral dimensions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52

6
Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52

6.1
Classical sequential growth models . . . . . . . . . . . . . . . . . . . . . . . .
53

6.2
Observables as beables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58

6.3
A route to quantisation: The quantum measure . . . . . . . . . . . . . . . . .
60

6.4
A continuum-inspired dynamics . . . . . . . . . . . . . . . . . . . . . . . . . .
61

7
Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67

7.1
The 1987 prediction for Λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68

8
Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71

A Notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71

1 Overview

In this review, causal set theory (CST) refers to the specific proposal made by
Bombelli, Lee, Meyer and Sorkin (BLMS) in their 1987 paper (Bombelli et al
1987). In CST, the space of Lorentzian geometries is replaced by the set of
locally finite posets, or causal sets. These causal sets encode the twin principles
of causality and discreteness. In the continuum approximation of CST, where
elements of the causal set set represent spacetime events, the order relation on
the causal set corresponds to the spacetime causal order and the cardinality of
an “order interval” to the spacetime volume of the associated causal interval.
This review is intended as a semi-pedagogical introduction to CST. The
aim is to give a broad survey of the main results and open questions and to
direct the reader to some of the many interesting open research problems in
CST, some of which are accessible even to the beginner.
We begin in Sect. 2 with a historical perspective on the ideas behind CST.
The twin principles of discreteness and causality at the heart of CST have both
been proposed – sometimes independently and sometimes together – starting
with Riemann (1873) and Robb (1914, 1936), and somewhat later by Zeeman
(1964); Kronheimer and Penrose (1967); Finkelstein (1969); Hemion (1988)
and Myrheim (1978), culminating in the CST proposal of BLMS (Bombelli
et al 1987). The continuum approximation of CST is an implementation of
a deep result in Lorentzian geometry due to Hawking et al (1976) and its


The causal set approach to quantum gravity
3

generalisation by Malament (1977), which states that the causal structure
determines the conformal geometry of a future and past distinguishing causal
spacetime. In following this history, the discussion will be necessarily somewhat
technical. For those unfamiliar with the terminology of causal structure we
point to standard texts (Hawking and Ellis 1973; Beem et al 1996; Wald 1984;
Penrose 1972).
In Sect. 3, we state the CST proposal and describe its continuum approxim-
ation, in which spacetime causality is equivalent to the order relation and finite
spacetime volumes to cardinality. Not all causal sets have a continuum approx-
imation – in fact we will see that most do not. Those that do are referred to
as manifold-like. Important to CST is its “Hauptvermutung” or fundamental
conjecture, which roughly states that a manifold-like causal set is equivalent
to the continuum spacetime, modulo differences up to the discreteness scale.
Much of the discussion on the Hauptvermutung is centered on the question
of how to estimate the closeness of Lorentzian manifolds or more generally,
causal sets. While there is no full proof of the conjecture, there is growing
body of evidence in its favour as we will see in Sect. 4. An important outcome
of CST discreteness in the continuum approximation is that it does not violate
Lorentz invariance as shown in an elegant theorem by Bombelli et al (2009).
Because of the centrality of this result we review this construction in some
detail. The combination of discreteness and Lorentz invariance moreover gives
rise to an inherent and characteristic non-locality, which distinguishes CST
from other discrete approaches. Following Sorkin (1997), we then discuss how
the twin principles behind CST force us to take certain “forks in the road” to
quantum gravity.
We present some of the key developments in CST in Sects. 4, 5 and 6. We
begin with the kinematical structure of theory and the program of “geometric
reconstruction” in Sect. 4. Here, the aim is to reconstruct manifold invariants
from order invariants in a manifold-like causal set. These are functions on the
causal set that are independent of the labelling or ordering of the elements
in the causal set. Finding the appropriate order invariants that correspond to
manifold invariants can be challenging, since there is little in the mathematics
literature which correlates order theory to Lorentzian geometry via the CST
continuum approximation. Extracting such invariants requires new technical
tools and insights sometimes requiring a rethink of familiar aspects of con-
tinuum Lorentzian geometry. We will describe some of the progress made in
this direction over the years (Myrheim 1978; Brightwell and Gregory 1991;
Meyer 1988; Bombelli and Meyer 1989; Bombelli 1987; Reid 2003; Major et al
2007; Rideout and Wallden 2009; Sorkin 2007b; Benincasa and Dowker 2010;
Benincasa 2013; Benincasa et al 2011; Glaser and Surya 2013; Roy et al 2013;
Buck et al 2015; Cunningham 2018a; Aghili et al 2018; Eichhorn et al 2018).
The correlation between order invariants and manifold invariants in the con-
tinuum approximation lends support for the Hauptvermutung and simultan-
eously establishes weaker, observable-dependent versions of the conjecture.
Somewhere between dynamics and kinematics is the study of quantum
fields on manifold-like causal sets, which we describe in Sect. 5. The simplest


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Sumati Surya

system is free scalar field theory on a causal set approximated by d-dimensional
Minkowski spacetime Md. Because causal sets do not admit a natural Hamilto-
nian framework, a fully covariant construction is required to obtain the quantum
field theory vacuum. A natural starting point is the advanced and retarded
Green functions for a free scalar field theory since it is defined using the causal
structure of the spacetime. The explicit form for these Green functions were
found for causal sets approximated by Md for d = 2, 4 (Daughton 1993; John-
ston 2008, 2010) as well as de Sitter spacetime (Dowker et al 2017). In trying
to find a quantisation scheme on the causal set without reference to the con-
tinuum, Johnston (2009) found a novel covariant definition of the discrete
scalar field vacuum, starting from the covariantly defined Peierls’ bracket for-
mulation of quantum field theory. Subsequently Sorkin (2011a) showed that the
construction is also valid in the continuum, and can be used to give an alternat-
ive definition of the quantum field theory vacuum. This Sorkin–Johnston (SJ)
vacuum provides a new insight into quantum field theory and has stimulated
the interest of the algebraic field theory community (Fewster and Verch 2012;
Brum and Fredenhagen 2014; Fewster 2018). The SJ vacuum has also been
used to calculate Sorkin’s spacetime entanglement entropy (SSEE) (Bombelli
et al 1986; Sorkin 2014) in a causal set (Saravani et al 2014; Sorkin and Yazdi
2018). The calculation in d = 2 is surprising since it gives rise to a volume law
rather than an area law. What this means for causal set entanglement entropy
is still an open question.
In Sect. 6, we describe the CST approach to quantum dynamics, which
roughly follows two directions. The first, is based on “first principles”, where
one starts with a general set of axioms which respect microscopic covariance
and causality. An important class of such theories is the set of Markovian
classical sequential growth (CSG) models of Rideout and Sorkin (Rideout and
Sorkin 2000a, 2001; Martin et al 2001; Rideout 2001; Varadarajan and Rideout
2006), which we will describe in some detail. The dynamical framework finds a
natural expression in terms of measure theory, with the classical covariant ob-
servables represented by a covariant event algebra A over the sample space Ωg
of past finite causal sets (Brightwell et al 2003; Dowker and Surya 2006). One
of the main questions in CST dynamics is whether the overwhelming entropic
presence of the Kleitman–Rothschild (KR) posets in Ωg can be overcome by
the dynamics (Kleitman and Rothschild 1975). These KR posets are highly
non-manifold-like and “static”, with just three “moments of time”. Hence, if
the continuum approximation is to emerge in the classical limit of the the-
ory, then the entropic contribution from the KR posets should be suppressed
by the dynamics in this limit. In the CSG models, the typical causal sets
generated are very “tall” with countable rather than finite moments of time
and, though not quite manifold-like, are very unlike the KR posets or even
the subleading entropic contributions from non-manifold-like causal sets Dhar
(1978, 1980). The CSG models have generated some interest in the mathemat-
ics community, and new mathematical tools are now being used to study the
asymptotic structure of the theory (Brightwell and Georgiou 2010; Brightwell
and Luczak 2012, 2011, 2015).


The causal set approach to quantum gravity
5

In CST, the appropriate route to quantisation is via the quantum meas-
ure or decoherence functional defined in the double-path integral formulation
(Sorkin 1994, 1995, 2007d). In the quantum versions of the CSG (quantum
sequential growth or QSG) models the transition probabilities of CSG are re-
placed by the decoherence functional. While covariance can be easily imposed,
a quantum version of microscopic causality is still missing (Henson 2005).
Another indication of the non-triviality of quantisation comes from a prosaic
generalisation of transitive percolation, which is the simplest of the CSG mod-
els. In this “complex percolation” dynamics the quantum measure does not
extend to the full algebra of observables which is an impediment to the con-
struction of covariant quantum observables (Dowker et al 2010c). This can be
somewhat alleviated by taking a physically motivated approach to measure
theory (Sorkin 2011b). An important future direction is to construct covariant
observables in a wider class of quantum dynamics and look for a quantum
version of coupling constant renormalisation.
Whatever the ultimate quantum dynamics however, little sense can be
made of the theory without a fully developed quantum interpretation for closed
systems, essential to quantum gravity. Sorkin’s co-event interpretation (Sorkin
2007a; Dowker and Ghazi-Tabatabai 2008) provides a promising avenue based
on the quantum measure approach. While a discussion of this is outside of the
scope of the present work, one can use the broader “principle of preclusion”,
i.e., that sets of zero quantum measure do not occur(Sorkin 2007a; Dowker
and Ghazi-Tabatabai 2008), to make a limited set of predictions in complex
percolation (Sorkin and Surya, work in progress).
The second approach to quantisation is more pragmatic, and uses the con-
tinuum inspired path integral formulation of quantum gravity for causal sets.
Here, the path integral is replaced by a sum over the sample space Ω of causal
sets, using the Benincasa–Dowker (BD) action, which limits to the Einstein–
Hilbert action (Benincasa and Dowker 2010). This can be viewed as an effect-
ive, continuum-like dynamics, arising from the more fundamental dynamics
described above. A recent analytic calculation in Loomis and Carlip (2018)
showed that a sub-dominant class of non-manifold-like causal sets, the bilayer
posets, are suppressed in the path integral when using the BD action, under
certain dimension dependent conditions satisfied by the parameter space. This
gives hope that an effective dynamics might be able to overcome the entropy
of the non-manifold-like causal sets.
In Surya (2012), Glaser and Surya (2016), and Glaser et al (2018), Markov
Chain Monte Carlo (MCMC) methods were used for a dimensionally restricted
sample space Ω2d of 2-orders, which corresponds to topologically trivial d = 2
causal set quantum gravity. The quantum partition function over causal sets
can be rendered into a statistical partition function via an analytic continu-
ation of a “temperature” parameter, while retaining the Lorentzian character
of the theory. This theory exhibits a first order phase transition (Surya 2012;
Glaser et al 2018) between a manifold-like phase and a layered, non-manifold-
like one. MCMC methods have also been used to examine the sample space Ωn
of n-element causal sets and to estimate the onset of asymptotia, character-


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Sumati Surya

ised by the dominance of the KR posets (Henson et al 2017). These techniques
have recently been extended to topologically non-trivial d = 2 and d = 3
CST (Cunningham and Surya, work in progress). While this approach gives
us expectation values of covariant observables which allows for a straightfor-
ward interpretation, relating it to the complex or quantum partition function
is non-trivial and an open problem.

In Sect. 7, we describe in brief some of the exciting phenomenology that
comes out of the kinematical structure of causal sets. This includes the mo-
mentum space diffusion coming from CST discreteness (“swerves”) (Dowker
et al 2004) and the effects of non-locality on quantum field theory (Sorkin
2007b), which includes a recent proposal for dark matter (Saravani and Af-
shordi 2017). Of these, the most striking is the 1987 prediction of Sorkin for
the value of the cosmological constant Λ (Sorkin 1991, 1997). While the ori-
ginal argument was a kinematic estimate, subsequently dynamical models of
fluctuating Λ were examined (Ahmed et al 2004; Ahmed and Sorkin 2013;
Zwane et al 2018) and have been compared with recent observations (Zwane
et al 2018). This is an exciting future direction of research in CST which inter-
faces intimately with observation. We conclude with a brief outlook for CST
in Sect. 8.

Finally, since this is an extensive review, to assist the reader we have made
a list of some of the key definitions, as well as the abbreviations in Appendix
A.

As is true of all other approaches to quantum gravity, CST is not as yet a
complete theory. Some of the challenges faced are universal to quantum grav-
ity as a whole, while others are specific to the approach. Although we have
developed a reasonably good grasp of the kinematical structure of CST and
some progress has been made in the construction of effective quantum dynam-
ics, CST still lacks a satisfactory quantum dynamics built from first principles.
Progress in this direction is therefore very important for the future of the pro-
gram. From a broader perspective, it is the opinion of this author that a deeper
understanding of CST will help provide key insights into the nature of quantum
gravity from a fully Lorentzian, causal perspective, whatever ultimate shape
the final theory takes.

It is not possible for this review to be truly complete. The hope is that the
interested reader will use it as a springboard to the existing literature. Several
older reviews exist with differing emphasis (Sorkin 1991, 2005b; Henson 2006b;
Dowker 2005; Sorkin 2009; Dowker 2005; Henson 2010; Wallden 2013), some
of which have an in depth discussion of the conceptual underpinnings of CST.
The focus of the current review is to provide as cohesive an account of the
program as possible, so as to be useful to a starting researcher in the field. For
more technical details, the reader is urged to look at the original references.


The causal set approach to quantum gravity
7

2 A historical perspective

One of the most important conceptual realisations that arose from the special
and general theories of relativity in the early part of the 20th century, was
that space and time are part of a single construct, that of spacetime. At a
fundamental level, one does not exist without the other. Unlike Riemannian
spaces, spacetime has a Lorentzian signature (−, +, +, +) which gives rise to
local lightcones and an associated global causal structure. The causal structure
(M, ≺) of a causal spacetime1 (M, g) is a partially ordered set or poset, with
≺ denoting the causal ordering on the “event-set” M.

Future Timelike

Past Timelike

Spacelike

Null

Fig. 1 The local lightcone of a Lorentzian spacetime.

Causal set theory (CST) as proposed in Bombelli et al (1987), takes the
Lorentzian character of spacetime and the causal structure poset in particular,
as a crucial starting point to quantisation. It is inspired by a long but sporadic
history of investigations into Lorentzian geometry, in which the connections
between (M, ≺) and the conformal geometry were eventually established. This
history, while not a part of the standard narrative of General Relativity, is
relevant to the sequence of ideas that led to CST. In looking for a quantum
theory of spacetime, (M, ≺) has also been paired with discreteness, though
the earliest ideas on discreteness go back to pre-quantum and pre-relativistic
physics. We now give a broad review of this history.

1 Henceforth, we will assume that spacetime is causal, i.e., without any closed causal
curves.


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Sumati Surya

The first few decades after the formulation of General Relativity were ded-
icated to understanding the physical implications of the theory and to finding
solutions to the field equations. The attitude towards Lorentzian geometry
was mostly practical: it was seen as a simple, though odd, generalisation of
Riemannian geometry.2 There were however early attempts to understand this
new geometry and to use causality as a starting point. Weyl and Lorentz (see
Bell and Kort´e 2016) used light rays to attempt a reconstruction of d di-
mensional Minkowski spacetime Md, while Robb (1914, 1936) suggested an
axiomatic framework for spacetime where the causal precedence on the collec-
tion of events was seen to play a critical role. It was only several decades later,
however, that the mathematical structure of Lorentzian geometry began to be
explored more vigorously.
In a seminal paper titled “Causality Implies the Lorentz Group”, Zeeman

(1964) identified the chronological poset (Md, ≺≺) in Md, where ≺≺ denotes
the chronological relation on the event-set Md. Defining a chronological auto-
morphism3 fa of Md as the chronological poset-preserving bijection

fa : Md → Md,
x ≺≺ y ⇔ fa(x) ≺≺ fa(y), ∀ x, y ∈ Md,
(1)

Zeeman showed that the group of chronological automorphisms GA is iso-
morphic to the group GLor of inhomogeneous Lorentz transformations and
dilations on Md when d > 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.


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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

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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

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20

30

40

50

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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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● Δ: ℓ=66

0
50
100
150
200
250

0.0

0.2

0.4

0.6

dh

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.


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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

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