intellecton/papers/references/Tegmark2000.txt

arXiv:quant-ph/9907009v2  10 Nov 1999

The Importance of Quantum Decoherence in Brain Processes

Max Tegmark
Institute for Advanced Study, Olden Lane, Princeton, NJ 08540; max@ias.edu
Dept. of Physics, Univ. of Pennsylvania, Philadelphia, PA 19104
(Submitted to Phys. Rev. E July 2 1999, accepted October 25)

Based on a calculation of neural decoherence rates, we ar-
gue that that the degrees of freedom of the human brain that
relate to cognitive processes should be thought of as a classical
rather than quantum system, i.e., that there is nothing funda-
mentally wrong with the current classical approach to neural
network simulations. We find that the decoherence timescales
(∼ 10−13 − 10−20 seconds) are typically much shorter than
the relevant dynamical timescales (∼ 10−3 − 10−1 seconds),
both for regular neuron firing and for kink-like polarization
excitations in microtubules. This conclusion disagrees with
suggestions by Penrose and others that the brain acts as a
quantum computer, and that quantum coherence is related
to consciousness in a fundamental way.

I. INTRODUCTION

In most current mainstream biophysics research on
cognitive processes, the brain is modeled as a neural net-
work obeying classical physics. In contrast, Penrose [1,2],
and others have argued that quantum mechanics may
play an essential role, and that successful brain simula-
tions can only be performed with a quantum computer.
The main purpose of this paper is to address this issue
with quantitative decoherence calculations.
The field of artificial neural networks (for an introduc-
tion, see, e.g., [4–6]) is currently booming, driven by a
broad range of applications and improved computing re-
sources. Although the popular neurological models come
in various levels of abstraction, none involve effects of
quantum coherence in any fundamental way. Encouraged
by successes in modeling memory, learning, visual pro-
cessing, etc. [7,8], many workers in the field have boldly
conjectured that a sufficiently complex neural network
could in principle perform all cognitive processes that we
associate with consciousness.
On the other hand, many authors have argued that
consciousness can only be understood as a quantum ef-
fect. For instance, Wigner [9] suggested that conscious-
ness was linked to the quantum measurement problem1,
and this idea has been greatly elaborated by Stapp [3].
There have been numerous suggestions that conscious-
ness is a macroquantum effect, involving superconduc-

1 Interestingly, Wigner changed his mind and gave up this
idea [10] after he became aware in of the first paper on deco-
herence in 1970 [11].

tivity [12], superfluidity [13], electromagnetic fields [14],
Bose condensation [15,16], superflourescence [17] or some
other mechanism [18,19]. Perhaps the most concrete one
is that of Penrose [2], proposing that this takes place
in microtubules, the ubiquitous hollow cylinders that
among other things help cells maintain their shapes. It
has been argued that microtubules can process informa-
tion like a cellular automaton [20], and Penrose suggests
that they operate as a quantum computer. This idea has
been further elaborated employing string theory methods
[21–27].
The make-or-break issue for all these quantum mod-
els is whether the relevant degrees of freedom of the
brain can be sufficiently isolated to retain their quan-
tum coherence, and opinions are divided. For instance,
Stapp has argued that interaction with the environment
is probably small enough to be unimportant for cer-
tain neural processes [28], whereas Zeh [29], Zurek [30],
Scott [31], Hawking [32] and Hepp [33] have conjectured
that environment-induced coherence will rapidly destroy
macrosuperpositions in the brain. It is therefore timely
to try to settle the issue with detailed calculations of the
relevant decoherence rates. This is the purpose of the
present work.
The rest of this paper is organized as follows. In Sec-
tion II, we briefly review the open system quantum me-
chanics necessary for our calculations, and introduce a
decomposition into three subsystems to place the prob-
lem in its proper context.
In Section III, we evaluate
decoherence rates both for neuron firing and for the mi-
crotubule processes proposed by Penrose et al., relegating
some technical details to the Appendix. We conclude in
Section IV by discussing the implications of our results,
both for modeling cognitive brain processes and for in-
corporating them into a quantum-mechanical treatment
of the rest of the world.

II. SYSTEMS AND SUBSYSTEMS

In this section, we review those aspects of quantum
mechanics for open systems that are needed for our cal-
culations, and introduce a classification scheme and a
subsystem decomposition to place the problem at hand
in its appropriate context.

1


A. Notation

Let us first briefly review the quantum mechanics of
subsystems. The state of an arbitrary quantum system
is described by its density matrix ρ, which left in isolation
will evolve in time according to the Schr¨odinger equation

˙ρ = −i[H, ρ]/¯h.
(1)

It is often useful to view a system as composed of two
subsystems, so that some of the degrees of freedom cor-
respond to the 1st and the rest to the 2nd. The state of
subsystem i is described by the reduced density matrix
ρi obtained by tracing (marginalizing) over the degrees
of freedom of the other: ρ1 ≡ tr 2ρ, ρ2 ≡ tr 1ρ. Let us
decompose the Hamiltonian as

H = H1 + H2 + Hint,
(2)

where the operator H1 affects only the 1st subsystem
and H2 affects only the 2nd subsystem. The interaction
Hamiltonian Hint is the remaining nonseparable part, de-
fined as Hint ≡ H − H1 − H2, so such a decomposition
is always possible, although it is generally only useful if
Hint is in some sense small.
If Hint = 0, i.e., if there is no interaction between
the two subsystems, then it is easy to show that ˙ρi =
−i[Hi, ρi]/¯h, i = 1, 2, that is, we can treat each subsys-
tem as if the rest of the Universe did not exist, ignoring
any correlations with the other subsystem that may have
been present in the full non-separable density matrix ρ.
It is of course this property that makes density matrices
so useful in the first place, and that led von Neumann
to invent them [34]: the full system is assumed to obey
equation (1) simply because its interactions with the rest
of the Universe are negligible.

B. Fluctuation, dissipation, communication and
decoherence

In practice, the interaction Hint between subsystems
is usually not zero. This has a number of qualitatively
different effects:

1. Fluctuation

2. Dissipation

3. Communication

4. Decoherence

The first two involve transfer of energy between the sub-
systems, whereas the last two involve exchange of infor-
mation. The first three occur in classical physics as well
- only the last one is a purely quantum-mechanical phe-
nomenon.
For example, consider a tiny colloid grain (subsystem
1) in a jar of water (subsystem 2). Collisions with water

molecules will cause fluctuations in the center-of-mass
position of the colloid (brownian motion). If its initial ve-
locity is high, dissipation (friction) will slow it down to
a mean speed corresponding to thermal equilibrium with
the water. The dissipation timescale τdiss, defined as the
time it would take to lose half of the initial excess energy,
will in this case be of order τcoll × (M/m), where τcoll is
the mean-free time between collisions, M the colloid mass
M and m is the mass of a water molecule. We will define
communication as exchange of information. The infor-
mation that the two subsystems have about each other,
measured in bits, is

I12 ≡ S1 + S2 − S,
(3)

where Si ≡ −tr iρi log ρi is the entropy of the ith subsys-
tem, S ≡ −tr ρ log ρ is the entropy of the total system,
and the logarithms are base 2. If this mutual informa-
tion is zero, then the states of the two systems are un-
correlated and independent, with the density matrix of
the separable form ρ = ρ1 ⊗ ρ2. If the subsystems start
out independent, any interaction will at least initially
increase the subsystem entropies Si, thereby increasing
the mutual information, since the entropy S of the total
system always remains constant.
This apparent entropy increase of subsystems, which
is related to the arrow of time and the 2nd law of of ther-
modynamics [35], occurs also in classical physics. How-
ever, quantum mechanics produces a qualitatively new
effect as well, known as decoherence [11,36,37], sup-
pressing off-diagonal elements in the reduced density ma-
trices ρi. This effect destroys the ability to observe long-
range quantum superpositions within the subsystems,
and is now rather well-understood and uncontroversial
[30,38–42] – the interested reader is referred to [43] and
a recent book on decoherence [44] for details.
For in-
stance, if our colloid was initially in a superposition of
two locations separated by a centimeter, this macrosu-
perposition would for all practical purposes be destroyed
by the first collision with a water molecule, i.e., on a
timescale τdec of order τcoll, with the quantum superpo-
sition surviving only on scales below the de de Broigle
wavelength of the water molecules [45,46].2 This means

2Decoherence picks out a preferred basis in the quantum-
mechanical Hilbert space, termed the “pointer basis” by
Zurek [36], in which superpositions are rapidly destroyed and
classical behavior is approached. This normally includes the
position basis, which is why we never experience superposi-
tions of objects in macroscopically different positions. Deco-
herence is quite generic. Although it has been claimed that
this preferred basis consists of the maximal set of commuting
observables that also commute with Hint (the “microstable
basis” of Omnes [43]), this is in fact merely a sufficient condi-
tion, not a necessary one. If [Hint, x] = 0 for some observable
x but [Hint, p] ̸= 0 for its conjugate p, then the interaction

2


that τdiss/τdec ∼ M/m in our example, i.e., that decoher-
ence is much faster than dissipation for macroscopic ob-
jects, and this qualitative result has been shown to hold
quite generally as well (see [43] and references therein).
Loosely speaking, this is because each microscopic par-
ticle that scatters off of the subsystem carries away only
a tiny fraction m/M of the total momentum, but essen-
tially all of the necessary information.

QUANTUM�
SYSTEM

NOT �
INDEPENDENT�
SYSTEM

IMPOSSIBLE

CLASSICAL�
SYSTEM

0.1
1

1

0.1

10

100

10
100
Dissipation time/Decoherence time

Dynamical time/Decoherence time

FIG. 1. The qualitative behavior of a subsystem depends on
the timescales for dynamics, dissipation and decoherence.
This
classification is by necessity quite crude, so the boundaries should
not be thought of as sharp.

C. Classification of systems

Let us define the dynamical timescale τdyn of a subsys-
tem as that which is characteristic of its internal dynam-
ics. For a planetary system or an atom, τdyn would be
the orbital frequency.
The qualitative behavior of a system depends on the
ratio of these timescales, as illustrated in Figure 1. If
τdyn ≪ τdec, we are are dealing with a true quantum sys-
tem, since its superpositions can persist long enough to
be dynamically important. If τdyn ≫ τdiss, it is hardly
meaningful to view it as an independent system at all,
since its internal forces are so week that they are dwarfed

will indeed cause decoherence for x as advertised. But this
will happen even if [Hint, x] ̸= 0 — all that matters is that
[Hint, p] ̸= 0, i.e., that the interaction Hamiltonian contains
(“measures”) x.

by the effects of the surroundings. In the intermediate
case where τdec ≪ τdyn <∼ τdiss, we have a familiar classi-
cal system.
The relation between τdec and τdiss depends only on
the form of Hint, whereas the question of whether τdyn
falls between these values depends on the normalization
of Hint in equation (2). Since τdec ∼ τdiss for microscopic
(atom-sized) systems and τdec ≪ τdiss for macroscopic
ones, Figure 1 shows that whereas macroscopic systems
can behave quantum-mechanically, microscopic ones can
never behave classically.

D. Three systems: subject, object and environment

Most discussions of quantum statistical mechanics split
the Universe into two subsystems [47]: the object under
consideration and everything else (referred to as the en-
vironment). Since our purpose is to model the observer,
we need to include a third subsystem as well, the subject.
As illustrated in Figure 2, we therefore decompose the
total system into three subsystems:

• The subject consists of the degrees of freedom as-
sociated with the subjective perceptions of the ob-
server. This does not include any other degrees of
freedom associated with the brain or other parts of
the body.

• The object consists of the degrees of freedom that
the observer is interested in studying, e.g., the
pointer position on a measurement apparatus.

• The environment consists of everything else, i.e.,
all the degrees of freedom that the observer is not
paying attention to. By definition, these are the
degrees of freedom that we always perform a partial
trace over.

3


SUBJECT
OBJECT

ENVIRONMENT

Hs
Ho

He

Hso

Hoe
Hse

Object �
decoherence

Subject�
decoherence,�
finalizing �
decisions

Measurement,�
observation,�
"wavefuntion �
collapse",�
willful action

(Always traced over)

(Always zero entropy)

FIG. 2. An observer can always decompose the world into three
subsystems: the degrees of freedom corresponding to her subjective
perceptions (the subject), the degrees of freedom being studied (the
object), and everything else (the environment). As indicated, the
subsystem Hamiltonians Hs, Ho, He and the interaction Hamilto-
nians Hso, Hoe, Hse can cause qualitatively very different effects,
which is why it is often useful to study them separately. This paper
focuses on the interaction Hse.

Note that the first two definitions are very restrictive.
Whereas the subject would include the entire body of
the observer in the common way of speaking, only very
few degrees of freedom qualify as our subject or object.
For instance, if a physicist is observing a Stern-Gerlach
apparatus, the vast majority of the ∼ 1028 degrees of
freedom in the the observer and apparatus are counted
as environment, not as subject or object.
The term “perception” is used in a broad sense in item
1, including thoughts, emotions and any other attributes
of the subjectively perceived state of the observer.
The practical usefulness in this decomposition lies in
that one can often neglect everything except the object
and its internal dynamics (given by Ho) to first order,
using simple prescriptions to correct for the interactions
with the subject and the environment.
The effects of
both Hso and Hoe have been extensively studied in the
literature. Hso involves quantum measurement, and gives
rise to the usual interpretation of the diagonal elements of
the object density matrix as probabilities. Hoe produces
decoherence, selecting a preferred basis and making the
object act classically if the conditions in Figure 1 are met.
In contrast, Hse, which causes decoherence directly in
the subject system, has received relatively little atten-

tion. It is the focus of the present paper, and the next
section is devoted to quantitative calculations of decoher-
ence in brain processes, aimed at determining whether
the subject system should be classified as classical or
quantum in the sense of Figure 1.
We will return to
Figure 2 and a more detailed discussion of its various
subsystem interactions in Section IV.

III. DECOHERENCE RATES

In this section, we will make quantitative estimates
of decoherence rates for neurological processes. We first
analyze the process of neuron firing, widely assumed to be
central to cognitive processes. We also analyze electrical
excitations in microtubules, which Penrose and others
have suggested may be relevant to conscious thought.

A. Neuron firing

Neurons (see Figure 3) are one of the key building
blocks of the brain’s information processing system. It is
widely believed that the complex network of ∼ 1011 neu-
rons with their nonlinear synaptic couplings is in some
way linked to our subjective perceptions, i.e., to the sub-
ject degrees of freedom. If this picture is correct, then if
Hs or Hso puts the subject into a superposition of two
distinct mental states, some neurons will be in a super-
position of firing and not firing. How fast does such a
superposition of a firing and non-firing neuron decohere?
Let us consider this process in more detail.
For in-
troductory reviews of neuron dynamics, the reader is re-
ferred to, e.g., [48–50].
Like virtually all animal cells,
neurons have ATP driven pumps in their membranes
which push sodium ions out of the cell into the surround-
ing fluids and potassium ions the other way. The former
process is slightly more efficient, so the neuron contains a
slight excess of negative charge in its “resting” state, cor-
responding to a potential difference U0 ≈ −0.07 V across
the axon membrane (“axolemma”). There is an inher-
ent instability in the system, however. If the potential
becomes substantially less negative, then voltage-gated
sodium channels in the axon membrane open up, allow-
ing Na+ ions to come gushing in. This makes the poten-
tial still less negative, causes still more opening, etc. This
chain reaction, “firing”, propagates down the axon at a
speed of up to 100 m/s, changing the potential difference
to a value U1 that is typically of order +0.03 V [49].
The axon quickly recovers. After less than ∼ 1 ms, the
sodium channels close regardless of the voltage, and large
potassium channels (also voltage gated, but with a time
delay) open up allowing K+ ions to flow out and restore
the resting potential U0. The ATP driven pumps quickly
restore the Na+ and K+ concentrations to their initial
values, making the neuron ready to fire again if triggered.
Fast neurons can fire over 103 times per second.

4


Na+
Na+

dendrites

axon

cell body

myelin�
insulation

fraction f�
not insulated

thickness h

Here�
if�
firing

Here�
if not�
firing

voltage�
sensitive�
gate

length�
L

axon�
membrane

pulse

di

re

ct

io

n

diameter d

FIG. 3. Schematic illustration of a neuron (left), a section of
the myelinated axon (center) and and a piece of its axon membrane
(right).
The axon is typically insulated (myelinated) with small
bare patches every 0.5 mm or so (so-called Nodes of Ranvier) where
the voltage-sensitive sodium and potassium gates are concentrated
[51,52]. If the neuron is in a superposition of firing and not firing,
then N ∼ 106 Na+ ions are in a superposition of being inside and
outside the cell (right).

Consider a small patch of the membrane, assumed to
be roughly flat with uniform thickness h as in Figure 3.
If there is an excess surface density ±σ of charge near
the inside/outside membrane surfaces, giving a voltage
differential U across the membrane, then application of
Gauss’ law tells us that σ = ǫ0E, where the electric field
strength in the membrane is E = U/h and ǫ0 is the vac-
uum permittivity.
Consider an axon of length L and
diameter d, with a fraction f of its surface area bare (not
insulated with myelin). The total active surface area is
thus A = πdLf, so the total number of Na+ ions that
migrate in during firing is

N = Aσ

q
= πdLfǫ0(U1 − U0)

qh
,
(4)

where q is the ionic charge (q = qe, the absolute value
of the electron charge). Taking values typical for central
nervous system axons [52,53], h = 8 nm, d = 10 µm,
L = 10 cm, f = 10−3, U0 = −0.07 V and U1 = +0.03 V
gives N ≈ 106 ions, and reasonable variations in our
parameters can change this number by a few orders of
magnitude.

B. Neuron decoherence mechanisms

Above we saw that a quantum superposition of the
neuron states “resting” and “firing” involves of order a
million ions being in a spatial superposition of inside and
outside the axon membrane, separated by a distance of
order h ∼ 10 nm. In this subsection, we will compute the
timescale on which decoherence destroys such a superpo-
sition.

In this analysis, the object is the neuron, and the su-
perposition will be destroyed by any interaction with
other (environment) degrees of freedom that is sensitive
to where the ions are located. We will consider the fol-
lowing three sources of decoherence for the ions:

1. Collisions with other ions

2. Collisions with water molecules

3. Coloumb interactions with more distant ions

There are many more decoherence mechanisms [44–46].
Exotic candidates such as quantum gravity [54] and
modified quantum mechanics [55,56] are generally much
weaker [46]. A number of decoherence effects may be even
stronger than those listed, e.g., interactions as the ions
penetrate the membrane — the listed effects will turn out
to be so strong that we can make our argument by sim-
ply using them as lower limits on the actual decoherence
rate.
Let ρ denote the density matrix for the position r of a
single Na+ ion. As reviewed in the Appendix, all three
of the listed processes cause ρ to evolve as

ρ(x, x′, t0 + t) = ρ(x, x′, t0)f(x, x′, t)
(5)

for some function f that is independent of the ion state
ρ and depends only on the interaction Hamiltonian Hint.
This assumes that we can neglect the motion of the ion
itself on the decoherence timescale — we will see that
this condition is met with a broad margin.

1. Ion–ion collisions

For scattering of environment particles (processes 1
and 2) that have a typical de Broigle wavelength λ, we
have [46]

f(x, x′, t) = e−Λt�
1−e−|x′−x|2/2λ2�

≈

�
e−|x′−x|2Λt/2λ2
for |x′ − x| ≪ λ,

e−Λt
for |x′ − x| ≫ λ.
(6)

Here Λ is the scattering rate, given by Λ ≡ n⟨σv⟩, where
n is the density of scatterers, σ is the scattering cross
section and v is the velocity. The product σv is aver-
aged over a the velocity distribution, which we take to
be a thermal (Boltzmann) distribution for correspond-
ing to T = 37◦C ≈ 310 K. The gist of equation (6) is
that a single collision decoheres the ion down to the
de Broigle wavelength of the scattering particle.
The
information I12 communicated during the scattering is
I12 ∼ log2(∆x/λ) bits, where ∆x is the initial spread in
the position of our particle.
Since the typical de Broigle wavelength of a Na+ ion
(mass m ≈ 23mp) or H2O molecule (m ≈ 18mp) is

5


λ =
2π¯h
√

3mkT
≈ 0.03 nm
(7)

at 310K, way smaller than the the membrane thickness
h ∼ 10 nm over which we need to maintain quantum
coherence, we are clearly in the |x′ − x| ≫ λ limit of
equation (6). This means that the spatial superposition
of an ion decays exponentially Λ−1, of order its mean
free time between collisions. Since the superposition of
the neuron states “resting” and “firing” involves N such
superposed ions, it thus gets destroyed on a timescale
τ ≡ (NΛ)−1.
Let us now evaluate τ. Coulomb scattering between
two ions of unit charge gives substantial deflection angles
(θ ∼ 1) with a cross section or order3

σ ∼
� gq2

mv2

�2
,
(9)

where v is the relative velocity and g ≡ 1/4πǫ0 is the
Coulomb constant. In thermal equilibrium, the kinetic
energy mv2/2 is of order kT , so v ∼
�

kT/m. For the
ion density, let us write n = ηnH2O, where the density
of water molecules nH2O is about (1 g/cm3)/(18mp) ∼
1023/cm3 and η is the relative concentration of ions (pos-
itive and negative combined). Typical ion concentrations
during the resting state are [Na+] =9.2 (120) mmol/l and
[K+] =140 (2.5) mmol/l inside (outside) the axon mem-
brane [48], corresponding to total Na+ + K+ concentra-
tions of η ≈ 0.00027 (0.00022) inside (outside). To be
conservative, we will simply use η ≈ 0.0002 throughout.
Ion–ion collisions therefore destroy the superposition on
a timescale

τ ∼
1

Nnσv ∼

�

m(kT )3

Ng2q4en
∼ 10−20 s.
(10)

2. Ion–water collisions

Since H2O molecules are electrically neutral, the cross-
section is dominated by their electric dipole moment
p ≈ 1.85 Debye ≈ (0.0385 nm) × qe. We can model this

3 If the first ion starts at rest at r1 = (0, 0, 0) and the sec-
ond is incident with r2 = (vt, b, 0), then a very weak scatter-
ing with deflection angle θ ≪ 1 will leave these trajectories
roughly unchanged, the radial force F = gq2/|r1 −r2|2 merely
causing a net transverse acceleration [57]

∆vy =
� ∞

−∞

�y · F

m dt =
� ∞

−∞

gq2b dt

[b2 + (vt)2]3/2 = 2gq2

mvb .
(8)

The approximation breaks down as the deflection angle θ ≈
∆vy/v approaches unity. This occurs for b ∼ gq2/mv2, giving
σ = πb2 as in equation (9).

dipole as two opposing unit charges separated by a dis-
tance y ≡ p/qe ≪ b, so summing the two corresponding
contributions from equation (8) gives a deflection angle

θ ≈ 2gqep

mv2b2 .
(11)

This gives a cross section

σ = πb2 ∼ gqep

mv2 .
(12)

for scattering with large (θ ∼ 1) deflections. Although σ
is smaller than for the case of ion–ion collisions, n is larger
because the concentration factor η drops out, giving a
final result

τ ∼
1

Nnσv ∼

√

mkT

Ngqepn ∼ 10−20 s
(13)

3. Interactions with distant ions

As shown in the Appendix, long-range interaction with
a distant (environment) particle gives

f(r, r′, t) = �p2 [M(r′ − r)t/¯h] ,
(14)

up to a phase factor that is irrelevant for decoherence.
Here �p2 is the Fourier transform of p2(r) ≡ ρ2(r, r), the
probability distribution for the location of the environ-
ment particle. M is the 3 × 3 Hessian matrix of second
derivatives of the interaction potential of the two parti-
cles at their mean separation. A slightly less general for-
mula was derived in the seminal paper [45]. For roughly
thermal states, ρ2 (and thus p) is likely to be well ap-
proximated by a Gaussian [58,59]. This gives

f(r, r′, t) = e− 1

2 (r′−r)tMtΣM(r′−r)t2/¯h2,
(15)

where Σ = ⟨r2rt
2⟩ − ⟨r2⟩⟨rt
2⟩ is the covariance matrix of
the location of the environment particle.
Decoherence
is destroyed when the exponent becomes of order unity,
i.e., on a timescale

τ ≡
�
(r′ − r)tMtΣM(r′ − r)
�−1/2 ¯h.
(16)

Assuming a Coulomb potential V = gq2/|r2 − r1| gives
M = (3�a�at − I)gq2/a3 where a ≡ r2 − r1 = a�a, |�a| =
1. For thermal states, we have the isotropic case Σ =
(∆x)2I, so equation (16) reduces to

τ =
¯ha3

gq2|r′ − r|∆x
�
1 + 3 cos2 θ
�−1/2 ,
(17)

where cos θ ≡ �a · (r′ − r)/|r′ − r|. To be conservative,
we take ∆x to be as small as the uncertainty principle
allows. With the thermal constraint (∆p)2/m <∼ kT on
the momentum uncertainty, this gives

6


∆x =
¯h

2∆p ∼
¯h
√

mkT
.
(18)

Substituting this into equation (17) and dividing by the
number of ions N, we obtain the decoherence timescale

τ ∼
a3√

mkT

Ngq2|r′ − r|.
(19)

caused by a single environment ion a distance a away.
Each such ion will produce its own suppression factor f,
so we need to sum the exponent in equation (15) over all
ions. Since the tidal force M ∝ a−3 causes the exponent
to drop as a−6, this sum will generally be dominated by
the very closest ion, which will typically be a distance
a ∼ n−1/3 away. We are interested in decoherence for
separations |r′ − r| = h, the membrane thickness, which
gives

τ ∼

√

mkT

Ngq2enh ∼ 10−19 s.
(20)

The relation between these different estimates is dis-
cussed in more detail in the Appendix.

C. Microtubules

Microtubules are a major component of the cytoskele-
ton, the “scaffolding” that helps cells maintain their
shapes.
They are hollow cylinders of diameter D =
24 nm made up of 13 filaments that are strung together
out of proteins known as tubulin dimers. These dimers
can make transitions between two states known as α
and β, corresponding to different electric dipole moments
along the axis of the tube. It has been argued that micro-
tubules may have additional functions as well, serving as
a means of energy and information transfer [20]. A model
has been presented whereby the dipole-dipole interac-
tions between nearby dimers can lead to long-range po-
larization and kink-like excitations that may travel down
the microtubules at speeds exceeding 1 m/s [60].
Penrose has gone further and suggested that the dy-
namics of such excitations can make a microtubule act
like a quantum computer, and that microtubules are the
site of of human consciousness [2]. This idea has been fur-
ther elaborated [21–24] employing methods from string
theory, with the conclusion that quantum superpositions
of coherent excitations can persist for as long as a second
before being destroyed by decoherence. See also [61,62].
This was hailed as a success for the model, the interpre-
tation being that the quantum gravity effect on micro-
tubules was identified with the human though process on
this same timescale.
This decoherence rate τ ∼ 1 s was computed assuming
that quantum gravity is the main decoherence source.
Since this quantum gravity model is somewhat contro-
versial [32] and its effect has been found to be more than

20 orders of magnitude weaker than other decoherence
sources in some cases [46], it seems prudent to evalu-
ate other decoherence sources for the microtubule case
as well, to see whether they are in fact dominant. We
will now do so.
Using coordinates where the x-axis is along the tube
axis, the above-mentioned models all focus on the time-
evolution of p(x), the average x-component of the electric
dipole moment of the tubulin dimers at each x. In terms
of this polarization function p(x), the net charge per unit
length of tube is −p′(x). The propagating kink-like exci-
tations [60] are of the form

p(x) =
� +p0
for x ≪ x0,

−p0
for x ≫ x0,
(21)

where the kink location x0 propagates with constant
speed and has a width of order a few tubulin dimers.
The polarization strength p0 is such that the total charge
around the kink is Q = − � p′(x)dx = 2p0 ∼ 940qe, due
to the presence of 18 Ca2+ ions on each of the 13 fila-
ments contributing to p0 [60].
Suppose that such a kink is in two different places
in superposition, separated by some distance |r′ − r|.
How rapidly will the superposition be destroyed by de-
coherence?
To be conservative, we will ignore colli-
sions between polarized tubulin dimers and nearby water
molecules, since it has been argued that these may be in
some sense ordered and part of the quantum system [24]
– although this argument is difficult to maintain for the
water outside the microtubule, which permeates the en-
tire cell volume. Let us instead apply equation (19), with
N = Q/qe ∼ 103. The distance to the nearest ion will
generally be less than a = R + n−1/3 ∼ 26 nm, where the
tubulin diameter D = 24 nm dominates over the inter-
ion separation n−1/3 ∼ 2 nm in the fluid surrounding
the microtubule. Superpositions spanning many tubuline
dimers (|r′ − r| ≫ D) therefore decohere on a timescale

τ ∼ D2√

mkT

Ngq2e
∼ 10−13 s.
(22)

due to the nearest ion alone. This is quite a conserva-
tive estimate, since the other nD3 ∼ 103 ions that are
merely a small fraction further away will also contribute
to the decoherence rate, but it is nonetheless 6-7 orders
of magnitude shorter than the estimates of Mavromatos
& Nanopoulos [25–27]. We will comment on screening
effects below.

1. Decoherence summary

Our decoherence rates are summarized in Table 1. How
accurate are they likely to be?
In the calculations above, we generally tried to be con-
servative, erring on the side of underestimating the deco-
herence rate. For instance, we neglected that N potas-
sium ions also end up in superposition once the neuron

7


firing is quenched, we neglected the contribution of other
abundant ions such as Cl− to η, and and we ignored col-
lisions with water molecules in the microtubule case.
Since we were only interested in order-of-magnitude
estimates, we made a number of crude approximations,
e.g., for the cross sections. We neglected screening ef-
fects because the decoherence rates were dominated by
the particles closest to the system, i.e., the very same par-
ticles that are responsible for screening the charge from
more distant ones.

Table 1. Decoherence timescales.

Object
Environment
τdec

Neuron
Colliding ion
10−20s
Neuron
Colliding H2O
10−20s
Neuron
Nearby ion
10−19s
Microtubule
Distant ion
10−13s

IV. DISCUSSION

A. The classical nature of brain processes

The calculations above enable us to address the ques-
tion of whether cognitive processes in the brain consti-
tute a classical or quantum system in the sense of Fig-
ure 1. If we take the characteristic dynamical timescale
for such processes to be τdyn ∼ 10−2 s − 100 s (the ap-
parent timescale of e.g., speech, thought and motor re-
sponse), then a comparison of τdyn with τdec from Table 1
shows that processes associated with either conventional
neuron firing or with polarization excitations in micro-
tubules fall squarely in the classical category, by a mar-
gin exceeding ten orders of magnitude. Neuron firing it-
self is also highly classical, since it occurs on a timescale
τdyn ∼ 10−3 − 10−4 s [53]. Even a kink-like microtubule
excitation is classical by many orders of magnitude, since
it traverses a short tubule on a timescale τdyn ∼ 5×10−7 s
[60].
What about other mechanisms?
It is worth noting
that if (as is commonly believed) different neuron fir-
ing patterns correspond in some way to different con-
scious perceptions, then consciousness itself cannot be
of a quantum nature even if there is a yet undiscovered
physical process in the brain with a very long decoherence
time. As mentioned above, suggestions for such candi-
dates have involved, e.g., superconductivity [12], super-
fluidity [13], electromagnetic fields [14], Bose condensa-
tion [15,16], superflourescence [17] and other mechanisms
[18,19]. The reason is that as soon as such a quantum
subsystem communicates with the constantly decohering
neurons to create conscious experience, everything deco-
heres.
How extreme variations in the decoherence rates can
we obtain by changing our model assumptions? Although
the rates can be altered by a few of orders of magnitudes
by pushing parameters such as the neuron dimensions,
the myelination fraction or the microtubule kink charge

to the limits of plausibility, it is clearly impossible to
change the basic conclusion that τdec ≪ 10−3 s, i.e., that
we are dealing with a classical system in the sense of Fig-
ure 1. Even the tiniest neuron imaginable, with only a
single ion (N = 1) traversing the cell wall during firing,
would have τdec ∼ 10−14 s.
Likewise, reducing the ef-
fective microtubule kink charge to a small fraction of qe
would not help.
How are we to understand the above-mentioned claims
that brain subsystems can be sufficiently isolated to
exhibit macroquantum behavior?
It appears that the
subtle distinction between dissipation and decoherence
timescales has not always been appreciated.

B. Implications for the subject-object-environment
decomposition

Let us now discuss the subsystem decomposition of
Figure 2 in more detail in light of our results. As the
figure indicates, the virtue of this decomposition into
subject, object and environment is that the subsystem
Hamiltonians Hs, Ho, He and the interaction Hamiltoni-
ans Hso, Hoe, Hse can cause qualitatively very different
effects. Let us now briefly discuss each of them in turn.
Most of these processes are schematically illustrated
in Figure 4 and Figure 5, where for purposes of illus-
tration, we have shown the extremely simple case where
both the subject and object have only a single degree of
freedom that can take on only a few distinct values (3
for the subject, 2 for the object). For definiteness, we
denote the three subject states |¨- ⟩, | ¨⌣⟩ and | ¨⌢⟩, and in-
terpret them as the observer feeling neutral, happy and
sad, respectively. We denote the two object states |↑⟩
and |↓⟩, and interpret them as the spin component (“up”
or “down”) in the z-direction of a spin-1/2 system, say a
silver atom. The joint system consisting of subject and
object therefore has only 2 × 3 = 6 basis states: |¨- ↑⟩,
|¨- ↓⟩, | ¨⌣↑⟩, | ¨⌣↓⟩, | ¨⌢↑⟩, | ¨⌢↓⟩. In Figures 4 and 5, we
have therefore plotted ρ as a 6 × 6 matrix consisting of
nine two-by-two blocks.

=
+

Object�
evolution
Object�
decohe-�
rence
Ho
(Entropy�
constant)
(Entropy�
increases)

Hoe

Observation/Measurement

(Entropy decreases)
Hso
�

2
1_
2
1_

8


FIG. 4. Time evolution of the 6×6 density matrix for the basis
states |¨- ↑⟩, |¨- ↓⟩, | ¨⌣↑⟩, | ¨⌣↓⟩, | ¨
⌢↑⟩, | ¨⌢↓⟩ as the object evolves in
isolation, then decoheres, then gets observed by the subject. The
final result is a statistical mixture of the states | ¨⌣↑⟩ and | ¨⌢↓⟩,
simple zero-entropy states like the one we started with.

1. Effect of Ho: constant entropy

If the object were to evolve during a time interval t
without interacting with the subject or the environment
(Hso = Hoe = 0), then according to equation (1) its
reduced density matrix ρo would evolve into UρoU † with
the same entropy, since the time-evolution operator U ≡
e−iHot is unitary.
Suppose the subject stays in the state |¨- ⟩ and the
object starts out in the pure state |↑⟩. Let the object
Hamiltonian Ho correspond to a magnetic field in the y-
direction causing the spin to precess to the x-direction,
i.e., to the state (|↑⟩+|↓⟩)/
√

2. The object density matrix
ρo then evolves into

ρo = U|↑⟩⟨↑|U † = 1

2(|↑⟩ + |↓⟩)(⟨↑| + ⟨↓|)

= 1

2(|↑⟩⟨↑| + |↑⟩⟨↓| + |↓⟩⟨↑| + |↓⟩⟨↓|),
(23)

corresponding to the four entries of 1/2 in the second
matrix of Figure 4.
This is quite typical of pure quantum time evolution: a
basis state eventually evolves into a superposition of ba-
sis states, and the quantum nature of this superposition
is manifested by off-diagonal elements in ρo. Another fa-
miliar example of this is the familiar spreading out of the
wave packet of a free particle.

2. Effect of Hoe: increasing entropy

This was the effect of Ho alone. In contrast, Hoe will
generally cause decoherence and increase the entropy of
the object. As discussed in detail in Section III and the
Appendix, it entangles it with the environment, which
suppresses the off-diagonal elements of the reduced den-
sity matrix of the object as illustrated in Figure 4. If Hoe
couples to the z-component of the spin, this destroys the
terms |↑⟩⟨↓| and |↓⟩⟨↑|. Complete decoherence therefore
converts the final state of equation (23) into

ρo = 1

2(|↑⟩⟨↑| + |↓⟩⟨↓|),
(24)

corresponding to the two entries of 1/2 in the third ma-
trix of Figure 4.

3. Effect of Hso: decreasing entropy

Whereas Hoe typically causes the apparent entropy of
the object to increase, Hso typically causes it to decrease.

Figure 4 illustrates the case of an ideal measurement,
where the subject starts out in the state |¨- ⟩ and Hso is of
such a form that gets perfectly correlated with the object.
In the language of Section II, an ideal measurement is a
type of communication where the mutual information I12
between the subject and object systems is increased to its
maximum possible value. Suppose that the measurement
is caused by Hso becoming large during a time interval so
brief that we can neglect the effects of Hs and Ho. The
joint subject+object density matrix ρso then evolves as
ρso �→ UρsoU †, where U ≡ exp
�
−i
�
Hsodt
�
. If observing
|↑⟩ makes the subject happy and |↓⟩ makes the subject
sad, then we have U|¨-↑⟩ = | ¨⌣↑⟩ and U|¨-↓⟩ = | ¨⌢↓⟩. The
state given by equation (24) would therefore evolve into

ρo = 1

2U(|¨- ⟩⟨¨- |) ⊗ (|↑⟩⟨↑| + |↓⟩⟨↓|)U †
(25)

= 1

2(U|¨-↑⟩⟨¨-↑|U † + U|¨-↓⟩⟨¨-↓|U †
(26)

= 1

2(| ¨⌣↑⟩⟨ ¨⌣↑| + | ¨⌢↓⟩⟨ ¨⌢↓ |),
(27)

as illustrated in Figure 4.
This final state contains a
mixture of two subjects, corresponding to definite but
opposite knowledge of the object state.
According to
both of them, the entropy of the object has decreased
from one bit to zero bits.
In general, we see that the object decreases its en-
tropy when it exchanges information with the subject
and increases when it exchanges information with the
environment.4 Loosely speaking, the entropy of an ob-
ject decreases while you look at it and increases while
you don’t5.

4If n bits of information are exchanged with the environ-
ment, then equation (3) shows that the object entropy will
increase by this same amount if the environment is in ther-
mal equilibrium (with maximal entropy) throughout. If we
were to know the state of the environment initially (by our
definition of environment, we do not), then both the object
and environment entropy will typically increase by n/2 bits.
5 Here and throughout, we are assuming that the total
system, which is by definition isolated, evolves according to
the Schr¨odinger equation (1). Although modifications of the
Schr¨odinger equation have been suggested by some authors,
either in a mathematically explicit form as in [55,56] or ver-
bally as a so-called reduction postulate, there is so far no
experimental evidence suggesting that modifications are nec-
essary. The original motivations for such modifications were

1. to be able to interpret the diagonal elements of the
density matrix as probabilities and

2. to suppress off-diagonal elements of the density matrix.

The subsequent discovery by Everett [64] that the probability
interpretation automatically appears to hold for almost all
observers in the final superposition solved problem 1, and is
discussed in more detail in, e.g., [29,66–74]. The still more

9


=
+

Subject�
evolution
Subject�
decohe-�
rence
Hs
(Snap �
decision)
�
Hse

�

2
1_
2
1_

FIG. 5. Time evolution of the same 6 × 6 density matrix as in
Figure 4 when the subject evolves in isolation, then decoheres. The
object remains in the state |↑⟩ the whole time. The final result is
a statistical mixture of the two states | ¨⌣↑⟩ and | ¨
⌢↑⟩.

4. Effect if Hs: the thought process

So far, we have focused on the object and discussed
effects of its internal dynamics (Ho) and its interactions
with the environment (Hoe) and subject (Hso). Let us
now turn to the subject and consider the role played by
its internal dynamics (Hs) and interactions with the en-
vironment (Hse).
In his seminal 1993 book, Stapp [3]
presents an argument about brain dynamics that can be
summarized as follows.

1. Since the brain contains ∼ 1011 synapses connected
together by neurons in a highly nonlinear fashion,
there must be a huge number of metastable rever-
berating patters of pulses into which the brain can
evolve.

2. Neural network simulations have indicated that the
metastable state into which a brain does in fact
evolves depends sensitively on the initial conditions
in small numbers of synapses.

3. The latter depends on the locations of a small num-
ber of calcium atoms, which might be expected to
be in quantum superpositions.

4. Therefore, one would expect the brain to evolve
into
a
quantum
superposition
of
many
such
metastable configurations.

5. Moreover, the fatigue characteristics of the synap-
tic junctions will cause any given metastable state

recent discovery of decoherence [11,36,37] solved problem 2,
as well as explaining so-called superselection rules for the first
time (why for instance the position basis has a special status)
[44].

to become, after a short time, unstable:
the
subject will then be forced to search for a new
metastable configuration, and will therefore con-
tinue to evolve into a superposition of increasingly
disparate states.

If different states (perceptions) of the subject correspond
to different metastable states of neuron firing patterns, a
definite perception would eventually evolve into a super-
position of several subjectively distinguishable percep-
tions.
We will follow Stapp in making this assumption about
Hs. For illustrative purposes, let us assume that this can
happen even at the level of a single thought or snap de-
cision where the outcome feels unpredictable to us. Con-
sider the following experiment: the subject starts out
with a blank face and counts silently to three, then makes
a snap decision on whether to smile or frown. The time-
evolution operator U ≡ exp
�
−i � Hsdt
�
will then have
the property that U|¨- ⟩ = (| ¨⌣⟩ + | ¨⌢⟩)/
√

2, so the sub-
ject density matrix ρs will evolve into

ρs = U|¨- ⟩⟨¨- |U † = 1

2(| ¨⌣⟩ + | ¨⌢⟩)(⟨ ¨⌣| + ⟨ ¨⌢|)

= 1

2(| ¨⌣⟩⟨ ¨⌣| + | ¨⌣⟩⟨ ¨⌢| + | ¨⌢⟩⟨ ¨⌣| + | ¨⌢⟩⟨ ¨⌢|),
(28)

corresponding to the four entries of 1/2 in the second
matrix in Figure 5.

5. Effect of Hse: subject decoherence

Just as Hoe can decohere the object, Hse can decohere
the subject. The difference is that whereas the object can
be either a quantum system (with small Hoe) or a classi-
cal system (with large Hoe), a human subject always has
a large interaction with the environment. As we showed
in Section III, τdec ≪ τdyn for the subject, i.e., the ef-
fect of Hse is faster than that of Hs by many orders of
magnitude. This means that we should strictly speaking
not think of macrosuperpositions such as equation (28)
as first forming and then decohering as in Figure 5 —
rather, subject decoherence is so fast that such superpo-
sitions decohere already during their process of forma-
tion. Therefore we are never even close to being able to
perceive superpositions of different perceptions. Reduc-
ing object decoherence (from Hoe) during measurement
would make no difference, since decoherence would take
place in the brain long before the transmission of the ap-
propriate sensory input through sensory nerves had been
completed.

C. He and Hsoe

The environment is of course the most complicated sys-
tem, since it contains the vast majority of the degrees of

10


freedom in the total system. It is therefore very fortu-
nate that we can so often ignore it, considering only those
limited aspects of it that affect the subject and object.
For the most general H, there can also be an ugly
irreducible residual term Hsoe ≡ H − Hs − Ho − He −
Hso − Hoe − Hse.

D. Implications for modeling cognitive processes

For the neural network community, the implication of
our result is “business as usual”, i.e., there is no need
to worry about the fact that current simulations do not
incorporate effects of quantum coherence. The only rem-
nant from quantum mechanics is the apparent random-
ness that we subjectively perceive every time the subject
system evolves into a superposition as in equation (28),
but this can be simply modeled by including a random
number generator in the simulation. In other words, the
recipe used to prescribe when a given neuron should fire
and how synaptic coupling strengths should be updated
may have to involve some classical randomness to cor-
rectly mimic the behavior of the brain.

1. Hyper-classicality

If a subject system is to be a good model of us, Hso
and Hse need to meet certain criteria: decoherence and
communication are necessary, but fluctuation and dissi-
pation must be kept low enough that the subject does
not lose its autonomy completely.
In our study of neural processes, we concluded that the
subject is not a quantum system, since τdec ≪ τdyn. How-
ever, since the dissipation time τdiss for neuron firing is
of the same order as its dynamical timescale, we see that
in the sense of Figure 1, the subject is not a simple clas-
sical system either. It is therefore somewhat misleading
to think of it as simply some classical degrees of freedom
evolving fairly undisturbed (only interacting enough to
stay decohered and occasionally communicate with the
outside world). Rather, the semi-autonomous degrees of
freedom that constitute the subject are to be found at a
higher level of complexity, perhaps as metastable global
patters of neuron firing.
These degrees of freedom might be termed “hyper-
classical”:
although
there
is
nothing
quantum-
mechanical about their equations of motion (except that
they can be stochastic), they may bear little resemblance
with the underlying classical equations from which they
were derived.
Energy conservation and other familiar
concepts from Hamiltonian dynamics will be irrelevant
for these more abstract equations, since neurons are en-
ergy pumped and highly dissipative. Other examples of
such hyper-classical systems include the time-evolution
of the memory contents of a regular (highly dissipative)

digital computer as well as the motion on the screen of
objects in a computer game.

2. Nature of the subject system

In this paper, we have tacitly assumed that conscious-
ness is synonymous with certain brain processes. This is
what Lockwood terms the “identity theory” [66]. It dates
back to Hobbes (∼1660) and has been espoused by, e.g.,
Russell, Feigl, Smart, Armstrong, Churchland and Lock-
wood himself. Let us briefly explore the more specific
assumption that the subject degrees of freedom are our
perceptions. In this picture, some of the subject degrees
of freedom would have to constitute a “world model”,
with the interaction Hso such that the resulting commu-
nication keeps these degrees of freedom highly correlated
with selected properties of the outside world (object +
environment). Some such properties, i.e.,

• the intensity of the electromagnetic on the retina,
averaged through three narrow-band filters (color
vision) and one broad-band filter (black-and-white
vision),

• the spectrum of air pressure fluctuations in the ears
(sound),

• the chemical composition of gas in the nose (smell)
and solutions in the mouth (taste),

• heat and pressure at a variety of skin locations,

• locations of body parts,

are tracked rather continuously, with the corresponding
mutual information I12 between subject and surround-
ings remaining fairly constant.
Persisting correlations
with properties of the past state of the surroundings
(memories) further contribute to the mutual information
I12. Much of I12 is due to correlations with quite subtle
aspects of the surroundings, e.g., the contents of books.
The total mutual information I12 between a person and
the external world is fairly low at birth, gradually grows
through learning, and falls when we forget. In contrast,
most innate objects have a very small mutual informa-
tion with the rest of the world, books and diskettes being
notable exceptions.
The extremely limited selection of properties that the
subject correlates with has presumably been determined
by evolutionary utility, since it is known to differ between
species: birds perceive four primary colors but cats only
one, bees perceive light polarization, etc. In this picture,
we should therefore not consider these particular (“classi-
cal”) aspects of our surroundings to be more fundamental
than the vast majority that the subject system is uncor-
related with. Morover, our perception of e.g. space is as
subjective as our perception of color, just as suggested
by e.g. [50].

11


3. The binding problem

One of the motivations for models with quantum co-
herence in the brain was the so-called binding problem.
In the words of James [75,76], “the only realities are the
separate molecules, or at most cells. Their aggregation
into a ‘brain’ is a fiction of popular speech”. James’ con-
cern, shared by many after him, was that consciousness
did not seem to be spatially localized to any one small
part of the brain, yet subjectively feels like a coherent
entity. Because of this, Stapp [3] and many others have
appealed to quantum coherence, arguing that this could
make consciousness a holistic effect involving the brain
as a whole.
However, non-local degrees of freedom can be impor-
tant even in classical physics, For instance, oscillations
in a guitar string are local in Fourier space, not in real
space, so in this case the “binding problem” can be solved
by a simple change of variables. As Eddington remarked
[77], when observing the ocean we perceive the moving
waves as objects in their own right because they display a
certain permanence, even though the water itself is only
bobbing up and down. Similarly, thoughts are presum-
ably highly non-local excitation patterns in the neural
network of our brain, except of a non-linear and much
more complex nature.
In short, this author feels that
there is no binding problem.

4. Outlook

In summary, our decoherence calculations have in-
dicated that there is nothing fundamentally quantum-
mechanical about cognitive processes in the brain, sup-
porting the Hepp’s conjecture [33]. Specifically, the com-
putations in the brain appear to be of a classical rather
than quantum nature, and the argument by Lisewski [78]
that quantum corrections may be needed for accurate
modeling of some details, e.g., non-Markovian noise in
neurons, does of course not change this conclusion. This
means that although the current state-of-the-art in neu-
ral network hardware is clearly still very far from be-
ing able to model and understand cognitive processes as
complex as those in the brain, there are no quantum me-
chanical reasons to doubt that this research is on the
right track.

Acknowledgements:
The author wishes to thank
the organizers of the Spaatind-98 and Gausdal-99 win-
ter schools, where much of this work was done, and
Mark Alford, Philippe Blanchard, Carlton Caves, Angel-
ica de Oliveira-Costa, Matthew Donald, Andrei Gruzi-
nov, Piet Hut, Nick Mavromatos, Henry Stapp, Hans-
Dieter Zeh and Woitek Zurek for stimulating discussions
and helpful comments. Support for this work was pro-
vided by the Sloan Foundation and by NASA though

grant NAG5-6034 and Hubble Fellowship HF-01084.01-
96A from STScI, operated by AURA, Inc. under NASA
contract NAS5-26555.

APPENDIX: DECOHERENCE FORMULAS

The quantitative effect of decoherence from both short
range interactions (scattering) and long-range interac-
tions was first derived in a seminal paper by Joos & Zeh
[45]. Since our application involved scattering between
particles of comparable mass, we used a generalized ver-
sion of these results that included the effect of recoil [46].
In this Appendix, we derive a slightly generalized formula
for long-range interactions, and briefly comment on the
relation between these short-range and long-range limit-
ing cases.

1. Decoherence due to tidal forces

Even if the dissipation and fluctuation caused by Hint
is dynamically unimportant, H1 and H2 can be neglected
in equation (2) when calculating the decoherence effect in
the many cases where the interaction Hamiltonian deco-
heres the object on a timescale far below the dynamical
time. In this approximation, we consider two particles
with an interaction H = Hint = V (r2 − r1) for some
potential V . According to equation (1), the two-particle
density matrix ρ therefore evolves as

ρ(r1, r′
1, r2, r′
2, t0 + t)

= ρ(r1, r′
1, r2, r′
2, t)e−i[V (r2−r1)−V (r′
2−r′
1)]/¯h.
(A1)

Following [45], we assume that the two particles are fairly
localized near their initial average positions

r0
i ≡ ⟨ri⟩0 = tr [riρi(t0)],
(A2)

i = 1, 2, and approximate the potential by its second
order Taylor expansion

V (r2 − r1) ≈ V (a) − F · (x2 − x1)

+ 1

2(x2 − x1)tM(x2 − x1).
(A3)

Here F ≡= −∇V (a) is the average force, M is the Hes-
sian matrix Mij ≡ ∂i∂jV (a) and a ≡ r0
2−r0
1. We have in-
troduced relative coordinates xi ≡ ri−r0
i . Assuming that
the two particles are independent initially as in [45], i.e.,
that ρ(t0) takes the separable form ρ(x1, x′
1, x2, x′
2, t0) =
ρ1(x1, x′
1, t0)ρ2(x2, x′
2, t0), this gives

ρ1(x1, x′
1, t0 + t) = tr 2ρ(t0 + t) =
�
ρ(x1, x′
1, x, x, t0 + t)d3x = ρ1(x1, x′
1, t0)f(x1, x′
1, t), (A4)

where

12


f(x1, x′
1, t) ≈

eiφ(x1,x′
1,t)
�
ρ2(x2, x′
2, t0)e−it(x′
1−x1)tMx2/¯hd3x2 =

eiφ(x1,x′
1,t)�p2[M(x′
1 − x1)t/¯h].
(A5)

Here the phase factor

eiφ(x,x′,t) ≡ e
i
¯h[F·(x′−x)+ 1

2 x′tMx′− 1

2 xtMx]
(A6)

is of no importance for decoherence, since it does not
suppress the magnitude |ρ1(x1, x′
1, t)| of the off-diagonal
elements – it merely causes momentum transfer related
to fluctuation and dissipation.
It is the other term
that causes decoherence. �p2 is the Fourier transform of
p2(x) ≡ ρ2(x, x, t0), the probability distribution for the
location of the environment particle.

2. Properties of the effect

Let us briefly discuss some qualitative features of equa-
tion (A5).
Since �p2(0) =
�
p2(x2)d3x2 = tr ρ2 = 1,
ρ1(x, x′) remains unchanged on the diagonal x = x′.
This is because Hint is not changing the position of our
our object particle, merely its momentum.
Since the
mean position ⟨x2⟩ =
�
p2x2d3x2 = tr [x2ρ2] = 0 van-
ishes (using equation (A2)), we have ∇�p2(0) = 0.
In
fact, |f| takes a maximum on the diagonal, and the
Riemann-Lebesgue Lemma shows that |f| = |�p2| ≤ 1
whenever x ̸= x′, with equality only for the unphys-
ical case where p2 is a delta function, i.e., where the
location of the environment particle is perfectly known.
∂i∂j|f(0)| = −M⟨x2xt
2⟩Mt2/2¯h2, so so the larger ⟨x2xt
2⟩
is (i.e., the more spread out the environment particle is),
the closer to the diagonal decoherence will suppress our
density matrix.
Since M is the shear matrix of the force field −∇V , we
see that it is tidal forces that are causing the decoherence
— the average force F simply contributes to the phase
factor eiφ. Specifically, the rate at which our object de-
grees of freedom r1 decohere grows with the tidal force
that it exerts on the environment: if the environment
particle is spread out with ⟨x2xt
2⟩ large, experiencing a
wide range of forces from the object, object decoherence
is rapid. In the opposite situation, where the object is
spread out and the environment is not, the object will
experience strong classical tidal forces but no decoher-
ence.

3. Relation between long-range and short-range
decoherence

Above we derived the effect of decoherence from long-
range tidal forces. Another interesting case that has been
solved analytically [45] is that of short-range interactions

that can be modeled as scattering events. If the scatter-
ing takes place during short enough a time interval that
we can neglect the internal dynamics of the object, then
its reduced density matrix changes as [46]

ρ1(r, r′) �→ ρ1(r, r′)�p
�r′ − r

¯h

�
,
(A7)

where p(q) is the probability distribution for the momen-
tum transfer q in the collision. This equation generalizes
the scattering result of [45] by including the effect of re-
coil. The larger the uncertainty in momentum transfer,
the stronger the decoherence effect becomes, since widen-
ing p narrows its Fourier transform �p. Changing the mean
momentum transfer ⟨q⟩ does not affect the decoherence,
merely contributes a phase factor just as F did above.
Typically, the last factor in equation (A7) destroys coher-
ence down to scales of order the de Broigle wavelength
of the scatterer, with directional modulations from the
angular dependence of the scattering cross section. Gen-
eralization to a steady flux of scattering particles [46]
gives equation (6).
Equation (A7) has striking similarities with the tidal
force result of equation (A5): in both cases, the density
matrix gets multiplied by the Fourier transform of a prob-
ability distribution.
If fact, up to uninteresting phase
factors, we can rewrite our equation (A5) in exactly the
form of equation (A7) by redefining p to be the probabil-
ity distribution for momentum transfer q = M(x2 −x1)t
due to tidal forces for a fixed x1, i.e.,

p(q) ≡ p2(x2)d3x2

d3q = p2(x1 + M−1q/t)

t3 det M
.
(A8)

Fourier transforming this expression and substituting the
result into equation (A7), we recover equation (A5) up
to a phase factor.
Perhaps the simplest way to understand all these re-
sults is in terms of Wigner functions [79]. If W(x1, p1) is
the Wigner phase space distribution for the object parti-
cle, then any of the momentum-transferring interactions
that we have considered will take the form

W(x1, p1) �→
�
W(x1, p1 − q)p(q, x1)d3q
(A9)

for some probability distribution p that may or may not
depend on x1. Since the density matrix

ρ1(x1, x′
1) =
�
W
�x1 + x′
1

2
, p
�
e−i(x−x′)·pd3p
(A10)

is just the Wigner function Fourier transformed in the
momentum direction (and rotated by 45◦), the convolu-
tion with p in equation (A9) reduces to a simple multi-
plication with �p in equation (A7).

13


[1] R. Penrose, The Emperor’s New Mind (Oxford, Oxford
Univ. Press, 1989).
[2] R. Penrose, in The Large, the Small and the Human
Mind, ed.
M. Longair (Cambridge, Cambridge Univ.
Press, 1997).
[3] H. P. Stapp, Mind, Matter and Quantum Mechanics
(Berlin, Springer, 1993).
[4] D. J. Amit, Modeling Brain Functions (Cambridge, Cam-
bridge Univ. Press, 1989).
[5] M. M´ezard, G. Parisi, and M. Virasoro, Spin Glass The-
ory and Beyond (Singapore, World Scientific, 1993).
[6] R. L. Harvey, Neural Network Principles (Englewood
Cliffs, Prentice Hall, 1994).
[7] F. H. Eeckman and J. M. Bower, Computation and Neu-
ral Systems (Boston, Kluwer, 1993).
[8] D. R. McMillen, G. M. T D’Eleuterio, and J. R. P
Halperin, Phys. Rev. E 59, 6 (1999).
[9] E. P. Wigner, in The Scientist Speculates: an Anthology
of Partly-Baked Ideas, p284-302, ed. I. J. Good (London,
Heinemann, 1962).
[10] J. Mehra and A. S. Wightman, The Collected Works of
E. P. Wigner, Vol. VI, p271 (Berlin, Springer, 1995).
[11] H. D. Zeh, Found. Phys. 1, 69 (1970).
[12] E. H. Walker, Mathematical Biosciences 7, 131 (1970).
[13] L. H. Domash, in Scientific Research on TM, ed. D. W.
Orme-Johnson and J. T. Farrow (Weggis, Switzerland,
Maharishi Univ. Press, 1977).
[14] H. P. Stapp, Phys. Rev. D 28, 1386 (1983).
[15] I. N. Marshall, New Ideas in Psychology 7, 73 (1989).
[16] D. Zohar, The Quantum Self (New York, William Mor-
row, 1990).
[17] H. Rosu, Metaphysical Review 3, 1, gr-qc/9409007
(1997).
[18] L. M. Ricciardi and H. Umezawa, Kibernetik 4, 44
(1967).
[19] A. Vitiello, Int. J. Mod. Phys. B9, 973-89 (1996).
[20] S. R. Hameroff and R. C. Watt, Journal of Theoretical
Biology 98, 549 (1982).
S. R. Hameroff, Ultimate Computing: Biomolecular Con-
sciousness
and
Nanotechnology
(Amsterdam, North-
Holland, 1987).
[21] D. V. Nanopoulos 1995, hep-ph/9505374
[22] N.
Mavromatos
and D.
V.
Nanopoulos
1995,
hep-
ph/9505401
[23] N. Mavromatos and D. V. Nanopoulos 1995, quant-
ph/9510003
[24] N. Mavromatos and D. V. Nanopoulos 1995, quant-
ph/9512021
[25] N. Mavromatos and D. V. Nanopoulos, Int. J. Mod. Phys
B 12, 517, quant-ph/9708003 (1998).
[26] N. Mavromatos and D. V. Nanopoulos 1998, quant-
ph/9802063
[27] N. Mavromatos 1999, J.
Bioelectrochemistry & Bioenergetics;48;273
[28] H. P. Stapp, Found. Phys. 21, 1451 (1991).
[29] H. D. Zeh, quant-ph/9908084, Epistemological Letters of
the Ferdinand-Gonseth Association 63:0 (Biel, Switzer-
land, 1981)
[30] W. H. Zurek, Phys. Today 44 (10), 36 (1991).
[31] A. Scott, J. Consciousness Studies 6, 484 (1996).

[32] S. Hawking, in The Large, the Small and the Human
Mind, ed.
M. Longair (Cambridge, Cambridge Univ.
Press, 1997).
[33] K. Hepp, in Quantum Future, ed. P. Blanchard and A.
Jadczyk (Berlin, Springer, 1999).
[34] J.
von
Neumann,
Matematische
Grundlagen
der
Quanten-Mechanik (Springer, Berlin, 1932).
[35] H. D. Zeh, The Arrow of Time, 3rd ed. (Springer, Berlin,
1999).
[36] W. H. Zurek, Phys. Rev. D 24, 1516 (1981).
[37] W. H. Zurek, Phys. Rev. D 26, 1862 (1982).
[38] W. H. Zurek, reprint LAUR 84-2750, in Non-Equilibrium
Statistical Physics, ed. G. Moore and M. O. Sculy (New
York, Plenum, 1984).
[39] E. Peres, Am. J. Phys. 54, 688 (1986).
[40] P. Pearle, Phys. Rev. A 39, 2277 (1989).
[41] M. R. Gallis and G. N. Fleming, Phys. Rev. A 42, 38
(1989).
[42] W. H. Unruh and W. H. Zurek, Phys. Rev. D 40, 1071
(1989).
[43] R. Omn`es, Phys. Rev. A 56, 3383 (1997).
[44] D. Giulini, E. Joos, C. Kiefer, J. Kupsch, I. O. Sta-
matescu, and H. D. Zeh, Decoherence and the Appear-
ance of a Classical World in Quantum Theory (Springer,
Berlin, 1996).
[45] E. Joos and H. D. Zeh, Z. Phys. B 59, 223 (1985).
[46] M. Tegmark, Found. Phys. Lett. 6, 571 (1993).
[47] R. P. Feynman, Statistical Mechanics (Reading, Ben-
jamin, 1972).
[48] B. Katz, Nerve, Muscle, and Synapse (New York,
McGraw-Hill, 1966).
[49] J. P. Schad´e and D. H. Ford, Basic Neurology, 2nd ed.
(Amsterdam, Elsevier, 1973).
[50] A. G. Cairns-Smith, Evolving the Mind (Cambridge,
Cambridge Univ. Press, 1996).
[51] P. Morell and W. T. Norton, Sci. Am. 242, 74 (1980).
[52] A. Hirano and J. A. Llena, in The Axon, ed. S. G. Wax-
man, J. D. Kocsis, and P. K. Stys (New York, Oxford
Univ. Press, 1995).
[53] J. M. Ritchie, in The Axon, ed. S. G. Waxman, J. D.
Kocsis, and P. K. Stys (New York, Oxford Univ. Press,
1995).
[54] J. Ellis, S. Mohanty, and Nanopoulos D V, Phys. Lett. B
221, 113 (1989).
[55] P. Pearle, Phys. Rev. D 13, 857 (1976).
[56] G. C. Ghirardi, A. Rimini, and T. Weber, Phys. Rev. D
34, 470 (1986).
[57] J. D. Jackson, Classical Electrodynamics (New York, Wi-
ley, 1975).
[58] W. H. Zurek, S. Habib, and J. P. Paz, Phys. Rev. Lett.
70, 1187 (1993).
[59] M. Tegmark and H. S. Shapiro, Phys. Rev. E 50, 2538
(1994).
[60] M. V. Satari´c, J. A. Tuszy´nski, and R. B. ˇZakula, Phys.
Rev. E 48, 589 (1993).
[61] H. C. Rosu, Phys. Rev. E 55, 2038 (1997).
[62] H. C. Rosu, Nuovo Cimento D 20, 369 (1998).
[63] H. S. Stapp 1999,
Attention, Intention, and Mind in Quantum Physics and
Quantum Ontology and Mind-Matter Synthesis, available

14


at www-physics.lbl.gov/ stapp/stappfiles.html.
[64] H. Everett III, Rev. Mod. Phys. 29, 454 (1957).
H. Everett III, The Many-Worlds Interpretation of Quan-
tum Mechanics, B. S. DeWitt and N. Graham (Prince-
ton, Princeton Univ. Press, 1986).
[65] J. A. Wheeler, Rev. Mod. Phys. 29, 463;1957 (1957).
L. M. Cooper and D. van Vechten, Am. J. Phys 37, 1212
(1969).
B. S. DeWitt, Phys. Today 23, 30 (1971).
[66] M. Lockwood, Mind, Brain and the Quantum (Cam-
bridge, Blackwell, 1989).
[67] D. Deutsch The Fabric of Reality (Allen Lane, New York,
1997).
[68] D. N. Page A 1995, gr-qc/9507025
[69] M. J. Donald 1997, quant-ph/9703008
[70] M. J. Donald 1999, quant-ph/9904001
[71] L. Vaidman 1996, quant-ph/9609006, Int. Stud. Phil.
Sci., in press
[72] T. Sakaguchi 1997, quant-ph/9704039
[73] M. Tegmark, quant-ph/9709032, Fortschr. Phys. 46, 855
(1997).
[74] M. Tegmark, gr-qc/9704009, Annals of Physics 270, 1
(1998).
[75] W. James, The Principles of Psychology (New York, Holt,
1890).
[76] W. James 1904, in The Writings of William James,
pp169-183, ed. J. J. McDermott (Chicago, Univ. Chicago
Press, 1977).
[77] A. Eddington, Space, Time & Gravitation (Cambridge,
Cambridge Univ. Press, 1920).
[78] A. M. Lisewski 1999, quant-ph/9907052
[79] E. P. Wigner, Phys. Rev. 40, 749 (1932).
M. Hillery, R. H. O’Connell, M. O. Scully & Wigner E
P, Phys. Rep. 106, 121 (1984).
Y. S. Kim and M. E. Noz, Phase Space Picture of Quan-
tum Mechanics: Group Theoretical Approach (Singapore,
World Scientific, 1991).

15