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\begin{document}
\title{Implications of quantum theory in the foundations of \mbox{statistical mechanics}}
\author{David Wallace\thanks{Centre for Quantum Computation, Oxford University,
Oxford OX1 3PU, U.K.; email david.wallace@merton.ox.ac.uk.
I would like to thank Michael Bowler, Harvey Brown, David Deutsch,
Artur Ekert and
Simon Saunders for encouraging comments and constructive criticism on earlier
versions of this work, Guido Bacciagaluppi, Hannah Barlow,
Clare Horsman, Lev Vaidman, Lorenza Viola and Wojciech Zurek for useful conversations,
an anonymous referee for extensive feedback, and in particular Jeremy
Butterfield for extensive and
detailed comments and advice.} }
\maketitle
\abstract{An investigation is made into how the foundations of
statistical mechanics are affected once we treat classical mechanics as
an approximation to quantum mechanics in certain domains rather than as
a theory in its own right; this is necessary if we are to understand
statistical-mechanical systems in our own world. Relevant structural
and dynamical differences are identified between classical and quantum mechanics (partly
through analysis of technical work on quantum chaos by other authors).
These imply that quantum mechanics significantly affects a number of foundational
questions, including the nature of statistical probability and the
direction of time.}
\section{INTRODUCTION}\label{intro}
\begin{quote}
Classicality simply does not follow ``as $\hbar\rightarrow 0$''
in most \emph{physically} interesting cases \ldots The Planck constant is
$\hbar = 1.05459 \times 10^{-27}$ erg s and --- \textit{licentia
mathematica} to vary it notwithstanding --- it is a \emph{constant}.
(W.\,H.\,Zurek and J.\,P.\,Paz\cite{zurek95})
\end{quote}
Why should we consider quantum issues when working in the foundations of statistical physics?
The simple (too simple) answer is that classical physics is false. If
our purpose, in doing foundational work, is to understand the actual
world, it is necessary to use a theory which validly describes that world.
Of course, nonrelativistic quantum mechanics is strictly speaking also
false. It is believed to be only a tractable limiting case of quantum
field theory, which in turn is expected someday to be replaced by
quantum gravity, and even that theory may not be the final word on
physics.
This is not simply a foundational dilemma: we face it at all levels in
physics. It is obviously impossible to work with theories which we do
not yet know, and usually computationally intractable to work with our
most fundamental known theories; yet if we do not do this then we are
working with a theory which is known to be false --- why, then, should
we believe its conclusions?
From a theoretical point of view, we address this problem by finding
subdomains of more fundamental theories in which they approximate our
less fundamental, but more tractable theories --- so general relativity
reduces to special relativity in regions where matter densities are low
and space is nearly flat, while special relativity in turn reduces to
Newtonian mechanics when relative velocities are small.
Delineating these subdomains, however, is problematic. Although we can
get a good general idea of their locations, it is generally not possible
to predict perfectly at what level we can explain some natural
phenomenon, and it is always possible to be surprised. (To take one
example, it turns out that the color of gold is due to relativistic
corrections to its electron orbits).
To see this approach in more detail, we can
distinguish three theories. The first is classical mechanics (CM),
treated as an exact theory of particles moving under Hamilton's
equations. The second is quantum mechanics (QM), which is presumed to
be approximated by CM in certain domains. Restricting QM to these
domains gives something which we could call \emph{classical-domain
quantum mechanics}, or CDQM --- this isn't really a new theory, of
course, just a subtheory of QM.
In this language, the approximation of QM by CM amounts to some
``approximate isomorphism'' $\iota$ between CM and CDQM:\footnote{If
desired, $\iota$ may instead be understood as an approximate isomorphism
between models of CM, and models of QM restricted to a certain domain.}
\be CM \overset{\iota}{\leftrightsquigarrow} CDQM \subseteq QM.\ee
$\iota$ is required to preserve most of the structural properties of CM,
but we will not require it to preserve all of them --- instead we take
it as an open question as to what structural features can be preserved
by $\iota.$
As physicists studying a particular class of quantum phenomena, we start by
tentatively deciding whether these phenomena lie in the
domain of CDQM. If so, we try applying CM to the phenomena, but in
doing so it is always necessary to remember the possibility that the
particular features of CM which are doing explanatory or predictive work
in our proposed account of the phenomena may not actually be preserved
by the isomorphism $\iota$. In this case, it is necessary make slight modifications to
modify the structure of the part of CM being used so as to better approximate CDQM, or
if necessary to abandon CM altogether and work entirely with QM.
Fleshing out the details of this analysis of theory approximation and
approximate structure-preservation lies rather beyond the scope of this
paper. See Wallace\cite{wallace} for (some) further discussion of the idea of an
``approximate isomorphism'' between theories; see also the literature on
structural realism (\egc, Worrall\cite{worrall}, Psillos\cite{psillos}, Ladyman\cite{ladyman}.)
Very much the same approach could be used to describe nonrelativistic
mechanics (NRM) and special relativity (STR). NRM is a limit of STR as
$c\rightarrow \infty$, but the value of $c$ is fixed and finite so this limiting relationship
cannot directly be the reason why NRM sometimes gives quite accurate
results. Instead we consider domains of STR in which typical energies
seem quite low, restrict STR to those domains (producing low-energy STR,
or LSTR), and construct an approximate isomorphism $\tau$ between NRM
and LSTR,
\be NRM \overset{\tau}{\leftrightsquigarrow} LSTR \subseteq STR.\ee
To count as an approximate isomorphism $\tau$ should preserve most of
the structure of NRM, but not necessarily all of it.
The movement of electrons in magnetic fields provides an example of this
process (in the context of relativistic and non-relativistic QM, not CM).
There, we begin by considering electrons with velocities far
below $c$, which we suppose are covered by LSTR. However, on
investigation we find that relativity predicts a coupling between the
magnetic field and the internal spin of the electron, and to maintain
the correspondence between NRM and LSTR in this domain we have to add
such a coupling term to the non-relativistic Hamiltonian. If we were
to allow the energy of the electron to rise too high, we would have to
give up on even this modified form of NRM and work directly with STR, of
course. In either case, there is not really any way internal to NRM for us to predict
difficulties with it; such predictions have to come from STR itself.
Is this approach appropriate to foundational work? Yes and no, but
mostly yes. There is genuine reason to be interested in the structure
of old theories even when they are known not to apply to the world:
trying to find the classical description of charged point particles, or
examining the nature of spacetime as given by Newtonian physics, are
interesting tasks in their own right and potentially might shed light on
other problems. However, by and large we are interested in foundational
problems --- and, generally, in problems in physics and in philosophy ---
because of what their solutions would tell us about the world. In that
case, clearly it is necessary to work with a theory which applies to the
world, or at least to that part of it under study.
What of statistical mechanics? We might hope that at least a large
portion of statistical mechanics could be understood using classical
concepts --- after all, classical statistical mechanics makes the
assumption that classical mechanics is valid, and presumably there is a
domain of phenomena in which that assumption is reasonable. There would
be a separate set of foundational problems in purely \emph{quantum}
statistical mechanics --- in the statistical mechanics of atomic nuclei,
or of the quark-gluon plasma, for instance --- but they could be
addressed after the classical case was properly understood. However,
this approach would depend upon the existence of domains in which
statistical mechanics applies but in which quantum mechanics makes no
appreciable difference to the underlying physics: in other words, domains of QM in
which statistical mechanics applies, which are approximately classical in the
sense that there exists an approximate isomorphism $\iota$ between the
classical and quantum descriptions of the domains, and in which those
features of classical mechanics relevant to statistical-mechanical
explanation and prediction are preserved by $\iota$.
It is the purpose of this paper to show that no such domains exist.
Statements of classical statistical physics such as `there exist
isolated systems whose dynamics are chaotic and given by Hamilton's
equations', or `ordinary macroscopic systems have some unknown state
which is approximately localized in momentum and position' turn out to
be, not just only approximately true, but universally false.
As such, it would seem necessary to allow for at least certain quantum-
mechanical phenomena in foundational work on statistical
physics.\footnote{It may seem that there is a danger of infinite regress
here: why not also allow for quantum field theory in foundational work,
and so on? This danger is always present --- in foundational work as
much as in mainstream physics. All we can do is try to work out the
correct level of theory at which to work, and be prepared to change our
strategy if it is shown that our theory does not apply. In fact, I
think there are good arguments that non-relativistic quantum statistical
mechanics is not significantly affected by QFT, but a clear
demonstration of a significant effect would indicate the falsity of
any such arguments.} Specifically, our attitude to foundational
problems in classical statistical mechanics should be that, if they are
preserved by $\iota$, they are to be taken seriously, but if instead
$\iota$ does not preserve them then they are artifacts of CM which ---
whatever their intellectual interest --- cannot be directly relevant to
statistical mechanics as it applies to the actual world.
In the following I shall attempt to sketch out the ways in which quantum
mechanics affects statistical physics, concentrating on those domains of
quantum statistical mechanics in which it might appear that classical
physics should be valid. In section \ref{equilibrium} I shall review
the classical description of statistical equilibrium, and attempt to
show how problematic it is to extend this description to quantum
mechanics --- and hence, \textit{a fortiori}, to CDQM; I shall then
give some suggestions as to the correct description of quantum
statistical equilibrium.
In section \ref{CDQM} I shall focus more specifically on CDQM, and
analyze --- with the help of the Wigner function --- in exactly
what sense certain quantum systems can be regarded as
approximately classical; in passing I shall make some comments on the
problem of justifying phase-space measures (in particular, the
justification for neglecting regions of Liouville measure zero) in
classical statistical mechanics.
A description of CDQM having been obtained, in section \ref{chaos} I
shall apply it to classically chaotic systems (systems describable by
classical statistical mechanics generally being agreed to be of this
type). I shall show (mostly by reference to existing work on chaos and
decoherence) that such systems typically display relevantly different
behavior when regarded as ``approximately classical'' quantum systems
from when they are regarded as fully classical, and that as such there
are properties of CM, highly relevant to foundational problems in
classical statistical mechanics, which are not preserved by the
approximate isomorphism between CM and CDQM. This has implications, in particular, for
the problem of the direction of time, and I will briefly describe these
implications. Section \ref{conclusion} is the conclusion.
The approximation of quantum
physics by classical physics is one of the most notoriously complex --- and
controversial --- of the approximation schemes discussed above;
it is extremely hard to discuss without taking a stance on the
interpretation of quantum mechanics. In this paper I presume, wherever interpretational
issues come up, that the wave-function of a closed system at all times evolves unitarily
(\iec, in accordance with the Schr\"{o}dinger equation). This is the viewpoint taken by
Everett\cite{everett} and my treatment is most naturally compatible
with decoherence-based Everett-type interpretations such as those
proposed by Saunders\cite{saunders,saundersprob},
Vaidman\cite{vaidman}, Zurek\cite{zurekprob}, and
myself\cite{wallace,wallacestructure}, but should also apply to other versions of
the Everett interpretation such as the many-minds theories
proposed by Lockwood\cite{lockwood} and Albert and
Loewer\cite{albertloewer}, and to some extent even to other unitary
interpretations like the de Broglie-Bohm pilot-wave
theory\cite{bohm,holland}.
Other theories will be less well described
by the approach I take, although some fairly interpretation-independent
comments will be made in section \ref{chaos}.
It may be the case that other interpretations do in fact have different
consequences for statistical mechanics, but if anything this would strengthen the arguments
for considering quantum mechanics,
since it would show that foundational issues in statistical mechanics are affected by
interpretative issues in quantum theory.
\section{DESCRIBING STATES AT EQUILIBRIUM}\label{equilibrium}
\subsection{Equilibrium in classical systems}
In classical equilibrium statistical mechanics, it is generally assumed that:
\begin{itemize}
\item The possible states of a classical system are given by the points in some phase space
\mc{P}.
\item At any given time $t$, the specific system under consideration has a determinate state given
by a specific point in \mc{P} --- though this point is assumed not to be exactly known.
\item At time $t$, the probability that this determinate state is in a given region of \mc{P} is
given by some probability distribution over \mc{P}.
\item The time-evolution of the system is deterministic (given by Hamilton's equations) and so
knowing the probability distribution at one time tells us what it is at
all other times.
\item A system is said to be \emph{at equilibrium} when the
probability distribution does not vary in time.
\end{itemize}
Fleshing out this program leads both to technical and to conceptual
difficulties. On the technical side, we would like to establish that
if some system's energy is required to be in some infinitesimal interval
($E,E+\dr{E}$), then the only possible equilibrium probability
distribution is the \emph{microcanonical} distribution, for which the
probability of finding the system to have a state in some region \mc{R}
of phase space (where all states in \mc{R} have allowed energies) is
proportional to the Liouville volume\footnote{Recall
that the Liouville measure on phase space equals the
Lebesque measure, if the coordinates used to define the latter are
chosen to be canonical.}
of \mc{R}. This result is
guaranteed to hold whenever a system is \emph{ergodic}, that is, when
all dynamical trajectories through points of energy $E$ pass arbitrarily close to all
other points with the same energy.
Ergodicity (or weaker, but possibly still acceptable, analogues
of ergodicity) is a plausible property of classical systems because
phase space has no natural concept of distance --- there is a
mathematical notion of volume (the Liouville volume) which is conserved
under all acceptable dynamics, but in general no conserved metric, so
initially very close points in phase space can be taken by the dynamics
to widely separated points. This means that classical mechanics is
compatible with very chaotic dynamics, which in turn might plausibly
lead to very wild motions in the phase space, and hence to the sort of
mixing which implies ergodicity --- although of course this is just a
heuristic and not a proof.
On the conceptual side, there are problems in understanding the nature
of the probability distribution which we have put on the phase space. It
is often interpreted as being a relative-frequency distribution over an
infinite ensemble of copies of the system --- but what is the connection
between this ensemble (which obviously cannot physically exist in a
finite classical universe) and the single systems which we in fact
observe?
\subsection{Equilibrium in isolated quantum systems}
In discussions of quantum statistical
mechanics one often\cite{baierlein,grandy,feynman,kittel} finds the
following account of equilibrium: if the system is at energy $E$,
and if there are $n$ orthogonal eigenstates $\ket{\phi_1},\ldots,
\ket{\phi_n}$ all of energy $E$, then the system is determinately in one of these states,
with probability $1/n$ for each. It is sometimes even claimed that
quantum mechanics simplifies the classical description by replacing a
continuum of classical states with finitely many quantum states.
It is easy to see that this description must be inadequate. Any
system for which $E$ is a degenerate eigenvalue of the Hamiltonian will
have a continuum of eigenstates with energy $E$, and there is no reason
to prefer one basis for the $E$-eigenspace over another. But if we
say that $\{\ket{\phi_i}\} $ is such a basis, then we are claiming that the
system is with certainty in state \ket{\phi_j}, for some $j$, and hence
that it has no chance at all to be in superpositions such as
\be \frac{1}{\sqrt{2}}(\ket{\phi_1}+\ket{\phi_2}).\ee
In any interpretation which treats the quantum wavefunction as real,
this is implausible, as the choice of the \ket{\phi_i} basis was made
arbitrarily.
More careful treatments (see, \eg, page 33 of Binney \textit{et al}.\,\cite{binney} or page
99 of Garrod\cite{garrod}) acknowledge this problem, and instead define the
quantum microcanonical distribution in a manner very similar to that
used in classical mechanics: a probability distribution is placed upon
all states in the energy-$E$ eigenspace, not just on a specific basis
of that subspace. Since there is a continuum of such states, this
requires a measure to be specified: the measure used is the volume
measure on the unit sphere in Hilbert space. This process is
superficially closely analogous to the procedure used in classical
physics (the Hilbert-sphere measure, like the Liouville measure, is preserved under
all allowed dynamics), and can be shown to give the same statistical results as the
naive account.
However, it is easy to see that the analogy with classical mechanics can
only be superficial. In the classical case a given energy-$E$ state will move
wildly around the energy-$E$ subspace of \mc{P} (this is what ergodicity is all about),
but a quantum state of energy $E$ is an eigenstate of energy, and hence will not change
at all with time. Hence not just the canonical distribution, but any
distribution at all within the quantum energy-$E$ subspace will be
preserved with time.
More generally, finding ergodic behavior in quantum systems --- not
just in energy eigenspaces --- can reasonably be predicted to be much
harder than in classical systems. The reason for this is that in
quantum mechanics, unlike classical mechanics, there exists a natural
distance measure --- given by the Hilbert-space norm --- which is
preserved by the quantum dynamics. This means that states which begin
close together (as defined by this measure) will forever after remain as
close together. This property of quantum systems, though not
ruling out a quantum analogue of ergodicity, clearly makes it less plausible
as an assumption about a given
system.
The conceptual questions regarding the probability distribution over
states are also different in the quantum case. In quantum statistical
mechanics we do not in fact work with probability measures over states,
but with probability density operators --- that is, with self-adjoint,
positive, trace-one operators on the Hilbert space of the quantum
system. This is of course possible because our ability to measure the
state is much more restricted in quantum than in classical mechanics: in
the latter case we can in principle measure the exact phase-space point
describing the system, whereas in the former case it is a consequence of
Gleason's theorem\cite{gleason} that any probability distribution of observation outcomes on the system
can be described by a single density operator.
This leads to an underdetermination of the probability distribution in
quantum mechanics. The space of possible probability measures over an
$n$-dimensional eigenspace is of course infinite-dimensional, whereas
the space of density operators describing the same subspace is only
$(n^2-1)$-(real)-dimensional. Hence there are a vast number of possible
probability distributions on pure quantum states which give the same
density operator --- in other words, which are empirically equivalent.
We briefly recall why this is so: given a probability distribution $p(i)$ over some
(not necessarily orthogonal) states $\{\ket{i}\}$, the density operator
is
\be \denop = \sum_ip(i)\proj{i}{i},\ee
but this map is not $1:1$ and in general a single density operator can
be represented in vastly many ways as a probability distribution over
non-orthogonal states. (Generically there is only one way to represent
it as a distribution over orthogonal states, but we have already argued
that restricting the probability distribution to a specific orthogonal
family cannot be justified.)
\subsection{Equilibrium in realistic quantum systems}\label{decoherence}
The previous section showed that a statistical mechanics of pure quantum
states would be disanalogous in a number of ways from the statistical
mechanics of classical systems. However, even the concept of putting a
probability distribution over the pure states is dubious, since a pure
state is not the most general state for a quantum system.
To recall why this is, let the Hilbert space of the system under question be
$\mc{H}_\mc{S}$. Unless our system is the entire Universe, there must
also be some Hilbert space $\mc{H}_\mc{E}$ of environmental degrees of freedom,
such that the Hilbert space of the Universe
is\footnote{This is something of an over-simplification, actually:
if we really are considering the state of the Universe then we can
expect it to break into wildly different branches, with the subsystem
structure varying from branch to branch; hence, the tensor-product decomposition above
is to be understood as applying only to the branch or set of branches in
which the system actually exists.}
\be \mc{H}_\mc{S}\otimes\mc{H}_\mc{E}.\ee
Then even if the combined state of system-plus-environment is a pure
state, it will not generically be a product state, but rather a highly
entangled joint state of $\mc{H}_\mc{S}\otimes\mc{H}_\mc{E}$. In such a
situation, our system can be assigned a state of its own (provided
we avoid carrying out any joint operations on system and environment
together) but this state will be a mixed state --- \ie a density operator over $\mc{H}_\mc{S}$
and not a pure state in $\mc{H}_\mc{S}$.
Furthermore, even if we do prepare the system in a pure state there is no reason
actually to expect it to remain pure. Although the interaction between system and
environment is by assumption weak, it has been repeatedly
shown\cite{jooszeh, zurek} that even very weak connections
lead to very rapid decoherence (\ie entanglement with the environment). To get some
insight into why this is, suppose that the system is initially described by a pure
state made up of fairly localized particles. The uncertainty principle causes the
wave-packets of the particles to spread out, and of course we may regard a spread-out
packet as a superposition of differently-located, localized packets. These packets will
exert a superposition of differently directed forces on the environment, causing it to
evolve differently for different packets. In dynamical terms this difference may be
tiny, leading to environmental states which differ by only nanometers. But all that
is important for decoherence is that different states of the system are coupled with
orthogonal states of the environment, and two non-overlapping wave-packets are just as
orthogonal when separated by manometers as by parsecs.
All this means that if we wish to place a probability distribution over
states of a quantum system it had better be a distribution over all
states --- pure and mixed --- of the system, not just over the pure
states. It is then rather unclear what that distribution should be, for
while there is a measure on the space of density operators
(generated by the inner product
$\langle\op{A},\op{B}\rangle=\tr(\opad{A}\op{B})$ on the vector space of
all linear operators) which is preserved under unitary transformations,
the decoherence process which sends pure states to mixed ones is not
unitary.
At this point one might object that the problem has been changed by
sleight-of-hand. After all, in classical statistical mechanics it is
generally assumed that a system is isolated from external influences, so
should we not make the same assumption of isolation in quantum
mechanics?
There are three problems with this view. Firstly, even if the influence
of an external environment is entirely blocked, a quantum system can
still be entangled with that environment --- so a mixed state is a
perfectly valid state for that system, and prima facie should be considered when
putting a probability distribution onto such states. (There is no classical analogue for
this: even when a classical system is interacting with an environment, its state is still
represented by a single point in phase space.)
Secondly, it is overwhelmingly difficult to isolate a quantum system
even approximately from its environment --- vastly more so than for a
classical system. To understand why this is so, recall that for a
classical system interference from the outside is generally in the form
of energy transfer between system and environment, which can usually be
neglected by making the characteristic energies within the system very large compared
with the energy transfer in system-environment interactions. In quantum mechanics,
by contrast, interference primarily takes the form of \emph{information} transfer,
and this cannot be eliminated in the same fashion. For instance, in
considering the motion of the Earth around the sun, energy
loss through damping from the microwave background radiation is utterly
negligible --- but if the Earth were in a coherent superposition of two macroscopically
separated states, a single microwave photon would suffice to decohere
them.
The third reason is that to recover classical mechanics from unitary quantum
mechanics, it is generally agreed that we need decoherence to prevent
the existence of unobserved coherent superpositions of macroscopically
distinct states. As such, at least whilst we are considering domains of
quantum mechanics which are in some sense ``approximately'' classical (and in particular,
those which are macroscopic)
we will be forced to include some decoherence. This point will be developed further
in section \ref{chaos}.
So we return to the problem of finding a probability distribution over
the pure and mixed states of the system --- that is, over its density
operators. Now, these density operators are conceptually speaking very
different entities from the `probability' density operators which were
introduced in the previous section. Those were used to describe a
probability distribution over states, to be used when the actual state
was unknown. The density operators describing states of a
system entangled with its environment, on the other hand, encode no
classical ignorance at all, but rather reflect the impossibility of
describing an entangled state in terms of its constituent systems alone.
We might call these latter density operators ``entanglement'' density
operators (they are also sometimes referred to as `improper mixtures').
So applying the methodology of classical statistical mechanics would
lead us to place a probability distribution over the
$(n^2-1)$-dimensional space of such entanglement density operators. But here the
underdetermination of the previous section returns with a vengeance, for
any such distribution can itself be described by a \emph{single} probability
density operator. The map is of form
\be p(\denop)\longrightarrow \int\mc{D}\denop \,\, p(\denop)\, \denop\ee
(where $p(\denop)$ is the given probability distribution over entanglement density
operators $\denop$) and is obviously many-to-one.
So although the empirical success of quantum statistical mechanics leads
us to expect this probability density operator to be proportional to the projection
operator onto the energy-$E$ subspace, it tells us nothing about the
physical interpretation of this operator --- at one extreme it might be a
probability distribution over unknown pure states, at the other extreme the
system might be known to be described by an entanglement density operator equal to
the probability density operator, so that there is no classical ignorance or classical probability
at all.
\subsection{Failure of the classical description}\label{failure}
In view of the time-independence of energy eigenstates, of the subsequent difficulty of
using any ergodic results to establish uniqueness of equilibrium distributions, of the
inappropriateness of describing a macroscopic quantum system as totally
isolated from its environment, and of the great extent to which empirically
accessible data underdetermines any probability distribution placed on
quantum states, we conclude that there is no reason to think that the
classical definitions and derivations of equilibrium apply to
quantum-mechanical systems. It follows that the features of classical
mechanics which allow such definitions and derivations cannot be
well-approximated in any approximately classical domain of quantum
theory. If it is accepted (as was argued for in section \ref{intro})
that classical mechanics tells us about the world only insofar as it is
approximated by quantum mechanics, it follows that the classical
derivation of equilibrium cannot be used to justify the nature of
equilibrium states in the world.
This leaves the canonical distribution in a rather precarious position:
virtually all attempts to derive it begin with classical physics and
attempt to extend the results to quantum physics by analogy, but the
arguments of this section imply that
\begin{itemize}
\item this extension to quantum mechanics doesn't really work (in other words, the
arguments used in the derivation don't work, even if the answer is
correct);
\item as such, the `classical' account cannot be a valid account of the
microcanonical distribution even in `classical' regimes of quantum physics. If it can
be made to work for genuinely classical systems (which is controversial, of course)
then this must be an artifact of classical mechanics and not something
which applies to the actual world.
\end{itemize}
This seems to leave an urgent need for a genuinely quantum account of
equilibrium. Though fulfilling this need in any detail lies rather
beyond this paper,
some tentative ideas will be advanced in section \ref{qisp}.
\subsection{The quantum interpretation of statistical
probability}\label{qisp}
I would like to propose the following conjecture, which I shall refer to
as the quantum interpretation of statistical probability (or QISP):
\begin{quote}
`Ignorance' probability, in the sense of a probability distribution over
a space of many possible states of a system, one of which is actual, has no place in statistical
mechanics. As such, the \emph{probability} density operator should be banished
from statistical mechanics. When a density operator is used to describe
a statistical system, it is to be understood as the determinate ---
though highly non-pure --- \emph{entanglement} density operator which describes
that specific system.
\end{quote}
My reasons for recommending this conjecture are as follows. The first
three reasons are basically conceptual; the other three are more dynamical, and probably
more important.
\begin{enumerate}
\item It would solve --- albeit by fiat --- the problem of
underdetermination of the probability distribution by the statistical
facts. If no ignorance probability distribution is being introduced into the
theory at all, then there is no reason to be embarrassed by the problems
which would arise if such a distribution \emph{were} to be introduced.
\item It would make the concept of `ensemble' rather less problematic.
In one sense, regarding the density operator as describing a single
system removes the ensemble concept entirely; in another (particularly
to a reader sympathetic to `many-worlds' language) it makes the ensemble
a physical entity rather than a theoretical abstraction.\footnote{Of
course, it is important not to confuse the ontological statement that an
ensemble is physical, with the epistemological one that we have access
to all of it: we are part of the environment of the system and as such
will ourselves be entangled with it (I am grateful to an anonymous
referee for this point). Also, if we are to regard the density operator
as representing an ensemble of Everett worlds, in a sense we return to the
underdetermination problem, for we may make a decomposition into worlds
in many ways. Of course this is nothing more than the preferred-basis
problem; see \cite{wallace} for a discussion of why we need not let it
get in the way of seeing a quantum state as genuinely describing a
multiplicity of worlds.}
This has consequences for the way we use the ensemble concept in statistical physics.
In classical statistical mechanics we are used to a number of properties and
descriptions applying not to the individual system (which is in some specific
unknown microstate) but to an ensemble, a supposed infinite collection of such
systems whose microstates are distributed over those compatible with the macrostate
according to some probability distribution. In particular, concepts like entropy
are defined with respect to the ensemble, and the change with time of a macrostate
is usually taken to apply to an ensemble rather than an individual system (to
bypass the Poincar\'{e} recurrence theorem, amongst other reasons). A general
discussion is given by Sklar\cite{sklar}.
In quantum mechanics, if QISP holds then it makes sense to describe a single system
as being in a macrostate (\ie described by an entanglement density operator), and
we should be able to assign macrostate properties such as entropy to that single
system. This may make it more coherent to describe a unique system as having a
certain probability distribution.
This redescription of single systems has relevance for the reduction of
thermodynamics to statistical mechanics. In thermodynamics quantities such as
entropy are properties of single systems, whereas in classical statistical
mechanics they are often taken to be properties of ensembles, or of regions of phase
space, rather than of individual systems. It appears that this does not hold in
quantum statistical mechanics: entropy may be defined for a single system as in
the thermodynamic case.
\item If QISP holds, then the (highly problematic\cite{sklar})
probabilities of statistical mechanics are to a large extent removed
from consideration, to be replaced with the probability intrinsic to
quantum mechanics. Admittedly, the extent to which this is a good thing
will depend upon how sanguine the reader is about our prospects for
understanding quantum probability. In a pilot-wave theory the
probabilities are actually introduced in virtually the same way as for
classical statistical mechanics, so not much is gained; the topic of
probability in Everett-type interpretations is much more controversial
(for some recent proposals on how it is to be understood, see
Deutsch\cite{deutsch}, Saunders\cite{saundersprob},
Vaidman\cite{vaidman} and Zurek\cite{zurekprob}). But
at any rate the problems with probability in statistical mechanics and
in quantum mechanics are prima facie two hard problems urgently in need
of solution; there is perhaps something to be said for any strategy
which replaces two problems with one.
\item QISP allows us to construct a `transcendental' account of equilibrium --- that is, a justification
of the equilibrium state independent of any causal story as to how systems get into equilibrium
in the first place --- for quantum mechanics which is in some way
analogous to the usual accounts in classical mechanics. Recall that in
the latter, the microcanonical distribution is to be justified as the only
probability distribution which is time-independent; heuristically we
expect this to be so because the only obvious conserved quantity for the system
is energy and we expect the dynamics of large systems to be unstable
enough to explore all regions not expressly forbidden by conservation
laws. But all the evidence from studies\footnote{We can identify at
least three separate research programs in
decoherence which support these claims. The use of the predictability
sieve\cite{zurek93a,zurek93b} as a method of picking out minimum-decoherence pure
states tends to show that, although there exist states (usually
coherent states) which are maximally resistant to decoherence,
nonetheless all the pure states eventually decohere. (Admittedly this
is only partial support, for it does not prove that there are no highly mixed states other than the
microcanonical ones which don't decohere.)
The algebraic approaches\cite{zanardi,klv} to studying decoherence
in small (multi-qubit) systems suggest that only symmetries provide a
method of stabilizing systems against decoherence. And the topic of
quantum analogues of chaotic systems, which is perhaps the most
suggestive for our purposes, will be discussed \textit{in extenso} in
section \ref{chaosa}, and shown to give strong support to the claims in (4).}
of decoherence suggests that
(in the absence of dissipation) the only density operators which are invariant
under decoherence are projections (and sums of projectors) onto
eigenspaces of the conserved quantities. For a system which has
only energy as a conserved quantity, this is equivalent to saying that
the only invariant density operators are microcanonical operators and
their sums.
\item Although the classical arguments alluded to in (3) above seem plausible,
it has generally been recognized in classical statistical mechanics that
more technical work is needed to establish them rigorously --- this is
where ergodicity comes in. Much progress has been made on these
technical questions (reviewed by Sklar\cite{sklar}) and despite the arguments
advanced above against the classical derivations of equilibrium, it is hard
to deny that there is something very suggestive about some of the
ergodic results --- and thus something unsatisfactory in simply writing
them off as an artifact of a superseded theory. It is then a virtue of
the QISP approach that it rehabilitates this technical work, maintaining
its connection with equilibrium even whilst to some extent abandoning
the conceptual arguments usually used to establish the connection. This
point cannot be developed further until we have discussed quantum
mechanics on phase space; we will return to it in section \ref{chaosb}.
\item The last point is perhaps the most telling: if we accept
the plausibility of (4) then the microcanonical density operator
(interpreted as an entanglement density operator) is the only state of
the system (at given energy) which is a valid equilibrium state --- all other states
evolve to that state, so any probability distribution over any other
states will not be an equilibrium distribution at all. Put another way:
it may well be that QISP holds automatically at equilibrium, because the dynamics of the
system force it upon us.
\end{enumerate}
To illustrate these points --- and in particular the last one --- we now
leave quantum mechanics in its full generality, and focus on those
regimes of the theory in which it may be said to approximate classical
physics.
\section{CLASSICAL-DOMAIN QUANTUM MECHANICS}\label{CDQM}
\subsection{Quantum mechanics on phase space}
Let us take stock. In section \ref{intro} it was argued that (despite its strict falsehood)
we are able to use classical mechanics as a theory in some circumstances
because quantum mechanics has regimes which are approximately isomorphic
to regimes of classical physics, and that we should only be interested
in the foundational problems of classical statistical mechanics \emph{per se} insofar as they
are taken by this approximate isomorphism into genuine foundational
problems of quantum statistical mechanics.
Section \ref{equilibrium} gave what might be called a `non-constructive'
proof that many such classical foundational problems are not in fact taken over to quantum
mechanics. The argument can be summarized thus:
\begin{enumerate}
\item quantum mechanics is conceptually
and structurally incompatible with the methods used to derive classical
statistical mechanics;
\item hence those methods do not apply to quantum
mechanics;
\item \textit{a fortiori} they cannot apply to the classical domains of
quantum mechanics;
\item hence problems with classical statistical mechanics which are
formulated in terms of these classical methods cannot apply to
classical-domain quantum mechanics.
\end{enumerate}
In the remainder of the paper, we shall be concerned with the more
`constructive' question of why the approximate isomorphism between
classical mechanics and classical-domain quantum mechanics
fails to map the classical derivations of statistical mechanics into
quantum mechanics. In doing so, we will shed some light on the QISP
hypothesis of section \ref{qisp}, and restore to some extent the
importance of the concept of classical ergodicity.
To do this, we need to spell out just what `classical-domain quantum
mechanics' actually looks like as a theory (more accurately, as a
subtheory of quantum mechanics). The requirements on CDQM are fairly
obvious: it should be a theory of quantum states approximately localized
in both position and momentum, and those states should approximately
speaking evolve in accordance with the classical equations of motion ---
or more precisely, in accordance with \emph{some} classical equations of
motion.
Of course, finding states which are \emph{exactly} localized in both
position and momentum is an impossible goal: position and momentum do
not commute. But equally, there are some states which should obviously be described
as effectively localized within a certain region both of position and momentum space:
a Gaussian wave-packet is an obvious example. There is of course an extensive literature
on approximate joint measurements of position and momentum, using the
positive-operator-valued-measure (POVM) formalism (see Busch, Grabowski and Lahti\cite{busch}
for a recent review). We shall not need the details of this formalism here, for it is enough
to know that there exists a well-defined notion of approximate measurement of phase space
position, and that we can use it to determine which states are approximately localized.
Rather as we would expect, the POVM formalism predicts that it makes
sense to describe states localized in phase-space volumes large in
comparison to $\hbar^n$ (where $2n$ is the dimensionality of phase
space) but that we cannot find a good concept of phase-space localization within
significantly smaller volumes.
The phase-space dynamics of states is usually studied by the following method: we look for
some representation of all the quantum states (not just the
approximately localized ones) in some way which makes their associated phase-space
probability distribution transparent; we then look at how that
distribution evolves under the quantum dynamics. In general it is
possible to represent both pure and mixed states in this fashion.
We can then insist on two criteria which the system must satisfy in
order to count as ``approximately classical''. Firstly, those pure
states which are approximately phase-space-localized must remain
approximately localized on relevant timescales: localized states must not
be delocalized by the dynamics. (We are not restricting our attention
to closed quantum systems, so we will permit localized pure states to be
taken to entanglement density operators made up from localized pure
states). Secondly, we require the states to obey approximately
classical dynamics, which is to say that the localization centers of
localized states are required to evolve in accordance with Hamilton's equations
for some classical Hamiltonian (which should presumably be related in a
relatively straightforward way to the quantum Hamiltonian).
\subsection{The Wigner function}\label{wigner}
There is a considerable tradition of constructing phase-space
representations
of quantum states, going back to work of Wigner.\cite{wigner} Wigner's
own solution to the problem is still the most widely used and will
suffice for our purposes: to every quantum state (pure or mixed) \denop,
we will assign a real function $W_\denop(q,p)$ on phase space. This function
(the \emph{Wigner function}) has some of the formal properties of a
classical probability distribution. In particular, it yields exactly the
right `marginals', \ie the probability distributions over position and
momentum separately:
\begin{equation}
\int \dr{p} W_{\denop}(q,p) = \matel{q}{\denop}{q}
\end{equation}
and similarly for momentum. However, it cannot be regarded even
formally as a probability, as it is sometimes negative!
We always knew, though, that there was no prospect of quantum states
yielding an exact phase-space probability distribution, because (a) the uncertainty
principle tells us that no
quantum state is perfectly localized in phase space, and (b) it is easy to exhibit pure
states which are not remotely localized. As explained in the previous
section, our goal is the much more modest one of providing a
probability distribution over the outcomes of \emph{approximate} joint
measurements of $q$ and $p$, and it can be shown\cite{busch} that if we
``blur'' the Wigner function over regions of volume $\sim\hbar^n$, we
obtain precisely such a probability distribution for the state.
The Wigner function is far from the only function of this kind: many
others (Husimi functions, P-distributions, Q-distributions, etc.\,) have
been proposed at various times, and none of them avoid some pathologies
when examined on scales small compared with $\hbar^n$; this is an
inevitable (and provable) consequence of the uncertainty principle. It
is tempting, but mistaken, to ask which function is the `correct'
representation on phase space. They are all representations of the same
quantum state, and when observed at appropriate scales they all give the
correct probability distribution with
respect to approximate phase-space measurements. I have chosen to work
with the Wigner function in this paper because it is the most commonly
used in investigations of quantum phase-space dynamics and quantum
chaos; this is in turn because of the relative simplicity of its
definition and equations of motion.\footnote{The equation motion for the
Wigner function is given by the Moyal bracket\cite{moyal}, a generalization
of the Poisson bracket. The Moyal bracket of two functions equals their
Poisson bracket plus some higher terms suppressed by powers of $\hbar$.}
With this technical machinery in hand, we can now state cleanly some
necessary conditions for classical results to be treated as relevant to
the actual quantum world. Results about the evolution of some classical probability
measure on phase space apply to classical-domain quantum mechanics, and
hence have some chance of applying to the actual world, only if:
\begin{enumerate}
\item There exists a Wigner function whose phase-space probability
distribution approximately matches the classical distribution, and which is the
Wigner function either of an approximately localized pure state, or of a
mixed state made up of such pure states.
\item The quantum dynamics of the system cause the Wigner function's
evolution in phase space approximately to match the evolution of the
classical distribution.
\item These quantum dynamics also do not cause localized states to
become delocalized, at least on the timescales relevant to the problem.
\item The classical results in which we are interested have not been
lost as a consequence of all the `approximately' caveats of the first
three points!
\end{enumerate}
It should be stressed that these are necessary but not sufficient
conditions. In particular, all of the above could hold for some system and yet it might
still be necessary to regard the Wigner function as being that of an
entanglement density operator, and thus as having nothing to do with
classical probability --- unlike the case in the classical distribution
being approximated.
We are now in a position to consider the dynamics of `approximately classical' systems and
compare them to genuinely classical ones. This will be the task of
section \ref{chaos}, but first we digress onto another foundational
problem of classical mechanics which does not apply to classical-domain
quantum mechanics: the `problem of measure zero'.
\subsection{Justifying phase space measures}
When doing classical statistical mechanics, to calculate probabilities it is necessary
to have both a function on phase space and a measure on it, and it is customary to use
the Liouville measure. There are various motivations for the naturalness of this choice:
it is the only measure which has the same functional form in any canonical coordinates;
it is the only measure preserved under all allowed (classical) phase-space evolutions;
the equations
of motion for probability distributions take on a particularly elegant form when Liouville
measure is used, etc. Furthermore, in general our choice of measure can quite legitimately
be made on grounds of convenience: given a probability distribution we can represent it with
any measure\footnote{Given a function $f$, a measure $\mu$, and a region $\mc{R}$ of phase space,
the probability of finding the system in \mc{R} is $\int_\mc{R} f \dr{\mu}$; clearly, then,
modifications of $\mu$ can be compensated for by modifications of $f$.} (provided we are willing to use singular functions like delta functions).
Unfortunately, there are some situations where the connection between the Liouville measure
and probability must be made \emph{a priori}: specifically, many classical dynamical results
which we would like to hold for all phase-space points actually hold for all points \emph{except
for a set of Liouville measure zero}. We would like to exclude these points by saying
that a given system has zero probability to be found at one of these points, but this
requires us to justify why this should be the case --- after all, there are plenty of
measures which assign nonzero measure to these points. (This `set of measure zero problem'
is discussed, in the classical context, by Sklar\cite{sklar}.)
We shall see in section \ref{chaos} that modifications to classical dynamics made by quantum
mechanics mean that we should be quite cautious about the validity of classical dynamical
arguments applied to ergodic systems, but there is a more fundamental reason why the set
of measure zero problem may be alleviated by quantum mechanics --- namely, the sets of
measure zero do not appear to have any quantum analogue. In the Wigner function formalism,
\emph{every} state occupies a phase-space region of non-zero Liouville measure;
individual points on the phase-space do not correspond to any physical states.
Hence any dynamical result
applying to all regions of nonzero measure would apply to all states.
Of course, as mentioned earlier the Wigner function is not the only way to do
quantum mechanics on phase space, and there are phase-space representations in which
states are contained in regions of zero measure.\footnote{Glauber's\cite{glauber}``P-representation'',
for instance, assigns a delta function to coherent states; I am grateful to an anonymous referee for
this reference.}
However, even in these cases only a
special subclass of states have zero measure; the generic state has nonzero measure.
Hence given any zero-measure state we can always find one arbitrarily close (as measured
by the Hilbert-space norm) of non-zero measure, and since quantum dynamics are linear a
sufficiently close quantum state will approximate the dynamics of the state in question
to any desired degree of accuracy.
Another way of seeing why the measure-zero problem disappears in quantum
mechanics is as follows: the problem assumes that arbitrarily small
regions of phase space can legitimately be discussed, whereas all of the
phase-space representations of quantum states only describe phase-space
locality when we avoid probing them on scales smaller than $\hbar^n$.
This is not a limitation of these quantum descriptions: it is not that
we are giving up on describing smaller phase-space volumes, but rather
that we are recognizing that quantum mechanics --- and thus the actual
world --- has no concept of phase-space localization on such small scales.
Thus the approximate isomorphism between classical mechanics proper and
the classical domains of quantum mechanics will not preserve the concept
of a measure-zero subset --- so, by the fourth criterion of section
\ref{wigner}, we should reject the measure-zero problem as a relevant
foundational problem for statistical mechanics as it applies to the
actual
world.
\section{QUANTUM DYNAMICS OF CLASSICALLY CHAO\-TIC SYSTEMS}\label{chaos}
\subsection{Quantum chaos}\label{chaosa}
In this section, it will be argued that `classical' statistical-mechanical
systems have their dynamical behavior profoundly modified by
quantum theory. This will form the basis of my `constructive' argument about the relevance
of quantum effects to classical statistical mechanics, as well as providing the promised (in section
\ref{failure} illustration of QISP and the arguments in favour of it.
At first sight it may seem trivially wrong that classical and quantum systems will have such profoundly
different dynamics. After all, we \emph{know} that
classical mechanics is an effective limiting case of quantum mechanics, and
furthermore can directly derive correspondence results: for instance, the Wigner function
obeys dynamical laws equal to the classical dynamics together with some apparently
negligible higher terms.\cite{peres}.
The reason why quantum effects are after all important is linked to the nature
of the dynamics of (classical) statistical-mechanical systems. To have any chance
of showing the dynamical mixing properties which such systems need to approach
equilibrium, their dynamics need to be highly chaotic: that is, trajectories
initially close to one another must rapidly diverge, so that an initially localized
probability distribution becomes spread across the whole phase space. In fact,
one can characterize a chaotic system by the exponentially rapid divergence of
its trajectories.
However, the concept of trajectory is foreign to quantum mechanics, as is the
idea of arbitrarily close initial states ending up widely separated: after all,
the Schr\"{o}dinger equation is unitary. To see the implications for quantum states
we need to consider the evolution of the whole classical probability distribution,
and compare that to the evolution of the Wigner function or another phase-space
representation of the quantum state.
The evolution of the classical distribution can be understood from two conflicting
considerations: the trajectories are diverging exponentially so that initially
close points are increasingly found very far away from one another in the phase
space; but the overall volume of the region where the distribution is non-zero
must be conserved, by Liouville's theorem.\footnote{Of course, this was understood
by Gibbs\cite{gibbs} more than a century ago, in the context of statistical
mechanics; see Sklar\cite{sklar} for a historical discussion.} This means that the
distribution becomes highly distorted, extending long thin filaments which grow
and develop fine structure at exponential rates. There is no dynamically preferred
length scale in classical mechanics, hence nothing to stop this fine structure
developing at arbitrarily small scales.
Quantum mechanics, however --- and hence its classical limit --- does have a preferred
length scale. As Peres says:
\begin{quote}
Any compact domain, obeying the Liouville equation of motion, is continuously
distorted and tends to project increasingly long and thin filaments. As time passes,
new, finer filaments emerge, whose volume is less than $\hbar^N$. The quantum density
\denop\ (or the Wigner function \ldots) cannot reproduce these minute details and smoothes
them away. We therefore expect the quantum dynamical evolution to be qualitatively
\emph{milder} than the classical one. (\cite[]{peres}, p. 303)
\end{quote}
How can we reconcile this with the observed accuracy of classical mechanics? The
details are both complicated and somewhat controversial, but appear to be as
follows (I follow Berry and co-workers\cite{berry78,berry81} and Zurek and
Paz\cite{zurek98,zurek94}; however see Casati and Chirikov\cite{casati} for a
criticism of Zurek and Paz's approach).
\begin{enumerate}
\item Suppose we begin with a quantum system in a pure state, and describe it by a
Wigner function on phase space. To begin with (\ie while the classical state has no
fine structure on scales smaller than $\hbar^n$) the Wigner function accurately approximates
the classical evolution.
\item After a time of approximately
\begin{equation}\label{timescale}
\tau_c \ln(I/\hbar),
\end{equation}
where $\tau_c$ is the timescale for exponential growth of classical filaments and
$I$ is the system's action\cite{berry78}, fine structure has developed on scales
too small for the Wigner function to follow; on a similar timescale\cite{zurek98},
the Wigner function will have become highly delocalized in position space in at
least some directions. Note that (due to the logarithm in (\ref{timescale}), which is
in turn due to the exponential growth rate of fine structure) this timescale will not
be comfortably large, even for macroscopic systems; it will instead be on roughly
the same timescale at which the classical state spreads across phase space, and
hence on the timescale of approach to equilibrium. (In fact, Zurek\cite{zurek98}
calculates the timescale for Hyperion, Saturn's classically chaotic moon, to become
delocalized across its entire orbit if allowed to evolve in isolation, and gets a
figure of about twenty years. Zurek's paper, describing the quantum mechanics of
planetary motion(!) is a salutary lesson in how careful we should be before deciding
that quantum effects are negligible.)
\item If the system were genuinely isolated (and given our assumption that even macroscopic
isolated systems have unitary dynamics)
then the system will genuinely cease to behave classically. However, in any
realistic case the presence of macroscopically delocalized states will by now have
led to environment-induced decoherence, effectively collapsing the wave-function
into a basis approximately localized in both position and momentum and changing it
from a pure to a mixed state. The effect of this on the Wigner function is to
prevent narrow filaments\cite{zurek94,zurek98} from getting \emph{too} narrow,
allowing them to continue to track
the classical evolution at the cost of no longer preserving the phase-space volume
of the state. Since phase-space volume is related to entropy (via $S = k \ln $(volume),
this increase in
phase-space volume corresponds to an increase in the fine-grained entropy of the
system.\footnote{It should be acknowledged that we are assuming here that phase-space entropy
is a good measure of the \emph{quantum} entropy $- \tr \denop \ln \denop$ of the mixed state; this seems
plausible, and is motivated by Zurek\cite{zurek98}, but probably
does not hold exactly.}
\item Of course, the fact that outside interference can increase the entropy of
a system is also true in classical statistical mechanics. The difference here is
that it seems reasonable to consider classically chaotic systems in isolation,
whereas if we isolate chaotic quantum systems they develop, in short order, the
pathology of macroscopically delocalized states. (Also phenomena which have a
totally negligible effect on classical dynamics --- the effect of the microwave
background radiation on the orbit of Jupiter, for instance --- lead to significant
levels of quantum decoherence. As was mentioned in section \ref{decoherence}, this is
because classical isolation is about limiting energy transfer, whilst quantum isolation
is about the vastly more difficult task of limiting information transfer.)
\end{enumerate}
This brief sketch cannot do justice to the technical work it attempts to describe,
but the basic consequences can be fairly simply described --- and moreover seem to
a large extent to be interpretation-independent: a chaotic quantum system which is
isolated and evolves unitarily will in fairly short order develop macroscopically
delocalized states and stop obeying the classical equations of motion. To prevent
this pathology we must invoke collapse of the wave-function (using whatever description
of collapse our particular interpretation gives us; in the Everett interpretation
it is environment-induced
decoherence). The collapse continually converts the (initially pure) quantum state to a
mixed state. This mixed state will spread out across the phase space at the same rate as
the classical probability distribution, but will not develop the arbitrarily small-scale
structure of the classical state. Instead, its filaments will have a minimum width,
and as such its effective phase-space volume will increase exponentially ---
which implies that its entropy will increase linearly.
\subsection{Foundational implications of quantum chaos}\label{chaosb}
With an account of quantum chaos available, we are now able to make a number of foundational
points, supporting and developing the arguments of section \ref{equilibrium} (and in particular
\ref{qisp}, in which the QISP was advocated).
\begin{enumerate}
\item The implications of these
results for isolated systems are startling. Imagine taking a classically chaotic system
(even a highly macroscopic one such as the Solar System), preparing it in an initially
well-localized state, isolating it completely (\textit{per impossibile})
from the rest of the Universe, and letting it evolve on timescales which
are of the same order as the system's classical dynamical timescale.
If the wave-function evolves unitarily, then the predictions made by
classical physics for this system will not be correct, or even
approximately correct: they will be totally wrong.
Here we have possibly the most powerful reason for including decoherence
(and thus, interaction with other systems) when considering statistical
systems, as was advocated in section \ref{decoherence}: there is just no such
thing as a totally isolated, approximately classical system. Such
systems are artifacts of classical mechanics proper: they do not exist
in classical-domain quantum mechanics, and hence do not exist in the
actual world.
\item Classically chaotic systems give us an example of the modification of CM
which might be necessary to
preserve its approximate isomorphism with CDQM, as mentioned in
section \ref{intro}. In chaotic systems CDQM is more closely isomorphic
not to CM but to what we could call \emph{irreversible classical mechanics},
or ICM. In ICM, there is negligible energy transfer into and out of the
system, and when coarse-grained on scales large with respect to
$\hbar^n$, the evolution of phase-space distributions is the same in ICM
as in CM --- but the fine-grained structure, produced on exponentially
small scales in CM, is washed out in ICM, and phase-space volume,
conserved in CM, increases exponentially in ICM.
In terms of the criteria laid out at the end of section \ref{wigner} for when classical
results are to be treated as relevant, the situation is as follows: we
are able to construct a Wigner function, and a quantum dynamics, which
satisfies the first three criteria (that is, remains comprised of
localized states, and approximately tracks the classical dynamics), but
in the process, time-reversibility and conservation of Liouville volume are
lost; thus, any classical result which depends upon time-reversibility and volume
conservation fails the
fourth criterion (that the relevant effects are not, in fact, lost due to
the approximations made).
\item Note that quantum mechanics has converted classical unpredictability into quantum
indeterminacy. Classically, in a chaotic system if we specify a state to any
finite accuracy we will sooner or later lose our ability to predict where the
state is. In quantum mechanics we can reliably predict what the future state
will be --- but our prediction is that it will be a mixed state spread across
the whole of the allowable region of phase space,
and that it is in principle impossible (no matter how accurately we knew the
initial state) to predict the result of a position or momentum measurement.
\item We are now in a position to illustrate point (6) in section \ref{qisp}: viz, that it may
be dynamically impossible to avoid imposing QISP, \ie replacing probability density operators
with entanglement density operators. We shall show that QISP must hold
for chaotic systems at equilibrium, as follows. Suppose we begin with a density operator
describing a uniform distribution across phase space. If we interpret this operator
as a \emph{probability} density operator over rather well-localized pure states, then the actual state of the system
will be relatively localized in phase space. Our ignorance about the results of position
or momentum measurements will be largely epistemic: we don't know the state, but if we did
then we could predict reasonably well the results of measurements.
However, this pure state will not stay pure for long. The Wigner function of the state
will begin to develop fine structure, which in conjunction with environmental entanglement
will force a transition from a pure to a mixed state. On a timescale of order a few
multiples of the timescale $\tau_c$, our pure state will have been converted to a mixed
state entirely delocalized in (the accessible region of) phase space: in other words, to
a mixed state formally identical to the original probability density operator.
As a consequence of this, \emph{all} of the original unknown states will evolve into the
\emph{same} entanglement density operator. There is no longer any ignorance-type probability,
so we can discard the probability distribution and just work with the entanglement density
operator. However, since this is mathematically identical to the original probability
density operator there is no practical change. The only change is in the nature of the
probabilities, which now must be interpreted in the QISP sense.
\item We are also now able to fulfil the promissory note of (5) in
section \ref{qisp}, and to rehabilitate the importance of the ergodicity concept. For
the quantum-mechanical results of this section rely upon the fact that
the classical evolution explores all of (a given energy surface in)
phase space --- in other words, upon classical ergodicity. If
significant regions of phase space remained unexplored by the classical
dynamics --- as is implied, for instance, by the KAM theorem (of Kolmogorov,
Arnold and Moser\cite{arnold}) --- then it seems less clear that
the Wigner function would spread across all of phase space even given
decoherence.
\end{enumerate}
\subsection{The arrow of time}\label{s7b}
The asymmetry between past and future remains one of the great puzzles of physics,
and is not confined to statistical mechanics: in addition to the increase of entropy
from past to future, `arrows of time' are also defined by phenomena as disparate as
the expansion of the universe and our own perception of temporal flow. It remains
an open question as to what extent these different temporal asymmetries can be related,
and (if they can be related) which should be viewed as fundamental.
Quantum mechanics seems to add another `arrow': the collapse of the wave-function
appears to be an explicitly time-asymmetric
process. Furthermore, it has long been recognized (since Von
Neumann\cite{vn}; see also \cite{zeh}) that the
pure-to-mixed-state transition associated with wave-function collapse is a process
which increases entropy; this hints at some sort of connection between the entropic
and quantum-mechanical arrows of time.
If the account of quantum chaos presented in section \ref{chaosa} is correct, the two are
in fact linked in a fairly straightforward manner: chaotic systems
will become macroscopically delocalized unless their wavefunction is collapsed,
but adding this collapse into such systems causes an increase in entropy ---
hence the arrow of time defined by the collapse process gives the direction of
entropic increase. (See \cite{zurek98} for further, and more
technical, discussion.)
Of course, different interpretations of quantum mechanics give different
explanations for the time asymmetry of wavefunction collapse. Theories of
dynamical collapse (of which the most well-developed are due to
Ghirardi, Rimini and Weber\cite{grw} and Pearle\cite{pearle}) are time-asymmetric
by \textit{fiat},
of course, and presumably could cause entropic increase in even an isolated
system (Albert\cite{albert} has developed ideas along these lines).
The Everett interpretation, on the other hand, incorporates no explicit
time-asymmetry,\footnote{Neither does the de~Broglie Bohm interpretation,
of course.} and so we have a problem analogous to that of statistical mechanics:
given time-symmetric dynamics how do we explain observed asymmetries? Specifically,
we need to explain why the branching structure of histories is indeed branching.
When we phrase this question in terms of subsystems it becomes the question of why
subsystems of the universe are less entangled in the past than in the future.
Explanations of this in terms of a very special initial state of the universe begin
to look indistinguishable from their counterparts in the foundations of statistical
mechanics.
The moral seems to be that the quantum-mechanical and entropic arrows of time are
closely linked. Depending on our interpretation we may solve one by \textit{fiat}
and thus solve the other; or we may recognize both as aspects of the same problem.
But it seems unwise to attempt to solve them separately.
\section{CONCLUSIONS}\label{conclusion}
If the arguments of this paper are valid, then some rather basic
concepts in classical statistical mechanics are significantly altered
once quantum effects are taken into account. In terms of the conceptual
structure of the theory, we have found that the existence of superposition and
entanglement
and the dual interpretations of the density operator
cast serious doubt upon the validity of some classical derivations of
equilibrium, and call into question the standard view of statistical
mechanics' phase-space distributions as probability distributions over
unknown but determinate microstates. On the dynamical side, the nature
of quantum chaos requires us to replace CM with an irreversible cousin,
drastically undermines the validity of treating systems as totally
isolated, and implies a close link between the statistical-mechanical and
quantum (`wave-packet-collapse') arrows of time.
These need not be seen as \emph{negative} consequences, of course. Quantum
considerations indicate ways to make progress on issues such as:
\begin{itemize}
\item the nature of ensembles;
\item the reduction of thermodynamics to statistical mechanics;
\item the concept of probability in
statistical mechanics;
\item the problem of measure zero;
\item the connection between ergodicity and equilibrium.
\end{itemize}
Furthermore, statistical mechanics may
offer a potential testing ground for rival interpretations of quantum mechanics.
If the way in which we formulate statistical mechanics is dependent on our interpretation
of quantum theory then it is \textit{a priori} possible that two interpretations, identical in their
predictions for any given experiment, will produce different statistical
mechanics. If so we have a potential test between them: failure to reproduce
the canonical ensemble is just as fatal for an interpretation as failure to
reproduce the two-slit experiment.
But in any case, whether or not moving to quantum mechanics helps
solve foundational problems in statistical physics, it is not an
optional move. It is certainly no goal of this paper to undermine the
superb and sustained work that has been done on understanding the
foundations of classical statistical mechanics, nor to claim that this
work is rendered useless by quantum considerations. But to see how it
applies to statistical-mechanical systems in the actual world, it is
necessary to see how --- and to what extent --- this classical
work can be transferred across to quantum systems. If the approximate isomorphism
between classical mechanics
and classical-domain quantum mechanics was such as to transfer all our
understanding of classical statistical mechanics unproblematically
across to at least certain domains of quantum theory, there would
have been no need to worry about quantum effects. As the isomorphism
is not in fact like this, we cannot justify the avoidance of quantum considerations.
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\end{document}