murray gell-mann - quasiclassical thermodynamic entropy

8/7/2019 Murray Gell-Mann - Quasiclassical Thermodynamic Entropy

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arXiv:quant-ph/0609190v3

16May2007

Quasiclassical Coarse Graining

and Thermodynamic Entropy

Murray Gell-Mann1, and James B. Hartle1,2,

1Santa Fe Institute, Santa Fe, NM 875012Department of Physics, University of California, Santa Barbara, CA 93106-9530

(Dated: February 1, 2008)

Our everyday descriptions of the universe are highly coarse-grained, following only a tiny fractionof the variables necessary for a perfectly fine-grained description. Coarse graining in classical physicsis made natural by our limited powers of observation and computation. But in the modern quantummechanics of closed systems, some measure of coarse graining is inescapable because there are nonon-trivial, probabilistic, fine-grained descriptions. This essay explores the consequences of thatfact. Quantum theory allows for various coarse-grained descriptions some of which are mutuallyincompatible. For most purposes, however, we are interested in the small subset of quasiclassicaldescriptions defined by ranges of values of averages over small volumes of densities of conservedquantities such as energy and momentum and approximately conserved quantities such as baryonnumber. The near-conservation of these quasiclassical quantities results in approximate decoher-ence, predictability, and local equilibrium, leading to closed sets of equations of motion. In anydescription, information is sacrificed through the coarse graining that yields decoherence and givesrise to probabilities for histories. In quasiclassical descriptions, further information is sacrificed inexhibiting the emergent regularities summarized by classical equations of motion. An appropriateentropy measures the loss of information. For a quasiclassical realm this is connected with the

usual thermodynamic entropy as obtained from statistical mechanics. It was low for the initial stateof our universe and has been increasing since.

I. INTRODUCTION

Coarse graining is of the greatest importance in theo-retical science, for example in connection with the mean-ing of regularity and randomness and of simplicity andcomplexity [1]. Of course it is also central to statisti-cal mechanics, where a particular kind of coarse grain-ing leads to the usual physico-chemical or thermody-namic entropy. We discuss here the crucial role that

coarse graining plays in quantum mechanics, making pos-sible the decoherence of alternative histories and enablingprobabilities of those decoherent histories to be defined.The quasiclassical realms that exhibit how classicalphysics applies approximately in this quantum universecorrespond to coarse grainings for which the historieshave probabilities that exhibit a great deal of approxi-mate determinism. In this article we connect these coarsegrainings to the ones that lead, in statistical mechanics,to the usual entropy.

Our everyday descriptions of the world in terms ofnearby physical objects like tables and clouds are exam-ples of very coarse-grainedquasiclassical realms. At any

one time, these everyday descriptions follow only a tinyfraction of the variables needed for a fine-grained descrip-tion of the whole universe; they integrate over the rest.Even those variables that are followed are not trackedcontinuously in time, but rather sampled only at a se-

Dedicated to Rafael Sorkin on his 60th birthday.Electronic address: [email protected] address: [email protected]

quence of times1.A classical gas of a large number of particles in a

box provides other excellent examples of fine and coarsegraining. The exact positions and momenta of all theparticles as a function of time give the perfectly fine-grained description. Useful coarse-grained descriptionsare provided by dividing the box into cells and specify-ing the volume averages of the energy, momentum, andparticle number in each. Coarse graining can also be ap-plied to ranges of values of the variables followed. Thisis an example in a classical context of what we will calla quasiclassical coarse graining. As we will describe inmore detail below, under suitable initial conditions andwith suitable choices for the volumes, this coarse grainingleads to a deterministic, hydrodynamic description of thematerial in the box. We need not amplify on the utilityof that.

In classical physics, coarse graining arises from practi-cal considerations. It might be forced on us by our punyability to collect, store, recall, and manipulate data. Par-ticular coarse grainings may be distinguished by theirutility. But there is always available, in principle, forevery physical system, an exact fine-grained description

that could be used to answer any question, whatever thelimitations on using it in practice.

In the modern formulation of the quantum mechanics

1 This is not the most general kind of coarse graining. One canhave instead a probability distribution characterized by certainparameters that correspond to the expected values of particularquantities, as in the case of absolute temperature and the energyin a Maxwell-Boltzmann distribution [1].
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of a closed system a formulation based on histories the situation is different. The probability that either oftwo exclusive histories will occur has to be equal to thesum of the individual probabilities. This will be impos-sible in quantum mechanics unless the interference termbetween the the two is made negligible by coarse grain-ing. That absence of interference is called decoherence.There is no (non-trivial) fine-grained, probabilistic de-

scription of the histories of a closed quantum-mechanicalsystem. Coarse graining is therefore inescapable in quan-tum mechanics if the theory is to yield any predictionsat all.

Some of the ramifications of these ideas are the follow-ing:

Quantum mechanics supplies probabilities for themembers of various decoherent sets of coarse-grained alternative histories different realms forshort. Different realms are compatible if each onecan be fine-grained to yield the same realm. (Wehave a special case of this when one of the realmsis a coarse graining of the other.) Quantum me-

chanics also exhibits mutually incompatible realmsfor which there is no finer-grained decoherent set ofwhich they are both coarse grainings.

Quantum mechanics by itself does not favor onerealm over another. However, we are interested formost purposes in the family of quasiclassical realmsunderlying everyday experience a very smallsubset of the set of all realms. Roughly speaking, aquasiclassical realm corresponds to coarse grainingwhich follows ranges of the values of variables thatenter into the classical equations of motion.

Having non-trivial probabilities for histories re-

quires the sacrifice of information through thecoarse graining necessary for decoherence.

Coarse graining beyond that necessary for decoher-ence is needed to achieve the predictability repre-sented by the approximate deterministic laws gov-erning the sets of histories that constitute the qua-siclassical realms.

An appropriately defined entropy that is a functionof time is a useful measure of the information lostthrough a coarse graining defining a quasiclassicalrealm.

Such a coarse graining, necessary for having bothdecoherence and approximate classical predictabil-ity, is connected with the coarse graining that de-fines the familiar entropy of thermodynamics.

The entropy defined by a coarse graining associatedwith a quasiclassical realm must be sufficiently lowat early times in our universe (that is, low for thecoarse graining in question) so that it tends to in-crease in time and exhibit the second law of ther-modynamics. But for this the relaxation time for

the increase of entropy must also be long comparedwith the age of the universe. From this point ofview the universe is very young.

The second law of thermodynamics, the family ofquasiclassical realms, and even probabilities of anykind are features of our universe that depend, not

just on its quantum state and dynamics, but alsocrucially on the inescapable coarse graining neces-sary to define those features, since there is no exact,completely fine-grained description.

Many of these remarks are contained in our earlierwork [2, 3, 4, 5, 6, 7, 8, 11, 12] or are implicit in it or arecontained in the work of others e.g. [13, 14] and the refer-ences therein. They are developed more extensively herein order to emphasize the central and inescapable roleplayed by coarse graining in quantum theory. What isnew in this paper includes the detailed analysis of thetriviality of decoherent sets of completely fine-grained

histories, the role of narrative, and, most importantly,the central result that the kind of coarse graining defin-ing the quasiclassical realms in quantum theory is closelyrelated to the kind of coarse graining defining the usualentropy of chemistry and physics.

The reader who is familiar with our previous work andnotation can immediately jump to Section III. But forthose who are not we offer a very brief summary in thenext section.

II. THE QUANTUM MECHANICS OF A

CLOSED SYSTEM

This section gives a bare-bones account of some essen-tial elements of the modern synthesis of ideas characteriz-ing the quantum mechanics of closed systems [7, 13, 14].

To keep the discussion manageable, we consider aclosed quantum system, most generally the universe, inthe approximation that gross quantum fluctuations in thegeometry of spacetime can be neglected. (For the gener-alizations that are needed for quantum spacetime see e.g.[15, 16].) The closed system can then be thought of as alarge (say > 20,000 Mpc), perhaps expanding box of par-ticles and fields in a fixed background spacetime. Every-

thing is contained within the box, in particular galaxies,planets, observers and observed , measured subsystems,and any apparatus that measures them. This is the mostgeneral physical context for prediction.

The fixed background spacetime means that the no-tions of time are fixed and that the usual apparatus ofHilbert space, states, and operators can be employed ina quantum description of the system. The essential theo-retical inputs to the process of prediction are the Hamil-tonian H governing evolution (which we assume for sim-


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plicity to be time-reversible2) and the initial quantumcondition (which we assume to be a pure state3 |).These are taken to be fixed and given.

The most general objective of quantum theory is theprediction of the probabilities of individual members ofsets of coarse-grained alternative histories of the closedsystem. For instance, we might be interested in alter-native histories of the center-of-mass of the earth in its

progress around the sun, or in histories of the correlationbetween the registrations of a measuring apparatus and aproperty of the subsystem. Alternatives at one momentof time can always be reduced to a set of yes/no ques-tions. For example, alternative positions of the earthscenter-of-mass can be reduced to asking, Is it in this re-gion yes or no?, Is it in that region yes or no?, etc.An exhaustive set of yes/no alternatives is represented inthe Heisenberg picture by an exhaustive set of orthogo-nal projection operators {P(t)}, = 1, 2, 3 . Thesesatisfy

P(t) = I, and P(t) P(t) = P(t) , (2.1)

showing that they represent an exhaustive set of exclusivealternatives. In the Heisenberg picture, the operatorsP(t) evolve with time according to

P(t) = e+iHt/hP(0) e

iHt/h . (2.2)

The state | is unchanging in time.Alternatives at one time can be described by express-

ing their projections in terms of fundamental coordinates,say a set of quantum fields and their conjugate momenta.In the Heisenberg picture these coordinates evolve intime. A given projection can be described in terms ofthe coordinates at any time. Thus, at any time it repre-

sents some alternative [8].An important kind of set of exclusive histories is spec-ified by sets of alternatives at a sequence of times t1 > L3.

Define a set of quasiclassical histories of the type

discussed in Section IIIA, using ranges of the(V(y, t), V(y, t), V(y, t)) at a sequence of times.Under the above assumptions it is possible to showthat the fluctuations in any of the quasiclassicalquantities are small and remain small, for instance

[V(y, t)]2/V(y, t)

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of defined as

S() = T r( log ) . (6.1)

This is zero when is a pure state, = ||, showingthat a pure state is a complete description of a quan-tum system. For a Hilbert space of finite dimension N,complete ignorance is expressed by = I/Tr(I). Themaximum value of the missing information is logN.

A coarse-grained description of a quantum system gen-erally consists of specifying the expected values of certainoperators Am, (m = 1, , M) in the mixed state .That is, the system is described by specifying

Am T r(Am) , (m = 1 , , M) . (6.2)

For example, the {Am} might be an exhaustive set oforthogonal projection operators {P} of the kind used inSection II to describe yes/no alternatives at one momentof time. These projections might be onto ranges of thecenter-of-mass position of the planet Mars. In a finer-grained quasiclassical description, of the kind discussedin Section V, they might be projections onto ranges ofvalues of the densities of energy, momentum, and otherconserved or nearly conserved quantities averaged oversmall volumes in the interior of Mars. In a coarse grain-ing of that, the {Am} might be the operators V(y, t),V(y, t), and V(y, t) themselves, so that the descriptionat one time is in terms of the expected values of variablesrather than expected values of projections onto ranges ofvalues of those variables. As these examples illustrate,the {Am} are not necessarily mutually commuting.

The measure of missing information in descriptions ofthe form (6.2) for a quantum system with density ma-trix is the maximum entropy over the effective densitymatrices that are consistent with the coarse-grained

description (e.g. [33, 34, 35]). Specifically,

S({Am}, ) T r( log ) , (6.3)

where maximizes this quantity subject to the con-straints

T r(Am) = Am T r(Am), m = 1, , M . (6.4)

The solution to this maximum problem is straightforwardto obtain by the method of Lagrange multipliers and is

= Z1 exp

M

m=1mAm

. (6.5)

Here, the {m} are c-number Lagrange multipliers de-termined in terms of and Am by the coarse grainingconstraints (6.4); Z ensures normalization. The missinginformation is then the entropy

S({Am}, ) = T r ( log ) . (6.6)

As mentioned earlier the As need not be commuting forthis construction. What is important is that the con-straints (6.4) are linear in .

Suppose that a coarser-grained description is charac-terized by operators {Am}, m = 1, , M < M suchthat

Am =mm

cmmAm , (6.7)

where the cmm are coefficients relating the operators ofthe finer graining to those of the coarser graining. For

example, suppose that the Am are averages over volumesV(y) of one of the quasiclassical variables. Averagingover bigger volumes V(z) that are unions of these wouldbe a coarser-grained description. The coefficients cmmwould become c(z, y) V(y)/V(z). Then, we obtain

S({Am}, ) S({Am}, ) , (6.8)

since there are fewer constraints to apply in themaximum-missing-information construction. Entropy in-creases with further coarse graining.

Among the various possible operators {Am} that couldbe constructed from those defining histories of the qua-siclassical realm, which should we choose to find a con-nection (if any), with the usual entropy of chemistry andphysics? Familiar thermodynamics provides some clues.The first law of thermodynamics connects changes in theexpected values of energy over time with changes in vol-ume and changes in entropy in accord with conservationof energy. This suggests that for our present purpose weshould consider an entropy defined at one time ratherthan for a history. It also suggests that we should con-sider the entropy defined by the expected values of quasi-classical quantities and not use the finer-grained descrip-tion by ranges of values of the the quantities themselves,which enter naturally into histories.

To see that this is on the right track, let us compute

the missing information specifying the expected values ofthe total energy H, total momentum P, and total con-served number N for our model system in a box. Thedensity matrix maximizing the missing information is14,from (6.5)

max = Z1 exp[(H U P N)] . (6.9)

This is the equilibrium density matrix (6.3). The entropyS defined by (6.6) is straightforwardly calculated fromthe (Helmholtz) free energy F defined by

T r{exp[(H U P N)]}

exp[(F U P N)] . (6.10)

We find

S = (H F) . (6.11)

14 Note that in (6.9) H means the Hamiltonian, not the enthalpy,and H is the same as what is often denoted by U in thermo-dynamics.


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This standard thermodynamic relation shows that wehave recovered the standard entropy of chemistry andphysics.

In an analogous way, the missing information canbe calculated for a coarse graining in which (y, t),(y, t), and (y, t) are specified at one moment oftime. The density matrix that maximizes the missing in-formation according to (6.3) is that for the assumed local

equilibrium (5.4). The entropy is

S =y

(y, t) [(y, t) (y, t)] , (6.12)

where (y, t) is the free energy density defined analo-gously to (6.10). The integrand of(6.12) is then naturallyunderstood as defining the average over small volumes ofan entropy density (x, t)..

What is being emphasized here is that, in these ways,the usual entropy of chemistry and physics arises nat-urally from the coarse graining that is inescapable forquantum-mechanical decoherence and for quasiclassical

predictability of histories, together with the assumptionof local equilibrium15.

VII. THE SECOND LAW OFTHERMODYNAMICS

We can now discuss the time evolution in our universeof the entropy described in the previous section by thecoarse graining defining a quasiclassical realm. Manypieces of this discussion are standard (see, e.g. [30]) al-though not often described in the context of quantumcosmology.

The context of this discussion continues to be ourmodel universe of fields in a box. For the sake of simplic-ity, we are not only treating an artificial universe in a boxbut also sidestepping vital issues such as the acceleratedexpansion of the universe, possible eternal inflation, thedecay of the proton, the evaporation of black holes, longrange forces, gravitational clumping, etc. etc. (See, forexample [36, 37, 38].)

For this discussion we will distinguish two connectedbut different features of the universe:

The tendency of the total entropy16 of the universeto increase.

15 In some circumstances (for example when we take gravitationinto account in metastable configurations of matter) there maybe other kinds of entropy that are appropriate such as q-entropywith q = 1. Such circumstances have been excluded from ourmodel for simplicity, but for more on them see e.g [45]

16 In the context of our model universe in a box the total entropy isdefined. For the more realistic cases of a flat or open universe thetotal entropy may be infinite and we should refer to the entropyof a large comoving volume.

The tendency of the entropy of each presently al-most isolated system to increase in the same direc-tion of time. This might be called the homogeneityof the thermodynamic arrow of time.

Evidently these features are connected. The first followsfrom the second, but only in the late universe when al-most isolated systems are actually present. In the earlyuniverse we have only the first. Together they may becalled the second law of thermodynamics.

More particularly, isolated systems described by qua-siclassical coarse grainings based on approximately con-served quantities can be expected to evolve toward equi-librium characterized by the total values of these con-served quantities. In our model universe in a box, theprobabilities of V(y, t), V(y, t), V(y, t), and their cor-relations are eventually given by the equilibrium densitymatrix (5.3). The conditions that determine the equi-librium density matrix are sums of the conditions thatdetermine the local equilibrium density matrix in theJaynes construction (6.4). The smaller number of condi-tions means that the equilibrium entropy will be larger

than that for any local equilibrium [cf. (6.8)].Two conditions are necessary for our universe to ex-

hibit a general increase in total entropy defined by qua-siclassical variables:

The quantum state | is such that the initial en-tropy is near the minimum it could have for thecoarse graining defining it. It then has essentiallynowhere to go but up.

The relaxation time to equilibrium is long com-pared to the present age of the universe so thatthe general tendency of its entropy to increase willdominate its evolution.

In our simple model we have neglected gravitation forsimplicity, but for the following discussion we restore it.Gravity is essential to realizing the first of these condi-tions because in a self-gravitating system gravitationalclumping increases entropy. The early universe is ap-proximately homogeneous, implying that the entropy hasmuch more room to increase through the gravitationalgrowth of fluctuations. In a loose sense, as far as grav-ity is concerned, the entropy of the early universe is lowfor the coarse graining we have been discussing. The en-tropy then increases. Note that a smaller effect in theopposite direction is connected with the thermodynamicequilibrium of the matter and radiation in the very earlyuniverse. See [39] for an entropy audit of the presentuniverse.

Coarse graining by approximately conserved quasiclas-sical variables helps with the second of the two conditionsabove relaxation time short compared to present age.Small volumes come to local equilibrium quickly. But theapproximate conservation ensures that the whole systemwill approach equilibrium slowly, whether or not suchequilibrium is actually attained. Again gravity is impor-tant because its effects concentrate a significant fraction


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of the matter into almost isolated systems such as starsand galactic halos, which strongly interact with one otheronly infrequently17.

Early in the universe there were no almost isolated sys-tems. They arose later from the condensation of initialfluctuations by the action of gravitational attraction18.They are mostly evolving toward putative equilibrium in

the same direction of time. Evidently this homogeneityof the thermodynamic arrow of time cannot follow fromthe approximately time-reversible dynamics and statis-tics alone. Rather the explanation is that the progeni-tors of todays nearly isolated systems were all far fromequilibrium a long time ago and have been running downhill ever since. This provides a stronger constraint onthe initial state than merely having low total entropy. AsBoltzmann put it over a century ago: The second law ofthermodynamics can be proved from the [time-reversible]mechanical theory, if one assumes that the present stateof the universe. . . started to evolve from an improbable[i.e. special] state [42].

The initial quantum state of our universe must be suchthat it leads to the decoherence of sets of quasiclassicalhistories that describe coarse-grained spacetime geome-try and matter fields. Our observations require this now,and the successes of the classical history of the universesuggests that there was a quasiclassical realm at a veryearly time. In addition, the initial state must be such thatthe entropy of quasiclassical coarse graining is low in thebeginning and also be such that the entropy of presentlyisolated systems was also low. Then the universe can ex-hibit both aspects of the second law of thermodynamics.

The quasiclassical coarse grainings are therefore dis-tinguished from others, not only because they exhibitpredictable regularities of the universe governed by ap-proximate deterministic equations of motion, but also be-cause they are characterized by a sufficiently low entropyin the beginning and a slow evolution towards equilib-rium, which makes those regularities exploitable.

This confluence of features suggests the possibility ofa connection between the dynamics of the universe andits initial condition. The no-boundary proposal for theinitial quantum state [43] is an example of just such aconnection. According to it, the initial state is com-

putable from the Euclidean action, in a way similar tothe way the ground state of a system in flat space can becalculated.

17 For thoughts on what happens in the very long term in an ex-panding universe see [40, 41].

18 Much later certain IGUSes create isolated systems in the labo-ratory which typically inherit the thermodynamic arrow of theIGUS and apparatus that prepared them.

VIII. CONCLUSIONS AND COMMENTS

Many of the conclusions of this paper can be foundamong the bullets in the Introduction. Rather than re-iterating all of them here, we prefer to discuss in thissection some broader issues. These concern the relationbetween our approach to quantum mechanics, based oncoarse-grained decoherent histories of a closed system,

and the approximate quantum mechanics of measuredsubsystems, as in the Copenhagen interpretation. Thelatter formulation postulates (implicitly for most authorsor explicitly in the case of Landau and Lifshitz [44]) aclassical world and a quantum world, with a movableboundary between the two. Observers and their measur-ing apparatus make use of the classical world, so that theresults of a measurement are ultimately expressed inone or more c-numbers.

We have emphasized that this widely taught interpre-tation, although successful, cannot be the fundamentalone because it seems to require a physicist outside thesystem making measurements (often repeated ones) of

it. That would seem to rule out any application to theuniverse, so that quantum cosmology would be excluded.Also billions of years went by with no physicist in theoffing. Are we to believe that quantum mechanics didnot apply to those times?

In this discussion, we will concentrate on how theCopenhagen approach fits in with ours as a set of spe-cial cases and how the classical world can be replacedby a quasiclassical realm. Such a realm is not postulatedbut rather is explained as an emergent feature of the uni-verse characterized by H, |, and the enormously longsequences of accidents (outcomes of chance events) thatconstitute the coarse-grained decoherent histories. The

material in the preceding sections can be regarded as adiscussion of how quasiclassical realms emerge.We say that a measurement situation exists if some

variables (including such quantum-mechanical variablesas electron spin) come into high correlation with a quasi-classical realm. In this connection we have often referredto fission tracks in mica. Fissionable impurities can un-dergo radioactive decay and produce fission tracks withrandomly distributed definite directions. The tracks arethere irrespective of the presence of an observer. Itmakes no difference if a physicist or other human or achinchilla or a cockroach looks at the tracks. Decoherenceof the alternative tracks induced by interaction with theother variables in the universe is what allows the tracksto exist independent of observation by an observer.All those other variables are effectively doing the ob-serving. The same is true of the successive positions ofthe moon in its orbit not depending on the presence ofobservers and for density fluctuations in the early uni-verse existing when there were no observers around tomeasure them.

The idea of collapse of the wave function correspondsto the notion of variables coming into high correlationwith a quasiclassical realm, with its decoherent histo-


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ries that give true probabilities. The relevant historiesare defined only through the projections that occur inthe expressions for these probabilities [cf (2.7)]. Withoutprojections, there are no questions and no probabilities.In many cases conditional probabilities are of interest.The collapse of the probabilities that occurs in their con-struction is no different from the collapse that occursat a horse race when a particular horse wins and future

probabilities for further races conditioned on that eventbecome relevant.The so-called second law of evolution, in which a

state is reduced by the action of a projection, and theprobabilities renormalized to give ones conditioned onthat projection, is thus not some mysterious feature ofthe measurement process. Rather it is a natural conse-quence of the quantum mechanics of decoherent histories,dealing with alternatives much more general than meremeasurement outcomes.

There is thus no actual conflict between the Copen-hagen formulation of quantum theory and the more gen-eral quantum mechanics of closed systems. Copenhagen

quantum theory is an approximation to the more generaltheory that is appropriate for the special case of measure-ment situations. Decoherent histories quantum mechan-ics is rather a generalization of the usual approximatequantum mechanics of measured subsystems.

In our opinion decoherent histories quantum theoryadvances our understanding in the following ways amongmany others:

Decoherent histories quantum mechanics extendsthe domain of applicability of quantum theory tohistories of features of the universe irrespective ofwhether they are receiving attention of observers

and in particular to histories describing the evolu-tion of the universe in cosmology.

The place of classical physics in a quantum universeis correctly understood as a property of a particu-lar class of sets of decoherent coarse-grained alter-native histories the quasiclassical realms [5, 11].In particular, the limits of a quasiclassical descrip-tion can be explored. Dechoherence may fail if thegraining is too fine. Predictability is limited byquantum noise and by the major branchings thatarise from the amplification of quantum phenomenaas in a measurement situation. Finally, we cannotexpect a quasiclassical description of the universein its earliest moments where the very geometry ofspacetime may be undergoing large quantum fluc-tuations.

Decoherent histories quantum mechanics providesnew connections such as the relation (which hasbeen the subject of this paper) between the coarsegraining characterizing quasiclassical realms andthe coarse graining characterizing the usual ther-modynamic entropy of chemistry and physics.

Decoherent histories quantum theory helps withunderstanding the Copenhagen approximation.For example, measurement was characterized as anirreversible act of amplification, the creation ofa record, or as a connection with macroscopicvariables. But these were inevitably impreciseideas. How much did the entropy have to increase,how long did the record have to last, what exactly

was meant by macroscopic? Making these ideasprecise was a central problem for a theory in whichmeasurement is fundamental. But it is less centralin a theory where measurements are just special,approximate situations among many others. Thencharacterizations such as those above are not false,but true in an approximation that need not be ex-actly defined.

Irreversibility clearly plays an important role in sci-ence as illustrated here by the two famous applica-tions to quantum-mechanical measurement situa-tions and to thermodynamics. It is not an absoluteconcept but context-dependent like so much else inquantum mechanics and statistical mechanics. It ishighly dependent on coarse graining, as in the caseof the document shredding [7]. This was typicallycarried out in one dimension until the seizure byIranian students of the U.S. Embassy in Tehranin 1979, when classified documents were put to-gether and published. Very soon, in many parts ofthe world, there was a switch to two-dimensionalshredding, which still appears to be secure today. Itwould now be labeled as irreversible just as the one-dimensional one was previously. The shredding andmixing of shreds clearly increased the entropy of thedocuments, in both cases by an amount dependent

on the coarse grainings involved. Irreversibility isevidently not absolute but dependent on the effortor cost involved in reversal.

.The founders of quantum mechanics were right in

pointing out that something external to the frameworkof wave function and Schrodinger equation is needed tointerpret the theory. But it is not a postulated classi-cal world to which quantum mechanics does not apply.Rather it is the initial condition of the universe that,together with the action function of the elementary par-ticles and the throws of quantum dice since the begin-

ning, explains the origin of quasiclassical realm(s) withinquantum theory itself.

APPENDIX A: TRIVIAL DECOHERENCE OFPERFECTLY FINE-GRAINED SETS OF

HISTORIES

A set of perfectly fine-grained histories trivially deco-heres when there is a different final projection for each


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history with a non-zero branch state vector. Equiva-lently, we could say that a set of completely fine-grainedhistories is trivial if each final projection has only onepossible prior history. The orthogonality of the finalalternatives then automatically guarantees the decoher-ence of the realm. Such sets are trivial in the sense thatwhat happens at the last moment is uniquely correlatedwith the alternatives at all previous moments. This ap-

pendix completes the demonstration, adumbrated in Sec-tion III.A, that completely fine-grained realms are trivial.The domain of discussion was described in Section

III.A. We consider sets of histories composed of com-pletely fine-grained alternatives at any one time speci-fied by a complete basis for Hilbert space. It is sufficientto consider a sequence of times tk, (k = 1, 2, , n) be-cause if such decoherent sets are trivial all finer-grainedsets will also be trivial. In Section III.A we denoted thebases by {|k, ik}. Here, in the hope of keeping the no-tation manageable, we will drop the first label and justwrite {|ik}. The condition for decoherence (2.8) is then

in, ,i1 |in, ,i1

= p(in, , i1) inin i1i1 (A.1)

where the branch state vectors |in, ,i1 are defined by(2.6) with alternatives at each time of the form ( 3.1) andwhere equality has replaced because we are insistingon exact decoherence.

Mathematically, the decoherence condition (A.1) canbe satisfied in several ways. Some of the histories couldbe represented by branch state vectors which vanish (zerohistories for short). These have vanishing inner productswith all branch state vectors including themselves. Theytherefore do not affect decoherence and their probabilitiesare equal to zero. Further discussion can therefore berestricted to the non-vanishing branches.

Decoherence in the final alternatives in is automaticbecause the projections Pin and Pin are orthogonal ifdifferent. Decoherence is also automatic if each non-zerohistory has its own final alternative or, equivalently, ifeach final alternative in has a unique prior history. Thendecoherence of the final alternatives guarantees the de-coherence of the set.

Two simple and trivial examples may help to makethis discussion concrete. Consider the set of completelyfine-grained histories defined by taking the same basis{|i} at each time. The resulting orthogonality of theprojections between times ensures that the only non-zerohistories have in = in1 = = i2 = i1 in an appropriate

notation. Evidently each in corresponds to exactly onechain of past alternatives and the decoherence condition(A.1) is satisfied.

A related trivial example is obtained choosing the{|ik} to be the state | and some set of orthogonalstates at each of the times t1, , tn1. That is, we aremindlessly asking the question, Is the system in the state| or not? over and over again until the last alterna-tive. The only non-zero branches are of the form

|inin|| | = |inin| . (A.2)

In this case, each final alternative is correlated with thesame set of previous alternatives. The set decoheres be-cause the only non-trivial branching is at the last time asthe equality in (A.2) shows. But then each history has aunique final end alternative and the set is thus trivial19

Trivial sets of completely fine-grained histories havemany zero-histories as in the above examples. To seethis more quantitatively, imagine for a moment that the

dimension of Hilbert space is a very large but finite num-ber N. A generic, completely fine-grained set with ntimes would consist of Nn histories. But there can be atmost N orthogonal branches no more than are be sup-plied just by alternatives at one time. Indeed the triviallydecohering completely fine-grained sets described abovehave at most N branches. Most of the possible historiesmust therefore be zero. Further, assuming that at leastone non-zero branch is added at each time, the numberof times n is limited to N.

Were a final alternative in a completely fine-grained setto have more than one possible previous non-zero history,the set would not decohere. To see this, suppose someparticular final alternative k

nhad two possible, non-zero,

past histories. Let the earliest (and possibly the only)moment the histories differ be time tj . The condition fordecoherence is [cf. (A.1)]

|i1i1|i

2 i

n1|knkn|in1 i2|i1i1|

= p(kn, , i1) in1

in1 i1i1 . (A.3)

Sum this over all the i1, , ij1 and i1, , ij1 to find

|ij M

ij , , in1, kn, in1, , ij

ij |

= p(kn, , ij) in1in1 ijij (A.4)

where we have employed an abbreviated notation for theproduct of all the remaining matrix elements. By hy-pothesis there are at least two chains (ij , , kn) differ-ing in the value ofij for which M does not vanish and thehistory is not zero. But decoherence at time tj then re-quires that ij | vanish for all but one ij , contradictingthe assumption that there are two non-zero histories.

The only decohering, perfectly fine-grained sets arethus trivial.

We now discuss the connection between this notion ofa trivial fine-grained realm and the idea of generalizedrecords introduced in [3, 12]. Generalized records of arealm are a set of orthogonal projections {R} that arecorrelated to a good approximation with the histories of

the realm {C} so that

RC | C| . (A.5)

19 Coarse-grained examples of both kinds of triviality can be given.Repeating the same Heisenberg picture sets of projections at alltimes is an example of the first. Histories defined by Heisenbergpicture sets of projections onto a subspace containing | andother orthogonal subspaces are examples of the second.


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Generalized records of a particular kind characterize im-portant physical mechanisms of decoherence. For in-stance, in the classic example of Joos and Zeh [19] a dustgrain is imagined to be initially in a superposition of twoplaces deep in intergalactic space. Alternative historiesof coarse-grained positions of the grain are made to de-cohere by the interactions with the 1011 photons of the3 cosmic background radiation that scatter every sec-

ond. Through the interaction, records of the positionsare created in the photon degrees of freedom, which arevariables not followed by the coarse graining.

But the trivial generalized records that characterizefine-grained realms are not of this kind. They are in vari-ables that are followed by the coarse graining. They couldnot be in other variables because the histories are fine-grained and follow everything. They are trivial recordsfor a trivial kind of realm20.

APPENDIX B: NO NON-TRIVIAL CERTAINTYARISING FROM THE DYNAMICS PLUS THE

INITIAL CONDITION

The Schrodinger equation is deterministic and the evo-lution of the state vector is certain. In this appendix weshow that this kind of determinism is the only source ofhistories that are certain (probability equal 1), coarse-grained or not, for a closed quantum-mechanical system.

Suppose we have a set of exactly decoherent historiesof the form (2.4), one member of which has probability1, i.e. the history is certain. Denote the class operatorfor this particular history by C1 and index its particularalternatives at each time so they are 1 = 2 = n =1. Thus we have

p(1) = C1|2

= 1 , p() = C|2

= 0 , = 1 .(B.1)

The second of these implies C| = 0 for = 1. Giventhat C = I, the first implies that C1| = |. Insummary we can write

C| =P1n(tn) P

11(t1)|

=n,1n1,1 1,1| . (B.2)

Summing both sides over this over all s except those attime tk, we get

Pkk(tk)| = k,1| . (B.3)

This means that all projections in a set of histories ofwhich one is certain are either onto subspaces which con-tain | or onto subspaces orthogonal to | . Effectivelythey correspond to questions that ask, Is the state still| or not?. That is how the certainty of unitary evolu-tion is represented in the Heisenberg picture, where |is independent of time.

Note that this argument doesnt exclude historieswhich are essentially certain over big stretches of time asin histories describing alternative values of a conservedquantity at many times. Then there would be, in gen-eral, initial probabilities for the value of the conservedquantity which then do not change through subsequenthistory.

ACKNOWLEDGMENTS

We thank J. Halliwell and S. Lloyd for useful recentdiscussions. We thank the Aspen Center for Physics forhospitality over several summers while this work was inprogress. JBH thanks the Santa Fe Institute for support-ing several visits there. The work of JBH was supportedin part by the National Science Foundation under grantPHY02-44764 and PHY05-55669. The work of MG-Mwas supported by the C.O.U.Q. Foundation, by InsightVenture Management, and by the KITP in Santa Bar-bara. The generous help provided by these organizationsis gratefully acknowledged.

20 In the general case of medium decoherence, we can construct pro-jection operators onto orthogonal spaces each of which includesone of the states C| [12], but again these operators are not

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murray gell-mann - quasiclassical thermodynamic entropy

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