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\begin{document}
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\title{Remarks on the Direction of Time in Quantum
Mechanics\thanks{Received...; Revised...}}
\author{Meir Hemmo\thanks{Send reprint requests to: Meir Hemmo,
Department of Philosophy, University of Haifa, Haifa 31905,
Israel; email: meir@research.haifa.ac.il.}\thanks{I thank
David Albert, Frank Arntzenius, Guido Bacciagaluppi, Jeremy Butterfield,
Itamar Pitowsky, Orly Shenker and Professor Dieter Zeh for helpful
comments on issues related to this paper.}
\\Department of Philosophy\\University of Haifa}
\date{}
\maketitle
\newpage
\begin{center}
Abstract
\end{center}
\noindent I argue that in the many worlds interpretation
of quantum mechanics time has no fundamental direction.
I further discuss a way to recover thermodynamics in this
interpretation using decoherence theory
(Zurek and Paz 1994). Albert's proposal to recover
thermodynamics from the collapse theory of Ghirardi, Rimini and
Weber (1986) is also considered.
\newpage
{\bf 1. Introduction.}
At face value the statistical frequencies obtained in quantum
mechanical measurements exhibit time asymmetry in the sense that they
invariably seem to depend on initial and not final states. As is well
known this cannot be taken to imply that time has an objective
direction (\ie that the spacetime structure in the future direction of
time is different from the past direction) for the simple reason
(among many others) that the asymmetric frequencies are accurately
produced also by theories with perfectly time symmetric dynamical laws
(\eg Aharonov and Vaidman's (1991) two-time theory, and Bohm's pilot
wave theory).
However, in standard quantum mechanics (\eg von Neumann's
(1932) formulation) the time asymmetry of the
frequencies is explained by appealing to the explicit time
asymmetry of the collapse of the wave function.
This can be seen in a measurement interaction as follows.
In the course of time the interaction has the form
\begin{equation}
\big(\sum_i\lambda_i\ket{\varphi_i}\big)\ten\ket{\psi_0}\rightarrow
\sum_i\lambda_i\ket{\varphi_i}\ten\ket{\psi_i},
\label{eq:meas}
\end{equation}
where the $\ket{\varphi_i}$ are the states of the system and the
$\ket{\psi_i}$ are the pointer states of the measuring apparatus.
The system starts out at the initial time in a pure state and
ends up at the final time in the mixed state:
\begin{equation}
\rho_s=\sum_i\abs{\lambda_i}^2\ket{\varphi_i}\bra{\varphi_i},
\label{eq:mix}
\end{equation}
where the Born rule probabilities $\abs{\lambda_i}^2$
are fixed by the initial and not final state.
If there is a real collapse, then the time evolution is truly stochastic,
and the collapse yields a time directed (pure-to-mixed) transition,
where at the final time the system
is actually in one of the pure states $\ket{\varphi_i}$, and the
$\abs{\lambda_i}^2$ correspond to the
genuine probabilities of each of the possible
trajectories of the system.
In a dynamical theory of the collapse, \eg the GRW theory (Ghirardi,
Rimini and Weber 1986), the above time asymmetry is built into the
equations of motion. The dynamics then become explicitly non-invariant
under time reversal. Moreover, the GRW transition probabilities
are non invariant under various manipulations in the forward but
presumably not in the backward direction of time. This has prompted
many authors (\eg Arntzenius 1997) to say that time may have an
objective direction according to the GRW theory in so far
as the dynamics is concerned. von Neumann (1932, Ch. 5)
argued that the quantum collapse brings about also an increase in the
quantum analogue of entropy $-k{\rm Tr}\rho {\rm ln}\rho$, and that this
should imply also the usual time asymmetry in thermodynamic evolutions,
\ie why thermodynamic systems are invariably observed to evolve
from low to high entropy states.
In the {\em Many Worlds} Interpretation (MWI) (following
Everett (1957), DeWitt and Graham (1973) and Deutsch (1985))
the question of the direction of time involves some conflicting
intuitions. On the one hand the dynamical equation of motion (\eg the
Schr\"odinger equation in the non relativistic case) of the universal
wave function, which we take to be lawlike in the MWI, is completely
deterministic and time reversal invariant, where by {\em time reversal
invariance} it is meant that the dynamics is invariant under complex
conjugation and temporal reflection. This means, taking for example
the Schr\"odinger equation, that $\psi(x,t)$ is a
solution if and only if $\psi^*(x,-t)$ is, for all
$\psi$. As a matter of principle this definition exhausts
the full empirical content of time symmetry in quantum mechanics, since
the Born rule predictions invariably assign the same probabilities to
$\psi$ and $\psi^*$, and so we shall work with it. Anyway,
this is the usual definition of time symmetry
in quantum mechanics.$^1$ And so in the MWI the overall dynamics of
the universal wave function
does not pick out a genuine direction of time. And this means that
the time asymmetric frequencies in the case of both quantum
and thermodynamic measurements may be explained only by
introducing some probabilistic assumptions about
the distribution of initial conditions, as in classical
statistical mechanics.
On the other hand, we will see that the pure-to-mixed
transitions associated with the quantum collapse are explained
in the MWI in terms of what seem to be time directed transitions
along the histories of the worlds. In particular, in the MWI
the time directed collapse of the wave
function in measurement-like interactions takes place as a
matter of fact at each world; \ie in some sense a single
collapse is replaced with many collapses. And so this presumably
may be taken to support the view that a genuine direction of
time is picked out by some form of a world-dependent dynamics.
In this paper I want to consider in more detail these two
intuitions in the MWI. To set the stage I
sketch in Section 2 one version of the MWI based on
decoherence theory. The direction of time in the context of pure
quantum mechanics is taken up in Section 3. Then, in
Section 4, I consider very roughly one way in which the
thermodynamic time asymmetry can also be recovered.
This is based on models developed by Zurek and Paz (1994)
in quantum decoherence theory. Finally (section 4),
I compare these results with an alternative proposal by Albert
(2000, Ch. 7) in the context of the GRW collapse theory.
{\bf 2. Worlds and Decoherence.}
In many versions of the MWI (\eg Saunders 1995;
Vaidman 2002) the splitting of the worlds is defined by the
process of {\em decoherence} of the wave function.
In standard models of decoherence theory (\eg Caldeira and Leggett
1983; Joos and Zeh 1985; Zurek, Habib and Paz 1993; Zeh 1989)
a macroscopic system interacts
with an environment that has many degrees of freedom. It is assumed that the
interaction depends on some preferred observable
$\Pi$ of the system (the eigenstates of which are called the {\em
decohering variables}), so that the interaction Hamiltonian
$H_{int}$ commutes (approximately) with $\Pi$ satisfying
\begin{equation}
[H_{int}, \Pi]\approx 0.
\label{eq:int}
\end{equation}
The initial state of system and environment is assumed to be
approximately a product state
\begin{equation}
\ket{\psi(x_1...x_N,t)}\ten\ket{E},
\label{eq:prod}
\end{equation}
where $\ket{\psi(x_1...x_N,t)}$ is some quantum state of
the system and $\ket{E}$ is some initial state of the environment.
This means, in particular, that the states of the system and of the
environment are separable (\ie not quantum mechanically
entangled). The Schr\"odinger equation yields for this
interaction the state:
\begin{equation}
\ket{\Psi(t)}=\sum_i\mu_i(t)\ket{\psi_i}\ten\ket{E_i(t)},
\label{eq:dec}
\end{equation}
where the kets $\ket{\psi_i}$ are assumed to be the eigenstates
of $\Pi$, and the $\ket{E_i(t)}$ are the {\em relative} states of the
environment. In the standard models it is shown that the scalar
products between different $\ket{E_i(t)}$ in (\ref{eq:dec})
decay exponentially satisfying
\begin{equation}
\braket{E_i (t+\Delta t)}{E_j (t+\Delta t)}\approx\delta_{ij}
\label{eq:Eij}
\end{equation}
after extremely short times $\Delta t$ (called {\em decoherence
times}). These times are typically short around
$10^{-23}{\rm sec}$.$^2$ It is further shown that
(\ref{eq:dec}) and (\ref{eq:Eij}) together imply that the reduced
state of the system approaches the diagonal form:
\begin{equation}
\rho_s(t)\approx\sum_i
\ket{\psi_i}\abs{\mu_i(t)}^2\bra{\psi_i},
\label{eq:diag}
\end{equation}
within times comparable to $\Delta t$ (so that formally the
amplitude squared measure behaves like classical probability;
see below). The approach of $\rho_s(t)$ to the diagonal form in
(\ref{eq:diag})) is highly invariant under changes of the initial
state of the system and of the environment.
The standard models usually assume that (approximate) position
picks out sets of preferred states in the Hilbert space of the
system, or more generally that preferred sets of states are fixed
by the dynamically conserved quantities (usually represented by
{\em coherent states}, \ie narrowly peaked Gaussians in
both position and momentum) (Zurek 1993). Zurek, Habib and Paz
(1993) have shown in some models that the $\ket{\psi_i}$ in
(\ref{eq:dec}) correspond to
the states that are maximally stable (and invariant)
under decoherence in the sense that all other states of the system
approach the diagonal form (\ref{eq:diag}).
Moreover, the coherent states minimize the production
of (von Neumann and linear) entropy (so that $\rho_s(t)$ becomes
maximally mixed when diagonalized by coherent states).
In this sense decohering systems are said to follow
quasi-classical trajectories and exhibit quasi classical
behavior.
As we sketched above the states $\ket{E_i(t)}$ of the
environment {\em relative} to the $\ket{\psi_i}$ in (\ref{eq:dec}) {\em
separate and don't reinterfere} again. This is because the $\ket{E_i(t)}$
remain over extremely long times (approximately) orthogonal during the
evolution of the quantum state. This means that the correlations between
the $\ket{\psi_i}$ and the $\ket{E_i(t)}$ are in fact stable over time
under the Schr\"odinger evolution. In this sense
one can say that the $\ket{E_i(t)}$) at later times are {\em records}
of the $\ket{\psi_i}$ that occurred at earlier times.
Here is one way to read the MWI.$^3$
Branches of the universal state $\ket{\Psi(t)}$ of the form
(\ref{eq:dec}) exist essentially at all times comparable with the
typical decoherence times. These branches can be
characterized as follows. (A) They have a product form
$\ket{\psi_i}\ten\ket{E_i(t)}$
which is essentially invariant under the time evolution of
$\ket{\Psi}$, and in particular, the branches don't reinterfere
over sufficiently long times.
(B) The $\ket{\psi_i}$ correspond to approximate
eigenstates of the classically conserved quantities
and typically they follow quasi classical trajectories.
(C) The quantum mechanical measure (given by the amplitude squared
$\abs{\mu_i(t)}^2$) defined over these branches exhibits formal
features of probability. This last point is crucial since
in the MWI the measure is supposed
to induce in some sense the {\em frequencies} along the
branches. Suppose now, following
Everett (1957), that all {\em branches} of the universal
$\ket{\Psi(t)}$ (relative to any choice of basis)
are equally real, and let us associate any such branch
structure with a set of {\em worlds}.$^4$ It follows that there are
sets of worlds associated with branches defined by decoherence, and so
these worlds may be taken to correspond to our experience.
There is a direct correspondence between branches
in the MWI and sets of decoherent
histories in the histories approach to quantum mechanics
(Griffiths (1984), Gell-Mann and Hartle (1993)). In this approach
the probabilities for individual histories are given by
\begin{equation}
\abs{C_{\alpha}\ket{\Psi}}^2,
\label{eq:gmh}
\end{equation}
where $C_{\alpha}$ is a quantum history$^5$, \ie a string of
projections at a sequence of times $t_1t_n$ after the last projection in the set there are sequences of
exhaustive and alternative projections that are in perfect correlations
with the histories in the set. In the context of the MWI
it is important to note
that this theorem doesn't uniquely single out branches that match
our experience: there are branchings associated with decoherent sets
of histories in which the projections correspond to superpositions of
quasi classical trajectories. But the converse is invariably true:
pointer basis histories which are recorded in the
relative states of the environment satisfy the decoherence condition
(\ref{eq:wd}).$^7$
As is well known the above sketch of the MWI is
not complete, because of the so-called probability and
preferred basis problems (see Vaidman 2002). But I shall
assume now that it is completeable, and focus on the question
of the direction of time.
{\bf 3. The Direction of Time.}
The branches associated with quasi classical behavior in the
sense described above don't reinterfere, because the $\ket{E_i(t)}$
don't due to decoherence.
In the version of the MWI sketched above this condition is taken
as a characteristic feature of the branches that are associated
with our worlds.
This has the consequence of an effective collapse along the quasi
classical branches. That is, the branches after a split evolve
independently of each other (as it were, in `parallel'). As
a result, a {\em time directed} collapse seems to have occurred
at each split (from the point of view of a branch).
This means that the branching associated with our worlds has {\em
effectively} a tree-like form in which on the backward (past)
direction of time a branch has a unique continuation, whereas on the
forward (future) direction it splits. In other words, the histories
corresponding to our worlds seem to follow a tree-like divergence
pattern. Note that the tree picture is a bit misleading: on the one hand
in the MWI the evolution of the total wave function is time reversible,
and in this sense we can retrodict a unique past (see below).
On the other given the present data {\em in a world} we cannot retrodict
a {\em unique} past. Hartle (1997) shows that, in general,
retrodictions of the past in the latter sense conditional on present
data are nonunique for almost any decoherent set of histories, even
though they are assigned probability one.
On this picture the process of decoherence itself exhibits time
asymmetry. This is because the interaction with the environment
increases the degree of mixing of the (reduced) state of the system
in the {\em forward} (and not in the backward) direction of time.
Also, the decoherence condition (\ref{eq:wd}) displays an asymmetry
in time. Take the set of histories $C_{\alpha}$ (call it the
{\em forward} set), and consider the {\em backward} set, \ie
the set of histories that unfold, as it were, backwards in time
from $t_n$ to $t_1$. Then for a given $\rho_{in}$ the decoherence
conditions of the forward and the backward sets are in general
{\em not} equal, and in particular the histories in the two sets
will not decohere together (Hartle 1997; Bacciagaluppi 2001).
Thus the branches corresponding to our histories can satisfy either
the forward decoherence condition or the backward one, but not both.
This time asymmetry is reflected in the probabilities for histories
(\ref{eq:gmh}) (and in the frequencies they should match along a
history), which are explicitly time asymmetric, because they invariably
depend on an {\em initial} or past state $\ket{\Psi}$, and not on a final
or future state.
The above time asymmetries may be taken to suggest that in the MWI
the direction of time as fixed by decoherence the is fundamental
(see Zeh 1992, Ch. 4) for an extensive discussion).
This intuition can be justified as follows. Along the branches
associated with our worlds we have an effective collapse onto quasi
classical states. This collapse is manifestly time directed. The tree-like
form of the branches in the case of decoherent sets of histories
distinguishes between the future and the
past directions of time. Furthermore, the time asymmetry of the
decoherence condition itself suggests that the direction of time we
experience (along our histories) is fixed by either the forward
condition of decoherence or the backward one (Bacciagaluppi 2001).
However, in the MWI the dynamical equations of motion of the universal
state are completely time symmetric.
In the MWI the collapse of the state has only an {\em effective}
status, no matter how we further choose to interpret it.$^8$
In fact, the appearance of a collapse (relative to a branch along a
given set of branches) is a straightforward result of the decoherence
condition. But, in general, it is the initial state of the universe
$\rho_{in}$ and the dynamical conditions that completely determine
whether or not a given set of histories is decoherent for any given
sequence of times. In particular, the dynamics and the initial
conditions fix completely whether any future (or past)
extensions of a given set of histories remain decoherent (\ie
whether the histories don't reinterfere in the future or past).
As long as the future extensions of our histories belong to a
decoherent set, the frequencies we shall observe will exhibit the above
time asymmetries. But this, in turn, is fully determined by the
initial conditions and the dynamics. The theory leaves, as it were,
no room for a fundamental direction of time.
In the case of decoherence through interactions with the environment,
the standard models usually make the following assumptions:
(I) The interaction between the system and environment takes the
form of (\ref{eq:int}). (II) The initial total state is factorizable
(\ie a state of {\em minimal} entanglement; see equation (\ref{eq:prod})).
(III) The phases of the environment states $\ket{E_i(t)}$ are uniformly
distributed. It is these three assumptions that imply
that the $\ket{E_i(t)}$ have low probability to reinterfere over
sufficiently long times (so that $\rho_s(t)$ keeps over time its
diagonal form (\ref{eq:diag})). That is, in the MWI
the direction of the evolution we perceive along our histories,
namely that the evolution is from pure to mixed states and not the
other way around, is not determined by the form of the dynamics (I)
alone, but also by the initial conditions (II) and (III).
Therefore, the direction of time fixed by decoherence through
interactions with the environment is not purely
dynamical, and in this sense it is not fundamental in the MWI.
Moreover, (II) is a highly implausible assumption, if there are enough
particles in the environment: that is, the set of factorizable states
in this case has measure zero in the total Hilbert space of system and
environment (but perhaps it is not necessary for decoherence;
see Arntzenius 1998).$^9$ Also, the statistical
distribution in (III) (although natural) is not unique, and in any
case it is not implied by the quantum dynamics, nor by the quantum
probabilities.
In the histories approach the time {\em symmetry} of the
dynamics is often stated by adding a final condition $\rho_{fin}$
to the probabilities (\ref{eq:gmh}) (Aharonov, Bergmann and Lebowitz
1964), and to the decoherence functional (\ref{eq:wd})
(Gell-Mann and Hartle 1993). And the time asymmetry
of the transition probabilities (and the observed frequencies along
our histories) is, again, traced back to some special
initial condition $\rho_{in}=\ket{\Psi}\bra{\Psi}$ that is consistent
with present data. From this perspective the above time asymmetries
seem to be a consequence of an asymmetry between initial and final
conditions. But, again, in the MWI this does not imply that the
direction of time is fundamental.$^{10}$
{\bf 4. The Thermodynamic Arrow.}
A detailed quantum mechanical analysis of the connection between
decoherence theory and thermodynamic behavior of a classical
chaotic system is given by Zurek and Paz (1994) (and also by Zurek, Habib
and Paz (1993)). I don't have space here to display in detail their results.
In their models they consider a chaotic system that is subjected to a
decoherence interaction with its environment, and show the following.
(I) The quesi classical states of the system picked out by
the decoherence interaction (see Section 2) are the most stable states
under which the production of the von Neumann entropy goes down to a
minimum. (II) After extremely short times (comparable to the decoherence
times; but see below) during the decoherence interaction the system
follows quasi-classical trajectories (this should be understood in the
context of a proper interpretation of quantum mechanics only).
(III) The quantum mechanical von Neumann entropy
$-k{\rm Tr}\rho {\rm ln}\rho$ typically increases as a monotonic function
of time, and moreover it is naturally interpreted in terms of increase in
phase space volume.
(IV) The rate of increase of the von Neumann entropy is fixed essentially
by the rate of divergence of the chaotic system (\eg by the Lyupanov
exponents), so that the classical predictions are recovered.
(V) The above results don't hold for closed (isolated)
systems (for more details, see Zurek and Paz 1994 and references
therein). This last point means that classical mechanics would be directly
{\em refuted} if only we could perfectly isolate a chaotic system from its
environment (\eg if the above models are right, the entropy of the
system would be constant throughout it's evolution even if the system
were to start in a non equilibrium state)$^{11}$.
Let us now suppose that these models of decoherence are
correct, and that they can be generalized to all realistic cases of
thermodynamic systems, such as gases spreading out in a container, or
gases embedded in clouds of other gases with much lighter molecules
(where the container and the lighter gases play the role of decohering
environments), etc. We want to consider the implications of these models
in the MWI regarding the question of the direction of time.
In the MWI the very definition of the branches corresponding to our
histories stipulates that they belong to a decoherent set of histories,
due to, \eg interactions of macroscopic systems with the environment.
In Section 2 we saw that in the MWI the dynamics of a macroscopic system
along our histories can be seen as corresponding to a dense sequence of
(effective) collapses separated by decoherence
times. At intermediate times between collapses, the system seems to evolve
(on each branch) from an effectively pure state to a quantum mechanical
mixed state (after a split) approaching a diagonal form as in
(\ref{eq:diag}). In the pointer basis expansion,
this mixture corresponds to outgoing branches on which the system is
described by quasi classical states (\eg coherent states, or more
generally, eigenstates of conserved quantities).$^{12}$
Actually we can say more, given the above results of Zurek and Paz
(1994). During the process of decoherence the (quasi) classical form of
the reduced state of the system $\rho_s(t)$ seems to play the role of a
quantum mechanical equilibrium state in two respects: (i)
all other states of the system evolve
under decoherence towards $\rho_s(t)$;
and (ii) $\rho_s(t)$ represents the most stable probability
distribution over the quantum states of the system.
And so in the MWI $\rho_s(t)$ may be taken to replace the standard
probability measure of classical statistical mechanics.$^{13}$
In this sense the probabilities of classical statistical mechanics
may be entirely reduced to the quantum
probabilities$.^{14}$ In fact, if the above models are correct, it is
plausible that the splitting of the branches reproduces the
standard probability measure of classical statistical mechanics.
Thus we may have a unified dynamical origin of the probabilities in
physics (compare this to Albert's GRW based approach to the
foundations of classical statistical mechanics (2000, Ch. 7); see the
next section).
Assuming that these results in decoherence theory are generic,
one may want to argue that in the MWI the {\em time asymmetry}
of thermodynamic evolutions is built into the dynamical equations of
motion, since the thermodynamic regularities are a consequence of the
dynamical evolution of the wave function in decoherence situations.
However, this is wrong. The argument is akin to the usual Loschmidt
objection from time reversibility (see also the end of Section 3).
Recall that the Schr\"odinger equation (and its
relativistic analogs) are time reversal invariant. This implies that
the evolution of the (total) quantum state of system and environment
in decoherence situations is in principle time reversible. Moreover,
the process of decoherence itself is a result of the time symmetric
dynamics only on the statistical hypothesis of a uniform probability
distribution over the phases of the environment states.
And so, as a matter of principle, there must be entropy
{\em decreasing} trajectories along which the quantum state
may evolve in the future direction of time.
This means that in the MWI whether or not entropy will actually
decrease in the future of our branches depends on initial
conditions. Entropy will invariably increase along branches
corresponding to the classical variables provided we assume that these
branches will not reinterfere in the future. If reinterference will
occur, the evolution will result (with certainty) for some initial
states in entropy decrease. In the MWI whether or not this will occur
depends enirely on initial conditions. In fact, there is no sense which
doesn't bear on statistical assumptions about the distribution
of initial states, in which reinterference of our branches is
unlikely at any finite time. No physical law in the MWI rules out the
possibility of a complex demon making an interference experiment
with our branches ten minutes from now. Such experiments may well
violate the second law of thermodynamics. In this sense it is not
fundamental, nor does it entail in the MWI that time has a direction.
{\bf 5. Conclusion.}
An alternative approach in which a fundamental direction of time,
in particular the thermodynamic arrow, is fixed by the dynamical
equations of motion in quantum mechanics has been proposed by
Albert (2000, Chapter 7). If the underlying quantum
mechanical theory is the collapse theory by GRW (see GRW 1986; Bell
1987), then the dynamical equations of motion of isolated systems are
time asymmetric (since the GRW equations of motion are non invariant
under time reversal).
A history, say of a thermodynamic system (closed or open) unfolds,
according to the GRW theory, in a time directed fashion fixed by
the direction of the dynamics. In addition, the dynamics
results in a random walk on a set of alternatives
with the probabilities given by the quantum mechanical
algorithm. For a given quantum state before a collapse,
the GRW dynamics gives a set of transition
probabilities over the possible states of the system immediately
after the collapse. Albert (2000, 155-156) argues that
given the GRW parameters for a collapse, the transition probabilities
in this theory plausibly entail the standard probability measure of
classical statistical mechanics.$^{15}$ If true, this means that
thermodynamic evolutions are with high probability time {\em
irreversible}, and that entropy decreasing trajectories are both
highly improbable and unstable. In this sense, Albert's approach may
underwrite by pure dynamical laws a fundamental direction for time,
the thermodynamic direction included (see also Arntzenius 1997;
Callender 1997). Moreover, it would also entail that the classical
ignorance-type probabilities are reduced to the quantum probabilities
(see below).
Since the GRW dynamics is time asymmetric,
Loschmidt-like objections (relying on time reversibility)
are not applicable to it.$^{16}$
Likewise in the GRW theory quantum mechanical reinterference
experiments of the kind considered above are not applicable,
because all but one of the branches of the quantum state really
vanish due to the GRW collapses. However, because of the time asymmetry
of the dynamics, the GRW theory can produce no {\em retrodictions} at
all. The retrodiction that entropy
decreases in the future-to-past direction of time can be
produced only by introducing a {\em past hypothesis} according to
which the {\em initial} macrostate of the universe was a state of
low entropy (see Albert 2000, Ch. 7). But due to the
{\em stochastic} nature of the GRW collapses, the past hypothesis
need only apply to the initial {\em macro} and not micro state of the
universe. In fact, if the GRW theory is true of our world, it is highly
plausible that no statistical assumptions at all are required about the
distribution of {\em micro} states in order to derive the thermodynamic
regularities.
In the MWI a past hypothesis of low entropy initial states is also
required, though here it is to block the usual Loschmidt objection from
time reversibility. However, in this context decoherence seems to yield
an interesting result. The initial {\em micro} conditions (see
(II) in Section 3) required in the decoherence models imply that the
initial total quantum state is of low von Neumann entropy. In
the model of decoherence described in Section 2 the condition of low
entanglement of the initial (product) state of system and environment
means that the initial state of the system is already a state of
minimal von Neumann entropy (which is equal to zero, if the initial
state of the universe is pure).$^{17}$ And so no past
hypothesis about the initial macro state of the universe, and in
particular no past hypothesis about the initial micro state of the
thermodynamic system, is required over and above the initial conditions
required for decoherence. In sum: no specific reference to any
{\em thermodynamic} feature of the initial micro (or macro) state of the
universe need be made, although assumptions about micro initial
conditions are required.
My conclusion about the connection between the quantum
probabilities and the direction of time is this.
In Albert's GRW-based approach there is a time directed collapse
which is built into the dynamics of the quantum wave function. This
dynamics, arguably, yields the thermodynamic regularities. And so,
in this theory the observed direction of thermodynamic processes
is fixed by the direction of time as defined by the dynamics. By the
laws of motion of the GRW theory, therefore, thermodynamic processes are
fundamentally irreversible. And since the GRW collapse is {\em stochastic},
the thermodynamic arrow is independent of initial
{\em micro} conditions.$^{18}$
In the MWI, by contrast, there is only an apparent connection between
the quantum probabilities and time direction. The thermodynamic
arrow may be recovered (effectively) along quasi classical histories
on the basis of the diffusion of correlations into the environment.
But as a matter of principle the process of decoherence is perfectly
time reversible. In particular, as we saw above, the recovery of
thermodynamic regularities in the MWI doesn't depend in any way on
whether or not the transitions along a history (when the wave
function splits) are stochastic. In fact, even if they
were truly stochastic, say denoting a chance
process$^{19}$, they will still have no impact on the question of
time reversibility. A similar analysis applies to Bohm's
theory and to modal theories (see Dieks and Vermaas 1998). It turns out
that in all those theories the extra dynamics of the hidden
variables are time reversal invariant. I conjecture that it {\em must}
be so.$^{20}$
If this is true, it means that in no collapse quantum mechanical
theories any stochastic dynamics over and above
the quantum wave function cannot determine a fundamental
direction of time. In quantum mechanics without collapse
all mechanical and thermodynamical processes are time reversible, just
because the the dynamical evolution of the wave function is.
The quantum probabilities may have an effect in determining
a direction of time only in so far as they are reflected in the
dynamics of the {\em wave function} (as in the GRW theory; see Albert
2000, Ch. 7). But, in the MWI this is not the case: the
transition probabilities along our histories have no impact on
the dynamics of the (total) wave function (no matter how they
may be interpreted). Hence, there can be no fundamental direction
of time in the MWI.
\newpage
\begin{center}
REFERENCES
\end{center}
\begin{flushleft}
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\end{flushleft}
\newpage
\begin{center}
FOOTNOTES
\end{center}
\begin{flushleft}
1. See (Albert, 2000, Ch. 1) for time symmetry in classical mechanics
and (Callender 2002) in quantum mechanics.
2. Also, in the standard models the decoherence times of the system are
much shorter than the dynamical times even for very weakly dissipative
systems. The relaxation times are typically extremely long,
in some models of the order $10^{40}\;{\rm sec}$.
3. This reading is based on (Bacciagaluppi and Hemmo 1995).
It is close to the versions by \eg Zeh (1973); Saunders (1995);
Vaidman (2002).
4. Perhaps a set of worlds should be associated not with an exact
branch structure, but rather with a {\em bundle} of branches all of
which elements are pairwise close to each other in some measure
theoretic sense (see Bacciagaluppi and Hemmo 1995).
5. We assume, for simplicity, throughout that the initial state of the
universe $\rho_{in}$ is pure $\rho_{in}=\ket{\Psi}\bra{\Psi}$.
6. See (Dowker and Kent 1996) for a critical evaluation of the
histories approach.
7. A direct proof is given by Zurek (1993); see also
(Hemmo 1996, Ch. 5).
8. In a relativistic setting, for example, the
collapse can be hyperplane-dependent, or it can be stipulated to
occur along future light cones (see Bacciagaluppi 2001).
9. This is the Haar measure defined on finite-dimensional Hilbert
spaces, which is invariant under the unitary transformations (see
Arntzenius 1998).
10. A more fundamental analysis of time asymmetry requires a background
(quantum) theory of space and time; see (Halliwell, Perez-Mercader and
Zurek 1992) for main stream papers on this topic.
11. I assume here that the behavior of the von Nuemann entropy
correpsonds to the behavior of thermodynamic entropy; this is
under dispute.
12. This should not hold in models of decoherence with {\em no}
fixed pointer basis, \eg extremely light gases, but also in this
case decoherence presumably produces the classical correlation
functions. I thank professor Dieter Zeh for pointing this out to me.
13. I Assume here that in the MWI $\rho_s(t)$ might be taken to
represent
a genuine probability distribution over the branches; this is under
dispute.
14. Compare a similar argument by Wallace (2001) and Hemmo and Shenker
(2001, 2003a). Classical approaches to equilibrium states
don't seem to be applicable in quantum mechanics (see Wallace 2001).
15. This is plausible, but not yet proved in the GRW theory.
16. For how Loschmidt-like reversals (\eg the {\em spin echo}
experiments) are explained in this theory, see (Albert 2000, Ch. 7;
Hemmo and Shenker 2003b).
17. I thank Guido Bacciagaluppi for raising this point, and Itamar
Pitowsky for discussions. See (Hemmo and Shenker 2003b).
18. Note that stochastic dynamics is compatible with time reversal
invariance, if, for example, the forward and the backward transition
probabilities turn out to be equal (see Callender 2002). But
this is not the case in the GRW theory.
19. The question is open; see fn. 12.
20. Assuming that hidden variables theories
are empirically equivalent to quantum mechanics, the flow of the
probabilities in any such theory must be controlled by the time
reversal invariant dynamics of the wave function; and nothing else.
This should hold also in the MWI and its many minds variants
\end{flushleft}
\end{document}