---
abstract: 'In this universe, governed fundamentally by quantum mechanical laws, characterized by indeterminism and distributed probabilities, classical deterministic laws are applicable over a wide range of time, place, and scale. We review the origin of these deterministic laws in the context of the quantum mechanics of closed systems, most generally, the universe as a whole. In this formulation of quantum mechanics, probabilities are predicted for the individual members of sets of alternative histories of the universe that decohere, for which there is negligible interference between pairs of histories in the set as measured by a decoherence functional. An expansion of the decoherence functional in the separation between histories allows the form of the phenomenological, deterministic equations of motion to be derived for suitable coarse grainings of a class of non-relativistic systems, including ones with general non-linear interactions. More coarse graining is needed to achieve classical predictability than naive arguments based on the uncertainty principle would suggest. Coarse graining is needed to effect decoherence, and coarse graining beyond that to achieve the inertia necessary to resist the noise that mechanisms of decoherence produce. Sets of histories governed largely by deterministic laws constitute the quasiclassical realm of everyday experience which is an emergent feature of the closed system’s initial condition and Hamiltonian. We analyse the question of the sensitivity of the existence of a quasiclassical realm to the particular form of the initial condition. We find that almost any initial condition will exhibit a quasiclassical realm of some sort, but only a small fraction of the total number of possible initial states could reproduce the everyday quasiclassical realm of our universe.'
author:
- 'James B. Hartle'
title: 'Quasiclassical Realms In A Quantum Universe[^1]'
---

\#1\#2[[\#1 \#2]{}]{}

Introduction {#sec:I}
============

In cosmology we confront a problem which is fundamentally different from that encountered elsewhere in physics. This is the problem of providing a theory of the initial condition of the universe. The familiar laws of physics describe evolution in time. The evolution of a plasma is described by the classical laws of electrodynamics and mechanics and the evolution of an atomic state by Schrödinger’s equation. These dynamical laws require boundary conditions and the laws which govern the evolution of the universe — the classical Einstein equation, for instance — are no exception. There are no particular laws governing these boundary conditions; they summarize our observations of the universe outside the subsystem whose evolution we are studying. If we don’t see any radiation coming into a room, then we solve Maxwell’s equations inside with no-incoming-radiation boundary conditions. If we prepare an atom in a certain way, then we solve Schrödinger’s equation with the corresponding initial condition.

In cosmology, however, by definition, there is no rest of the universe to pass the specification of the boundary conditions off to. The boundary conditions must be part of the laws of physics themselves. Constructing a theory of the initial condition of the universe, effectively its initial quantum state, and examining its observational consequences is the province of that area of astrophysics that has come to be called quantum cosmology.[^2] This talk will consider one manifest feature of the quantum universe and its connection to the theory of the initial condition. This is the applicability of the deterministic laws of classical physics to a wide range of phenomena in the universe ranging from the cosmological expansion itself to the turbulent and viscous flow of water through a pipe. This quasiclassical realm[^3] is one of the most immediate facts of our experience. Yet what we know of the basic laws of physics suggests that we live in a quantum mechanical universe, characterized by indeterminacy and distributed probabilities, where classical laws can be but approximations to the unitary evolution of the Schrödinger equation and the reduction of the wave packet. What is the origin of this wide range of time, place, and scale on which classical determinism applies? How can we derive the form of the phenomenological classical laws, say the Navier-Stokes equations, from a distantly related fundamental quantum mechanical theory which might, after all, be heterotic, superstring theory? What features of these laws can be traced to their quantum mechanical origins? It is such old questions that will be examined anew in this lecture from the perspective of quantum cosmology, reporting largely on joint work with Murray Gell-Mann [@GH93a].

Standard derivations of classical behavior from the laws of quantum mechanics are available in many quantum mechanics texts. One popular approach is based on Ehrenfest’s theorem relating the acceleration of the expected value of position to the expected value of the force: $$m\ \frac{d^2\langle x\rangle}{dt^2} = - \left\langle\frac{\partial
V}{\partial x}\right\rangle\ ,
\label{oneone}$$ (written here for one-dimensional motion). Ehrenfest’s theorem is true in general, but for certain states, typically narrow wave packets, we may approximately replace the expected value of the force with the force evaluated at the expected value of position, thereby obtaining a classical equation of motion for that expected value: $$m\ \frac{d^2\langle x\rangle}{dt^2} = - \frac{\partial V(\langle x
\rangle)}{\partial x}\ .
\label{onetwo}$$ This equation shows that the center of a narrow wave packet moves on an orbit obeying Newton’s laws. More precisely, if we make a succession of position and momentum measurements that are crude enough not to disturb the approximation that allows (1.2) to replace (1.1), the expected values of the results will be correlated by Newton’s deterministic law.

This kind of elementary derivation is inadequate for the type of classical behavior that we hope to discuss in quantum cosmology for the following reasons:

-   The behavior of expected or average values is not enough to define classical behavior. In quantum mechanics, the statement that the moon moves on a classical orbit is properly the statement that, among a set of alternative histories of its position as a function of time, the probability is high for those histories exhibiting the correlations in time implied by Newton’s law of motion and near zero for all others. To discuss classical behavior, therefore, we should be dealing with the probabilities of individual time histories, not with expected or average values.

-   The Ehrenfest theorem derivation deals with the results of “measurements” on an isolated system with a few degrees of freedom. However, in quantum cosmology we are interested in classical behavior in much more general situations, over cosmological stretches of space and time, and over a wide range of subsystems, [*independent*]{} of whether these subsystems are receiving attention from observers. Certainly we imagine that our observations of the moon’s orbit, or a bit of the universe’s expansion, have little to do with the classical behavior of those systems. Further, we are interested not just in classical behavior as exhibited in a few variables and at a few times of our choosing, but in as refined a description as possible, so that classical behavior becomes a feature of the systems themselves and not a choice of observers.

-   The Ehrenfest theorem derivation relies on a close connection between the equations of motion of the fundamental action and the phenomenological deterministic laws that govern classical behavior. But when we speak of the classical behavior of the moon, or of the cosmological expansion, or even of water in a pipe, we are dealing with systems with many degrees of freedom whose phenomenological classical equations of motion may be only distantly related to the underlying fundamental theory, say superstring theory. We need a derivation which derives the [*form*]{} of the equations as well as the probabilities that they are satisfied.

-   The Ehrenfest theorem derivation posits the variables — the position $x$ — in which classical behavior is exhibited. But, as mentioned above, classical behavior is most properly defined in terms of the probabilities and properties of histories. In a closed system we should be able to [*derive*]{} the variables that enter into the deterministic laws, especially because, for systems with many degrees of freedom, these may be only distantly related to the coördinates entering the fundamental action.

Despite these shortcomings, the elementary Ehrenfest analysis already exhibits two necessary requirements for classical behavior: Some coarseness is needed in the description of the system as well as some restriction on its initial condition. Not every initial wave function permits the replacement of (\[oneone\]) by (\[onetwo\]) and therefore leads to classical behavior; only for a certain class of wave functions will this be true. Even given such a suitable initial condition, if we follow the system too closely, say by measuring position exactly, thereby producing a completely delocalized state, we will invalidate the approximation that allows (1.2) to replace (1.1) and classical behavior will not be expected. Some coarseness in the description of histories is therefore needed. For realistic systems we therefore have the important questions of [*how restricted*]{} is the class of initial conditions which lead to classical behavior and [*what*]{} and [*how large*]{} are the coarse grainings necessary to exhibit it.

Before pursuing these questions in the context of quantum cosmology I would like to review a derivation of classical equations of motion and the probabilities they are satisfied in a simple class of model systems, but before doing [*that*]{} I must review, even more briefly, the essential elements of the quantum mechanics of closed systems [@Gri84; @Omnsum; @GH90a].

[=2.00in ]{}

The Quantum Mechanics of Closed Systems {#sec:II}
=======================================

Most generally we aim at predicting the probabilities of alternative time histories of a closed system such as the universe as a whole. Alternatives at a moment of time are represented by an exhaustive set of orthogonal projection operators $\{P^k_{\alpha_k} (t_k)\}$. For example, these might be projections on a set of alternative intervals for the center of mass position of a collection of particles, or projections onto alternative ranges of their total momentum. The superscript denotes the set of alternatives a certain set of position ranges or a certain set of momentum ranges, the discrete index $\alpha_k = 1,2,3 \cdots$ labels the particular alternative, a particular range of position, and $t_k$ is the time. A set of alternative histories is defined by giving a series of such alternatives at a sequence of times, say $t_1, \cdots, t_n$. An individual history is a sequence of alternatives $(\alpha_1, \cdots, \alpha_n)\equiv
\alpha$ and is represented by the corresponding chain of projections. $$C_\alpha \equiv P^n_{\alpha_n} (t_n) \cdots P^1_{\alpha_1} (t_1)\ .
\label{twoone}$$ Such a set is said to be “coarse-grained” because the $P$’s do not restrict all possible variables and because they do not occur at all possible times.

The decoherence functional $$D\left(\alpha^\prime, \alpha\right) = Tr\,\bigl[C_{\alpha^\prime} \rho
C^\dagger_\alpha\bigr]
\label{twotwo}$$ measures the amount of quantum mechanical interference between pairs of histories in a universe whose initial condition is represented by a density matrix $\rho$. When, for a given set, the interference between all pairs of distinct histories is sufficiently low, $$D\left(\alpha^\prime, \alpha\right) \approx  0\quad , \quad {\rm all}
\ \alpha^\prime \not=
\alpha 
\label{twothree}$$ the set of alternative histories is said to [*decohere*]{}, and probabilities can be consistently assigned to its individual members. The probability of an individual history $\alpha$ is just the corresponding diagonal element of $D$, [*viz.*]{} $$p(\alpha) = D(\alpha, \alpha)\ . 
\label{twofour}$$

Describe in terms of operators, check decoherence and evaluate probabilities — that is how predictions are made for a closed system, whether the alternatives are participants in a measurement situation or not.

When the projections at each time are onto the ranges $\{\Delta_\alpha\}$ of some generalized coördinates $q^i$ the decoherence functional can be written in a convenient path integral from

$$D\left(\alpha^\prime, \alpha\right)  = \int_{\alpha^\prime}
\delta q^\prime \int_\alpha \delta q\, \delta \bigl(q^\prime_f
- q_f\bigr)
 e^{i(S[q^\prime(\tau)] - S[q(\tau)])/\hbar} \rho \left(q^\prime_0,
q_0\right) 
\label{twofive}$$

where the integral is over the paths that pass through the intervals defining the histories (Fig. 1). This form will be useful in what follows.

Classical Behavior in a Class of Model Quantum Systems {#sec:III}
======================================================

The class of models we shall discuss are defined by the following features:

-   We restrict attention to coarse grainings that follow a fixed subset of the fundamental coördinates $q^i$, say the center of mass position of a massive body, and ignore the rest. We denote the followed variables by $x^a$ and the ignored ones by $Q^A$ so that $q^i=(x^a, Q^A)$. We thus posit, rather than derive, the variables exhibiting classical behavior, but we shall derive, rather than posit, the form of their phenomenological equations of motion.

-   We suppose the action is the sum of an action for the $x$’s, an action for the $Q$’s, and an interaction between them that is the integral of a local Lagrangian free from time derivatives. That is, $$S[q(\tau)] = S_{\rm free} [x(\tau)] + S_0 [Q(\tau)] + S_{\rm int}
    [x(\tau), Q(\tau)] 
    \label{threeone}$$ suppressing indices where clarity is not diminished.

-   We suppose the initial density matrix factors into a product of one depending on the $x$’s and another depending on the ignored $Q$’s which are often called the “bath” or the “environment”.

$$\rho\left(q^\prime_0, q_0\right) = \bar\rho \left(x^\prime_0,
x_0\right)\, \rho_B \left(Q^\prime_0, Q_0\right)\ . 
\label{threetwo}$$

Under these conditions the integral over the $Q$’s in (2.5) can be carried out to give a decoherence functional just for coarse-grained histories of the $x$’s of the form:

$$D\left(\alpha^\prime, \alpha\right)  =  \int_{\alpha^\prime} \delta x^\prime  
\int_{\alpha} \delta x\, \delta\bigl(x^\prime_f - x_f\bigr)
 \exp
\biggl\{i\Bigl(S_{\rm free} [x^\prime (\tau)]
 - S_{\rm free} [x(\tau)]
 + W
\left[x^\prime (\tau), x(\tau)\right]\Bigr)/\hbar\biggr\}\,
\bar\rho\left(x^\prime_0, x_0\right) 
\label{threethree}$$

where $W [x^\prime(\tau), x(\tau)]$, called the Feynman-Vernon influence phase, summarizes the results of integrations over the $Q$’s.

[=2.00in ]{}

The influence phase $W$ generally possesses a positive imaginary part [@Bru93]. If that grows as $|x^\prime-x|$ increases, it will effect decoherence because there will then be negligible contribution to the integral (3.3) for $x^\prime \not= x$ or $\alpha^\prime \not= \alpha$. That, recall, is the definition of decoherence (2.3). Let us suppose this to be the case, as is true in many realistic examples. Then we can make an important approximation, which is a [*decoherence*]{} [*expansion*]{}. Specifically, introduce coördinates which measure the average and difference between $x^\prime$ and $x$ (Fig. 2) $$X = \half \left(x^\prime + x\right)\ , \quad \xi = x^\prime - x\ .
\label{threefour}$$

The integral defining the diagonal elements of $D$, which are the probabilities of the histories, receives a significant contribution only for small $\xi(t)$. We can thus expand the exponent of the integrand of (\[threethree\]) in powers of $\xi(t)$ and legitimately retain only the lowest, say up to quadratic, terms. The result for the exponent is

$$\begin{aligned}
S[x(\tau) & + & \xi(\tau)/2] - S[x(\tau) -\xi(\tau)/2] + W[x(\tau), \xi(\tau)]
\nonumber\\
& = & -\xi_0 P_0
 +  \int^T_0 dt\, \xi(t)\ \left[\frac{\delta S}{\delta X(t)} +
\left(\frac{\delta W}{\delta\xi(t)}\right)_{\xi(t)=0}\right]
 + \half \int^T_0 dt' \int^T_0 dt\ \xi(t^\prime)
\left(\frac{\delta^2 W}{\delta \xi(t')
\delta \xi(t)}\right)_{\xi(t)=0}
\ \xi(t) + \cdots\ . 
\label{threefive}\end{aligned}$$

The essentially unrestricted integrals over the $\xi(t)$ can then be carried out to give the following expression for the probabilities

$$p(\alpha) = \int_\alpha \delta X\, ({\rm det}\ K_I/4\pi)^{-\half}
 \exp\Bigl[-\frac{1}{\hbar} \int^T_0 dt' \int^T_0 dt\ {\cal E} 
(t', X(\tau)]\,  K^{\rm inv}_I \left(t', t; X(\tau)\right]\ {\cal E}
(t, X(\tau)]\Bigr]\ \bar w \left(X_0, P_0\right)\ . 
\label{threesix}$$

Here, $${\cal E}(t, X(\tau)] \equiv  \frac{\delta S}{\delta X(t)} +
\left\langle F(t, X(\tau)]\right\rangle 
\label{threeseven}$$ where $\langle F(t, X (\tau)]\rangle$ has been written for $(\delta
W/\delta\xi(t))_{\xi=0}$ because it can be shown to be the expected value of the force arising from the ignored variables in the state of the bath. $K^{\rm
inv}(t', t, X(\tau)]$ is the inverse of $(2\hbar/i)(\delta^2
W/\delta\xi(t')\delta\xi(t))$ which turns out to be real and positive. Finally $\bar w(X,P)$ is the Wigner distribution for the density matrix $\bar\rho$: $$w(X, P) = \frac{1}{2\pi} \int d\xi\, e^{i P\xi /\hbar}\rho (X+\xi/2, X-
\xi/2)\ .
\label{threeeight}$$ This expression shows that, when $K^{\rm inv}_I$ is sufficiently large, the probabilities for histories of $X(t)$ are peaked about those which satisfy the equation of motion $${\cal E} (t, X(\tau)] = \frac{\delta S}{\delta X(t)} + \langle F(t,
X(\tau)]\rangle = 0\ . 
\label{threenine}$$ and the initial conditions of these histories are distributed according to the Wigner distribution. The Wigner distribution is not generally positive, but, up to the accuracy of the approximations, this integral of it must be [@Hal92].

Thus we derive the form of the phenomenological equations of motion for this class of models. It is the equation of motion of the fundamental action $S[X(t)]$ corrected by phenomenological forces arising from the interaction with the bath. These depend not only on the form of the interaction Hamiltonian but also on the initial state of the bath, $\rho_B$. These forces are generally non-local in time, depending at a given instant on the whole trajectory $X(\tau)$. It can be shown that quantum mechanical causality implies that they depend only on part of path $X(\tau)$ to the past of $t$. Thus quantum mechanical causality implies classical causality.

It is important to stress that the expansion of the decoherence functional has enabled us to consider the equations of motion for fully non-linear systems, not just the linear oscillator models that have been widely studied.

The equation of motion (\[threenine\]) is not predicted to be satisfied [*exactly*]{}. The probabilities are [*peaked*]{} about ${\cal E}=0 $ but distributed about that value with a width that depends on the size of $K^{\rm inv}$. That is quantum noise whose spectrum and properties can be derived from (\[threethree\]). The fact that both the spectrum of fluctuations and the phenomenological forces can be derived from the same influence phase is the origin of the fluctuation dissipation theorem for linear systems.

Simple examples of this analysis are the linear oscillator models that have been studied using path integrals by Feynman and Vernon [@FV63], Caldeira and Leggett [@CL83], Unruh and Zurek [@UZ89], and many others. For these, the $x$’s describe a distinguished harmonic oscillator linearly coupled to a bath of many others. If the initial state of the bath is a thermal density matrix, then the decoherence expansion is exact. In the especially simple case of a cut-off continuum of bath oscillators and high bath temperature, there are the following results: The imaginary part of the influence phase is given by $$ImW[x'(\tau),x(\tau)]= \frac{2M\gamma kT_B}{\hbar} \int^T_0 dt 
\left(x^\prime(t) -
x(t)\right)^2 
\label{threeten}$$ where $M$ is the mass of the $x$-oscillator, $\gamma$ is a measure of the strength of its coupling to the bath, and $T_B$ is the temperature of the bath. The exponent of the expression (\[threeseven\]) giving the probabilities for histories is $$-\frac{M}{8\gamma kT_B} \int^T_0 dt\, \left[\ddot X + \omega^2  X +
2\gamma \dot X\right]^2 
\label{threeeleven}$$ where $\omega$ is the frequency of the $x$-oscillator renormalized by its interaction with the bath. The phenomenological force is friction, and the occurrence of $\gamma$, both in that force and the constant in front of (\[threeeleven\]), whose size governs the deviation from classical predictability, is a simple example of the fluctuation-dissipation theorem.

In this simple case, an analysis of the the requirements for classical behavior is straightforward. To achieve decoherence we need high values of $\gamma kT_B$. That is, strong coupling is needed if interference phases are to be dissipated efficiently into the bath. However, the larger the value of $\gamma kT_B$ the smaller the coefficient of front of (\[threeeleven\]), decreasing the size of the exponential and [*increasing*]{} deviations from classical predictability. This is reasonable: the stronger the coupling to the bath the more noise is produced by the interactions that are carrying away the phases. To counteract that, and achieve a sharp peaking about the classical equation of motion, $M$ must be large so that $M/\gamma kT_B$ is large. That is, high inertia is needed to resist the noise that arises from the interactions with the bath.

Thus, much more coarse graining is needed to ensure classical predictability than naive arguments based on the uncertainty principle would suggest. Coarse graining is needed to effect decoherence, and coarse graining beyond that to achieve the inertia necessary to resist the noise that the mechanisms of decoherence produce.

Quasiclassical Realms in Quantum Cosmology {#sec:IV}
==========================================

As observers of the universe, we deal every day with coarse-grained histories that exhibit classical correlations. Indeed, only by extending our direct perceptions with expensive and delicate instruments can we exhibit [*non*]{}-classical behavior. The coarse grainings that we use individually and collectively are, of course, characterized by a large amount of ignorance, for our observations determine only a very few of the variables that describe the universe and those only very imprecisely. Yet, we have the impression that the universe exhibits a much finer-grained set of histories, [*independent of our choice*]{}, defining an always decohering “quasiclassical realm”, to which our senses are adapted but deal with only a small part of. If we are preparing for a journey to a yet unseen part of the universe, we do not believe that we need to equip our spacesuits with detectors, say sensitive to coherent superpositions of position or other unfamiliar quantum operators. We expect that histories of familiar quasiclassical operators will decohere and exhibit patterns of classical correlation there as well as here.

Roughly speaking, a quasiclassical realm is a set of decohering histories, that is maximally refined with respect to decoherence, and whose individual histories exhibit as much as possible patterns of deterministic correlation. At present we lack satisfactory measures of maximality and classicality with which to make the existence of one or more quasiclassical realms into quantitative questions in quantum cosmology [@GH90a; @PZ93]. We therefore do not know whether the universe exhibits a [*unique*]{} class of roughly equivalent sets of histories with high levels of classicality constituting the quasiclassical realm of familiar experience, or whether there might be other essentially inequivalent quasiclassical realms [@GH94]. However, even in the absence of such measures and such analyses, we can make an argument for the form of at least some of the operators we expect to occur over and over again in histories defining one kind of quasiclassical realm — operators we might call “quasiclassical”. In the earliest instants of the history of the universe, the coarse grainings defining spacetime geometry on scales above the Planck scale must emerge as quasiclassical. Otherwise, our theory of the initial condition is simply inconsistent with observation in a manifest way. Then, when there is classical spacetime geometry we can consider the conservation of energy, and momentum, and of other quantities which are conserved by virtue of the equations of quantum fields. Integrals of densities of conserved or nearly conserved quantities over suitable volumes are natural candidates for quasiclassical operators. Their approximate conservation allow them to resist deviations from predictability caused by “noise” arising from their interactions with the rest of the universe that accomplish decoherence. Such “hydrodynamic” variables [*are*]{} among the principal variables of classical theories.

This argument is not unrelated to a standard one in classical statistical mechanics that seeks to identify the variables in which a hydrodynamic description of non-equilibrium systems may be expected. All isolated systems approach equilibrium — that is statistics. With certain coarse grainings this approach to equilibrium may be approximately described by hydrodynamic equations, such as the Navier-Stokes equation, incorporating phenomenological descriptions of dissipation, viscosity, heat conduction, diffusion, etc. The variables that characterize such hydrodynamic descriptions are the local quantities which very most [*slowly*]{} in time — that is, averages of densities of approximately conserved quantities over suitable volumes. The volumes must be large enough that statistical fluctuations in the values of the averages are small, but small enough that equilibrium is established within each volume in a time short compared to the dynamical times on which the variables vary. The constitutive relations defining coefficients of viscosity, diffusion, etc. are then defined and independent of the initial condition, permitting the closure of the set of hydrodynamic equations. Local equilibrium being established, the further equilibration of the volumes among themselves is described by the hydrodynamic equations. In the context of quantum cosmology, coarse grainings by averages of densities of approximately conserved quantities not only permit local equilibrium and resist gross statistical fluctuations leading to high probabilities for deterministic histories as in this argument, they also, as described above, resist the fluctuations arising from the mechanicsms of decoherence necessary for predicting probabilities of any kind in quantum mechanics.

In this way we can sketch how a quasiclassical realm consisting of histories of ranges of values of quasiclassical operators, extended over cosmological dimensions both in space and in time, but highly refined with respect to those scales, is a feature of our universe and thus must be a prediction of its quantum initial condition. It may seem strange to attribute the classical behavior of everyday objects to the initial condition of the universe some 12 billion years ago, but, in this connection, two things should be noted: First, we are not just speaking of the classical behavior of a few objects described in a very coarse graining of our choosing, but of a much more refined feature of the universe extending over cosmological dimensions and indeed including the classical behavior of the cosmological geometry itself all the way back to the briefest of moments after the big bang. Second, at the most fundamental level the [*only*]{} ingredients entering into quantum mechanics are the theory of the initial condition and the theory of dynamics, so that [*any*]{} feature of the universe must be traceable to these two starting points and the accidents of our particular history. Put differently (neglecting quantum gravity) the possible classical behavior of a set of histories represented by strings of projection operators as in (\[twoone\]) does not depend on the operators alone except in trivial cases. Rather, like decoherence itself, classicality depends on the relation of those operators to the initial state $|\Psi\rangle$ through which we calculate the decoherence and probabilities of sets of histories by which classical behavior is defined.

Yet it is reasonable to ask — how sensitive is the existence of a quasiclassical realm to the particular form of the initial condition? In seeking to answer this question it is important to recognize that there are two things it might mean. First, we might ask whether [*given*]{} an initial state $|\Psi\rangle$, there is always a set of histories which decoheres and exhibits deterministic correlations. There is, trivially. Consider the set of histories which just consists of projections down on ranges $\{\Delta E_\alpha\}$ of the [*total*]{} energy (or any other conserved quantity) at a sequence of times $$C_\alpha = P^H_{\alpha_n} (t_n) \cdots P^H_{\alpha_1} (t_1) 
\ .\label{fourone}$$ Since the energy is conserved these operators are independent of time, commute, and $C_\alpha$ is merely the projection onto the intersection of the intervals $\Delta E_{\alpha_`}, \cdots, \Delta
E_{\alpha_n}$. The set of histories represented by (\[fourone\]) thus [*exactly*]{} decoheres $$D\left(\alpha^\prime, \alpha\right) = Tr\bigl[C_{\alpha^\prime}
|\Psi\rangle\langle\Psi|C^\dagger_\alpha\bigr] =
\langle\Psi|C^\dagger_\alpha C_{\alpha^\prime} |\Psi \rangle \propto
\delta_{\alpha\alpha^\prime}\ ,
\label{fourtwo}$$ and exhibits deterministic correlations — the total energy today is the same as it was yesterday. Of course, such a set is far from maximal, but imagine subdividing the total volume again and again and considering the set of histories which results from following the values of the energy in each subvolume over the sequence of times. If the process of subdividing is followed until we begin to lose decoherence we might hope to retain some level of determinism while moving towards maximality. Thus, it seems likely that, for most initial $|\Psi\rangle$, we may find [*some*]{} sets of histories which constitute a quasiclassical realm.

However, we might ask about the sensitivity of a quasiclassical realm to initial condition in a different way. We might fix the chains of projections that describe [*our*]{} highly refined quasiclassical realm and ask for how many [*other*]{} initial states does this set of histories decohere and exhibit the same classical correlations. This amounts to asking, for a given set of alternative histories $\{C_\alpha\}$, how many initial states $|\Psi\rangle$ will have the same decoherence functional? Expand $|\Psi\rangle$ in some generic basis in Hilbert space, $|i\rangle$: $$|\Psi\rangle = \sum\nolimits_i c_i | i\rangle\ .
\label{fourthree}$$ The condition that $|\Psi\rangle$ result in a given decoherence functional $D(\alpha^\prime, \alpha)$ is $$\sum\nolimits_{ij} c^*_i c_j\ \bigl\langle i|C^\dagger_{\alpha^\prime}
C_\alpha | j\bigr\rangle = D\left(\alpha^\prime, \alpha\right)\ .
\label{fourfour}$$ Unless the $C_\alpha$ are such that decoherence and correlations are trivially implied by the operators (as is the above example of chains of projections onto a total conserved energy), the matrix elements $\langle
i|C^\dagger_{\alpha^\prime}\, C_\alpha | j\rangle$ will not vanish indentically. Equation (\[fourfour\]) is therefore (number of histories $\alpha)^2$ equations for (dimension of Hilbert space) coefficients. When that dimension is made finite, say by limiting the total volume and energy, we expect a solution only when $$\left({\rm number\ of\ histories}\atop{\rm in\ the\ quasiclassical
\ realm}\right)^2 \ltwid \left({\rm dim}
\ {\cal H}\right)\ .
\label{fourfive}$$ As the set of histories becomes increasingly refined, so that there are more and more alternative cases, the two sides may come closer to equality. The number of states $|\Psi\rangle$ which reproduce the [*particular*]{} maximal quasiclassical realm of our universe may thus be large but still small compared to the total number of states in Hilbert space.

The Main Points Again {#sec:V}
=====================

-   Classical behavior of quantum systems is defined through the probabilities of deterministic correlations of individual time histories of a closed system.

-   Classical predictability requires coarse graining to accomplish decoherence, and coarse graining beyond that to achieve the necessary inertia to resist the noise which mechanisms of decoherence produce.

-   The maximally refined quasiclassical realm of familiar experience is an emergent feature, not of quantum evolution alone, but of that evolution, coupled to a specific theory of the universe’s initial condition. Whether the whole closed system exhibits a quasiclassical realm like ours, and indeed whether it exhibits more than one essentially inequivalent realm, are calculable questions in quantum cosmology if suitable measures of maximality and classicality can be supplied.

-   A generic initial state will exhibit some sort of quasiclassical realm, but the maximally refined quasiclassical realm of familiar experience will be an emergent feature of only a small fraction of the total possible initial states of the universe.

Most of this paper reports joint work with M. Gell-Mann. The author’s research was supported in part by NSF grant PHY90-08502.

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[^1]: Talk given at the Lanczos Centenary Meeting, North Carolina State University, December 15, 1993

[^2]: For a recent review see [@Hal91]

[^3]: Earlier work, e.g. [@GH90a] called this the ‘quasiclassical domain’, but this risks confusion the usage in condensed matter physics.
