Classicality criteria

Classicality criteriaNuno Costa Dias Citation: Journal of Mathematical Physics 43, 5882 (2002); doi: 10.1063/1.1516626 View online: http://dx.doi.org/10.1063/1.1516626 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/43/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Quantum mechanics of damped systems J. Math. Phys. 44, 3718 (2003); 10.1063/1.1599074 Quantal two-center Coulomb problem treated by means of the phase-integral method. II. Quantization conditionsin the symmetric case expressed in terms of complete elliptic integrals. Numerical illustration J. Math. Phys. 42, 5077 (2001); 10.1063/1.1399295 From classical to quantum mechanics: “How to translate physical ideas into mathematical language” J. Math. Phys. 42, 3983 (2001); 10.1063/1.1386410 Classical and quantum mechanics of jointed rigid bodies with vanishing total angular momentum J. Math. Phys. 40, 2381 (1999); 10.1063/1.532871 Intrinsic resonance representation of quantum mechanics J. Chem. Phys. 106, 8564 (1997); 10.1063/1.473911

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Classicality criteriaNuno Costa Diasa)

Departamento de Matema´tica, Universidade Luso´fona de Humanidades e Tecnologias, Av.Campo Grande, 376, 1749-024 Lisboa, Portugal

~Received 2 November 2000; accepted 27 August 2002!

We present two possible criteria quantifying the degree of classicality of an arbi-trary ~finite dimensional! dynamical system. The inputs for these criteria are theclassical dynamical structure of the system together with the quantum and theclassical data providing the two alternative descriptions of its initial time configu-ration. It is proved that a general quantum system satisfying the criteria up to someextend displays a time evolution consistent with the classical predictions up tosome degree and thus it is argued that the criteria provide a suitable measure ofclassicality. The features of the formalism are illustrated through two simpleexamples. ©2002 American Institute of Physics.@DOI: 10.1063/1.1516626#

I. INTRODUCTION

It is generally accepted that quantum mechanics yields the most fundamental description of allphysical systems. Such status requires the theory to provide a suitable description of every dayclassical like phenomena. However, quantum mechanics faces a considerable number of problemsto explain the emergence of a classical domain. In a broad sense, this is called the problem of thesemiclassical limit of quantum mechanics.1–6

This is a difficult topic not least because there are a variety of different ways of relatingclassical and quantum mechanics. The standard approach is to take the limit\→0, or equivalently,the limit of a large number of particlesN→`. Under some general conditions one can derive theclassical evolution from the quantum dynamics.7 Alternatively, one may start with a set of quan-tum initial data with minimal spread and show that its quantum time evolution remains verypeaked around the classical orbits. Coherent states8 ~and squeezed states9,10! have been extensivelystudied in this context.11–17 It has been proved, using several different approaches and for a largeclass of dynamical systems, that coherent states evolve, up to a prescribed error and during aprescribed lapse of time, peaked around the classical paths.14,16–18Coherent states methods havealso been used to prove that in the limit\→0 and for times shorter than Eherenfest’s time, thequantum and the classical predictions coincide exactly.10,18–20Related results have been obtainedusing the tools of pseudodifferential calculus21–23and microlocal analysis.24–26These have playedan important part in several attempts to understand the semiclassical behavior of classically cha-otic systems.24,27

Two other significant approaches are the Wigner~or more generally the quasidistributions!28

and the dechoerent histories3,4,29approaches. The aim is to~either by identifying proper quantumphase space distributions, or by coupling the quantum system to a suitable environment! recoverthe classical statistical predictions from the quantum formulation of the system. All these viewsare of course not antagonic, but complementary, meaning that the classical behavior can emerge ina variety of different regimes.

Despite the importance of these results there are still several open problems in the field of thesemiclassical limit of quantum mechanics. The common view is that classical mechanics is anapproximate description of quantum mechanics. This approximation is not always valid. In fact,one expects it to be only valid for some particular set of quantum initial data and even in this casenot for all dynamical systems~consider for instance a particle hitting a potential barrier!. An

a!Electronic mail: [email protected]

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 43, NUMBER 12 DECEMBER 2002

58820022-2488/2002/43(12)/5882/20/$19.00 © 2002 American Institute of Physics


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important question is then what is the degree of validity of the classical approximation? Clearlythis question does not have a single answer. It depends on the dynamical structure of the systemand on the quantum and also the classical data providing the two alternative descriptions of theinitial time configuration. Let us assume that we are given this information. Can we compare theclassical and the quantum descriptions of the dynamical system and produce a statement about thedegree of validity of the classical description? Or, in other words, can we produce a statementabout the degree of classicality of the quantum description? The answer to these questions shouldbe affirmative since we are given all the necessary ingredients. However, and despite the fact thatthe pertinence of these questions have been recognized by several authors,3–5 the problem ofestablishing a measure of classicality is still lacking an unique solution.

Closely related to this topic is the problem of establishing a clear relation between the quan-tum initial state and the classical like behavior. Very few studies elaborate on this subject. Instead,one typically assumes a particular quantum initial state and works from there to prove the ‘‘clas-sicality’’ of the quantum dynamics. A weak point in the procedure is that we are still lacking aprecise definition of ‘‘classicality.’’ For instance in Refs. 10, 14, and 16 the aim is to provideminimal bounds for the spreading of the time evolution of a specific quantum initial data~typicallya coherent or squeezed state!. But the questions that remain are: are these results extendable to awider class of quantum initial states, or more generally, what is the relation between the spreadingof the wave function through time and the quantum initial state? And further, how is the spreadingof the wave function related to classicality?

In this paper we attempt to provide a partial answer to these questions. In general terms westudy the inverse problem of Refs. 10, 14, and 16: we start by imposing the bounds~which will berelated to the error margins of the classical time evolution! and attempt to identify those initialquantum states which display a time evolution that satisfies these bounds for all times. In doing sowe identify a~possible! set of general conditions an arbitrary, finite dimensional quantum systemshould satisfy so that it evolves in agreement with the classical predictions. These conditions willbe of the form of a sequence of restrictions on the initial quantum state~the higher the classicalitythe more stringent the conditions! and provide the basic mathematical structure to construct ameasure of classicality.

More precisely our approach will be as follows: We work in the context of the standardCopenhagenformulation of quantum mechanics30–33 and consider a general dynamical systemwith N degrees of freedom. Its initial time configuration might be described by a set of quantuminitial data or, alternatively, by a set of classical initial data. The classical description is given bya set of valuesai

0 for a complete set of classical observablesai ( i 51¯ 2N), together with theirassociated error marginsd i . The quantum description is given by the initial time wave functionuc&. Classical mechanics states that the output of a measurement of an observableai will belong tothe interval@ai

02d i ,ai01d i #. Quantum mechanics, in turn, states that there is some probability

pi<1 that a measurement of the observableai yields a value inside the former interval. Clearlythe two statements are not equivalent. However, there are a number of fairly intuitive ways bywhich one can measure the consistency~i.e., the agreement! of the two former descriptions of theinitial time configuration. We will propose two suchconsistency criteriain Sec. II.

Unfortunately, the fact that the two descriptions of the initial time configuration are consistentup to some degree does not imply that the classical and the quantum descriptions of a futureconfiguration will also be consistent up to the same degree. Our aim is then to identify theconditions that should be satisfied by the initial time configuration so that the degree of consis-tency is preserved through time evolution. These conditions are obtained~they are given by a setof relations between the classical initial data (ai

0 ,d i) and the spreads of the initial wave functionuc&! and used to construct two alternative classicality criteria. Given the classical initial data theseconditions constitute a sequence of growing restrictions on the functional form of the initial datawave function. Whenuc& satisfies then first former conditions we say thatuc& is n-order classical.When uc& satisfies the full set of conditions we say thatuc& is a classical limit initial data wavefunction.

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The main property of the classicality criteria is that if the classical and the quantum initial datasatisfy the classicality criteria up to some degree then the time evolution of the classical initial data~obtained using classical mechanics! and the time evolution of the quantum initial data~obtainedusing quantum mechanics! will display a minimum degree of consistency for all times. Hence, theclassicality criteria supply a suitable~although clearly not unique! measure of classicality thatmight be computed, for an arbitrary dynamical system, at the kinematical level.

These results are not limited to some specific type of systems displaying a particular dynami-cal behavior or having some particular quantum initial data. On the contrary, for an arbitrarydynamical system the classicality criteria can be used to determine the set of quantum initial datathat displays an-order classical time evolution~a quantum time evolutionn-order consistent withthe classical predictions!. This will be done explicitly for the two physical examples presented inSecs. VI and VII.

Further applications of the criteria are given in Refs. 34, 35, and 36.

II. CONSISTENCY CRITERIA

Let us consider a general dynamical system withN degrees of freedom. The phase spaceT* Mhas the structure of the cotangent bundle of the configuration spaceM , and henceforth will be alsoassumed to have the structure of aflat sympletic manifold. Therefore a global Darboux chart canbe defined inT* M . Let us choose a set of canonical variablesai ,i 51¯2N ~where ai5qi ,i51¯N and ai5pi 2N ,i 5N11¯2N). They yield the sympletic structure as the 2-formw5dqi∧dpi .

The classical description of a specific time configuration of the system is given by a set ofvaluesai

0 for the complete set of observablesai together with the associated error marginsd i . Theclassical statement is that a measurement ofai will yield a valueai

1 belonging to the classical errorinterval I i5@ai

02d i ,ai01d i #.

Alternatively, quantum mechanics describes the same configuration of the system with a wavefunction uc& belonging to the physical Hilbert spaceH and the quantum statement is that there issome probabilityp(ai)5(kuâi ,kuc&u2 that a measurement of the observableai yields the valueai ~where the statesuai ,k& form a complete set of eigenvectors ofai that spans the Hilbert spaceH, ai are the associated eigenvalues andk is the degeneracy index!. For notational conveniencewe will assume thatA displays a discrete spectrum, but all results can be easily rewritten for thecontinuous case.

A straightforward way of measuring the consistency between the two descriptions is given bythe following criterion.

Definition: First consistency criterion.Let us calculate the probabilitiespi—generated by thewave functionuc& in the representation of each of the observablesai , i 51¯2N—that a measure-ment of ai yields a value belonging to the classical error intervalI i5@ai

02d i ,ai01d i #, i.e.,

pi5 (aiPI i ,k

uâi ,kuc&u2. ~1!

The minimum value of the set$pi : i 51¯2N% provides a suitable measure of the consistencybetween the classical and the quantum descriptions of the configuration of the dynamical system.If this minimum value above is for instancep0 we say that the classical and the quantum descrip-tions arep0-consistent. Furthermore, ifp051 then the two descriptions are fully consistent anduc& is named aclassical limit wave function. h

Another, probably less intuitive, consistency criterion is given by the following definition:Definition: Second consistency criterion.Let 0<p,1 be an arbitrary probability and letM

PN. To each classical observableai we associate the set of intervals of the form:

5884 J. Math. Phys., Vol. 43, No. 12, December 2002 Nuno Costa Dias


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I i~p,M !5Fai02

d i

~12p!1/(2M ) ,ai01

d i

~12p!1/(2M )G . ~2!

In each of the former intervals we can calculate the probabilitypi generated by the wave functionuc&, in the representation of the corresponding quantum observableai :

pi~p,M !5 (aiPI i (p,M ),k

uâi ,kuc&u2. ~3!

For given values of the classical and quantum dataai0 , d i , anduc& this probability is an exclusive

function ofp andM . The second consistency criterion is defined as follows: The classical and thequantum data, describing a given configuration of the dynamical system, areM -order consistentifand only if for all 0<p,1 and for alli 51•••2N the conditionpi(p,M )>p is satisfied, i.e.,

(aiPI i (p,M ),k

uâi ,kuc&u2>p, ;pP[0,1[ ,; i 51•••2N , ~4!

where I i(p,M ) is given by ~2!. If M5` the classical and the quantum descriptions are fullyconsistent, in which caseuc& is a classical limitwave function. h

The former criterion provides a measure of how peaked is the wave function—in the repre-sentation of each of the quantum observables—around the classical error margin of the corre-sponding classical observable. This will became clear in the sequel~16!.

As a first remark let us point out that the two alternative definitions of a classical limit wavefunction ~using the first and the second consistency criterion! are, in fact, equivalent, i.e.p51⇔M51`, which in turn, is equivalent to the statement that

uc&5 (aiPI i ,k

âi ,kuc&uai ,k&, ; i 51•••2N , ~5!

where, as beforeI i5@ai02d i ,ai

01d i #, (ai0 ,d i) is the classical data anduai ,k& is the general

eigenstate of the operatorai .As a second remark let us notice that in some cases it is not possible to construct a wave

function satisfying Eq.~5! exactly. One should keep in mind that Eq.~5! stands for the limitM→` or p→1 which may, or may not admit an explicit realization in terms of some wave functionuc&.

Some emphasis will be put in the study of the time evolution of a general classical limit initialdata wave function. Although, for some systems, we might be unable to writeuc& @satisfying~5!#explicitly, its time evolution can be obtained, in the Heisenberg picture, using the standard rules ofquantum mechanics. This study will prove to be useful allowing us to develop a set of techniquesthat will later be used to obtain the classicality criteria and, in the sequel, to study physical systemswith general initial data.

III. ERROR KET FRAMEWORK

Let us introduce dynamics by specifying a general HamiltonianH. The quantum time evolu-tion of an arbitrary fundamental observableA ~with A5ai for somei 51¯2N) is given by

A~ t !5 (n50

`1

n! S t

i\ D n

@ ¯ @@A,H#,H# ¯ #. ~6!

Alternatively, the standard classical treatment of the same system provides the predictions



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A~ t !5 (n50

`tn

n!$ ¯ $$A,H%,H% ¯ %,

dA~ t !5 (n51

1`1

n! (k1 , . . . ,kn51

2N U ]nA~ t !

]ak1. . . ]akn

Udk1¯ dkn

, ~7!

where the error marginsdkiare taken at the initial time. Notice that, in general, it is possible to

obtain more accurate predictions than those of~7!, i.e., with smaller error margins. Such estimateshave the disadvantage of requiring a case by case evaluation, while Eq.~7! is valid in general.

To avoid some possible misunderstandings we will, from now on, focus on the case of thesecond consistency criterion. In Sec. V our results will then be rewritten using the language of thefirst consistency criterion. Let us then assume that the classical and the quantum initial data areM -consistent. The question we would like to answer is then: which are the conditions that shouldbe satisfied so that the quantum description of the system at a timet—given by the initial datawave functionuc& in the representation of the observablesai(t) ~6!—is alsoM -consistent with theclassical description given by~7!?

This is not an easy task. We start by presenting a framework that will later be used to give aprecise answer to this question.

A. Definition and properties of zE‹ and D

Let us start by introducing the relevant definitions. LetA be an operator acting on the quantumHilbert spaceH. Let ua,k& be a complete set of eigenvectors ofA, with associated eigenvaluesaandk being the degeneracy index. This set forms a complete orthogonal basis ofH. Finally, let uc&be the wave function describing the system.

Definition: Error Ket.We define thenth-order error ketuEn(A,c,a0)&, as the quantity

uEn~A,c,a0!&5~A2a0!nuc&, ~8!

wherenPN, a0PC andA does not need to be self-adjoint. The error braÊn(A,c,a0)u is definedaccordingly to the definition of the error ket.

Let now Az , z51,...,n be a set of operators acting onH. For each value ofz let uaz ,kz& be acomplete set of eigenvectors ofAz , with eigenvaluesaz , and kz being the degeneracy index.Moreover let az

0PC. The nth-order mixed error ketuE(A1 ,A2 , . . . ,An ,c,a10 ,a2

0 , . . . ,an0)&, is

defined as the quantity

uE~A1 ,A2 , . . . ,An ,c,a10 ,a2

0 , . . . ,an0!&5~An2an

0! ¯ ~A22a20!~A12a1

0!uc&. ~9!

When there is no risk of confusion we will use the short notationsuEn& or uEAn& for the nth-order

error ket anduEA1 ,A2 , . . , An& for the mixed error ket. h

We shall now study some of the properties of this quantity.~a! Explicit form of the error ket. Let us start with the 1st-order error ket. We have:

uE(A,c,a0)&5(a,k(a2a0)â,kuc&ua,k&. This result is easily extended to the case of thenth-order mixed error ketuEA1 ,A2 , . . ., An

&:

uEA1 ,A2 , . . ., An&5 (

a1 ,k1(

a2 ,k2

. . . (an ,kn

~a12a10!~a22a2

0! ¯ ~an2an0!â1 ,k1uc&â2 ,k2ua1 ,k1&

3â3 ,k3ua2 ,k2& . . . ân ,knuan21 ,kn21&uan ,kn&, ~10!

and if A15A25 ¯ 5An5A we haveuEA1 ,A2 , . . ., An&5uEA

n& which give us the explicit expres-sion for thenth-order error:



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uEAn&5~A2a0!nuc&5(

a,k~a2a0!nâ,kuc&ua,k&. ~11!

~b! Relation between the nth-order error ket and the mean second-order deviation. Let uscalculate the value ofEA

n uEAn&:

ÊAn uA

n&5(a,k

~~a2a0!* ~a2a0!!nuâ,kuc&u2, ~12!

and if A is self-adjoint anda05^cuAuc& thenÊAn uEA

n& is just the mean 2nth-order deviation ofA:(DA)2n ~if n51 thenÊA

1 uEA1& is just the mean square deviation!.

~c! Nth-order error ket of a classical limit initial data wave function. Let us consider the casein which the initial time configuration of the dynamical system is described by a classical limitinitial data wave functionuc&. Let A be a fundamental observable of the system and let theclassical initial data associated toA be given bya0 with error margindA . From Eqs.~5! and~12!it is straightforward to obtain the relation:

ÊAn uEA

n&5(a,k

ua2a0u2nuâ,kuc&u2<dA2n(

a,kuâ,kuc&u25dA

2n , ~13!

which is valid for allnPN and for all fundamental observablesA5ai( i 51¯2N).~d! Probabilistic distribution function from the nth-order error ket. Let A be self-adjoint, let

uc& be the state of the system and leta0 be a real number. GivenEn(A,c,a0)uEn(A,c,a0)&, toeach ‘‘quantity of probability’’ 0<p,1 we can associate an intervalI n around a0, I n5@a0

2Dn ,a01Dn#, such that the probability of obtaining a valueaPI n from a measurement ofA isat leastp. The range of the intervalI n is dependent ofA,c and a0 only through the value ofÊA

n uEAn&. The quantityDn(ÊA

n uEAn&,p) is named thenth-order spreadof the wave function. Let us

show that ifDn5Dn(A,c,a0,p) is given by

Dn~A,c,a0,p!5S ÊAn uEA

n&12p D 1/2n

~14!

then the probability of obtaining a valueaPI n from a measurement ofA is at leastp. From~12!,~14! we have

~Dn!2n~12p!5(k,a

ua2a0u2nuâ,kuc&u2,

> (k,a¹I n

ua2a0u2nuâ,kuc&u2>~Dn!2n (k,a¹I n

uâ,kuc&u2, ~15!

and this implies:(k,a¹I nuâ,kuc&u2<12p, which is the result we were looking for. IfA is not

self-adjoint the former result can also be obtained, but in this caseI n is a ball of radiusDn in thecomplex plane.

~e! Corollary of (d). A straightforward consequence of the previous result is the following: Iffor some positive integerM the condition:

ÊM~ ai ,c,ai0!uEM~ ai ,c,ai

0!&<d i2M ~16!

holds for all i 51•••2N then the classical and the quantum descriptions@given by (ai0 ,d i ,i

51¯ 2N) and uc&, respectively# are, accordingly to the second consistency criterion,M -orderconsistent.



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B. Time evolution of zE‹ and D

The aim is now to determine thenth-order error ket and thenth-order spread associated to theoperatorA(t) given in ~6!, as a function of the error kets and spreads associated with the initialtime operators. The calculations might seem, in a first reading, complicated and cumbersome.Nevertheless, the final result will be simple and physically appealing.

1. Evolving the error ket

We will start by calculating the 1st-order error ket associated with the sum and with theproduct of two arbitrary operatorsA andB and with the product of an operator by a scalar. Let thestate of the system beuc&, let uE(A,c,a)& anduE(B,c,b& be the first-order error kets associated toA and B and leta,b,cPC.

Theorem: The first-order error kets associated to the operatorsA1B, cA, and AB are,respectively,

uE~A1B,c,a1b!&5uE~A,c,a!&1uE~B,c,b!&, ~17!

uE~cA,c,ca!&5cuE~A,c,a!&, ~18!

uE~AB,c,ab!&5auE~B,c,b!&1buE~A,c,a!&1uE~B,A,c,b,a!&. ~19!

Proof: For the sum of operators we have

uEA1B&5~A1B2~a1b!!uc&5~A2a!uc&1~B2b!uc&5uEA&1uEB&. ~20!

The proof of the product by a scalar and of the product of two operators follows the same lines,this time using the relations (cA2ca)5c(A2a) and

~AB2ab!5a~B2b!1b~A2a!1~A2a!~B2b!, ~21!

that when applied to the stateuc& provide the desired results. h

Let us now extend these results to the case of several products and sums of fundamentaloperators. LetA be a general hermitian operator displayed as a sum of multiple products offundamental operators and letA0 be the classical function that is functionally identical toA:

A5(i 51

n

ci Bi5(i 51

n

ci)j 51

m

xi j , A05(i 51

n

ciBi05(

i 51

n

ci)j 51

m

xi j , ~22!

where xi j is one of the fundamental operators (xi j P$ I ,q1 ,. . ., qN ,p1 , . . ., pN% whereN is thedimension of the classical system!, n andm are arbitrary positive integers,ci are complex num-bers andBi , Bi are multiple products of the fundamental observables and multiple products of thecorresponding classical observables, respectively. The map fromA to A0 will be namedunquan-

tization. Clearly, the procedure~22! is beset by order problems~i.e., if A is displayed in differentorders we get differentA0). The consequences of this will be discussed later on.

The aim now is to expand (A2A0) in terms of (xi j 2xi j ). The first step is to set (A2A0)5( i 51

n ci(Bi2Bi0). Using Eq.~21! we get

Bi2Bi05 xi1)

j 52

m

xi j 2xi1)j 52

m

xi j

5xi1S )j 52

m

xi j 2)j 52

m

xi j D 1)j 52

m

xi j ~ xi12xi1!1~ xi12xi1!S )j 52

m

xi j 2)j 52

m

xi j D , ~23!



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and using~21! once again to expand () j 52m xi j 2) j 52

m xi j ) in terms of (xi22xi2) and () j 53m xi j

2) j 53m xi j ) and so on, after using the relation~21! m times we will obtain

Bi2Bi05 (

L51

1`

(j 1, j 2 •••, j L51

m ]LBi0

]xi j 1. . . ]xi j L

~ xi j 12xi j 1

! . . . ~ xi j L2xi j L

! ~24!

where the sum inL can be truncated at themth term. The next step is to multiply each term iniby ci and after sum ini . We get

A2A05 (L51

1`

(i 51

n

(j 1, j 2 •••, j L51

m]LA0

]xi j 1. . . ]xi j L

~ xi j 12xi j 1

! . . . ~ xi j L2xi j L

!. ~25!

Notice that this is just a ‘‘Taylor expansion’’ of an operator around the classical observable withthe same functional form. The sums in the expansions~24! and ~25! are taken overj 1, j 2 ¯

, j L to preserve the order in which the operatorsxi j appear inBi since in general (xik2xik)( xi j

2xi j )Þ( xi j 2xi j )( xik2xik). Because of this the analysis of the expansion~25! is tricky. The orderin which the classical variables appear inA0 is relevant. For instance ifA5 pq2qp then A0

5pq2qp and this form, and notA050, should be the one used to calculate the partial derivativesin ~25!.

Let us now consider the operator (A)15( i 51n ci(P j 51

m xi j )1 obtained from a general hermitianoperatorA ~22! by a term by term symmetrization: (A)15( i 51

n ci(Bi)1 , where (Bi)1 is thecompletly symmetric operator obtained fromBi . In general (A)1ÞA but the classical observablesobtained fromA and (A)1 , given by ~22!, are identical, i.e.,A05(A)1

0 . For this operator theexpansion~25! might be written

~A!12A05 (L51

1`1

L! (i 51

n

(j 1 , j 2 ,..,j L51

m]LA0

]xi j 1. . . ]xi j L

~ xi j 12xi j 1

! . . . ~ xi j L2xi j L

! ~26!

Our next step is to understand under which conditions, if any, is the expansion~26! a validapproximation to the expansion~25!. To do this the first step is to define the unquantization mapmore precisely.

Definition: Unquantization V0 . Let A(H) be the algebra of linear operators acting on thephysical Hilbert spaceH and letA(T* M ) be the algebra of complex functions over the classicalphase spaceT* M . V0 is the map:

V0 :A~H!→A~T* M !; A05V0~A!, ~27!

that satisfies the followingrequirements.~a! The action ofV0 on a fundamental operator provides the corresponding classical funda-

mental observable:V0(qi)5qi andV0( pi)5pi , i 51¯N. MoreoverV0( I )51.~b! The action ofV0 on a general operatorA, displayed in an arbitrary order, is given by

V0(A)5V0(AR) whereAR5A but displayed in an order in which~i! A is the sum of an hermitianterm with an anti-Hermitian term and~ii ! all the commutators present inA have been resolved (AR

does not contain antisymmetric terms!.If A is displayed in the required order,AR then~c! V0 is linear, i.e., ifAR5bB1cC thenV0(AR)5bV0(B)1cV0(C); b,cPC.~d! The product rule is valid: ifAR5¯1BC1¯ then V0(AR)5V0(¯)1V0(B)V0(C)

1V0(¯). h

One should notice that the mapV0 is not equivalent to the procedure~22!. However, for ageneral operatorA, all the classical observablesV0(A) can also be obtained using the procedure



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~22! ~by displaying A in appropriate order!. Therefore all the results that were valid forA0

obtained in~22!, are also valid forA05V0(A). Further properties of the mapV0 will be discussedin Sec. IV.

Let us proceed with the analysis of expansions~25! and~26!. We realize that ifA is displayedin the required orderAR then the difference betweenA and (A)1 is, at the most, proportional toa factor of\2. SinceA05V0(A) is identical toV0((A)1) we conclude that ifA05V0(A) then thedifference between the right hand sides of equations~25! and~26! is at the most proportional to afactor of \2 which, in the context of the results of this paper is of negligible magnitude. Inconclusion, ifA05V0(A) then the right-hand side of~26! is a valid approximation toA2A0.

Let us proceed: sincexi j —in the expansion~26!—is one of the 2N fundamental operators theformer expansion can be cast in the form

A2A05 (L51

1`1

L! (j 1 , . . . ,j L51

2N]LA0

]aj 1,. . .,]aj L

~ a j 12aj 1

! . . . ~ a j L2aj L

!. ~28!

Finally, if we apply this expansion to the quantum stateuc&, we get

uE~A,c,A0!&5 (L51

1`1

L! (j 1 , . . . ,j L51

2N]LA0

]aj 1. . . ]aj L

uEaj 1 , . . . ,ajL&. ~29!

The generalization of the former set of results to the case of themth-order error ket,uEAm&5(A

2A0)muc& can be obtained by exponentiating the expansion~28! to the mth power. Up to thelowest order we get

uEAm&5~A2A0!muc&5 (

k1 , . . . ,km51

2N S )i 51

m]A0

]aki

D uEak1 , . . . ,akm&, ~30!

which is our final result concerning the error ket of an operatorA functional of the fundamentaloperators. Notice that the former results are valid in general, irrespectively of the specific func-tional form of the wave functionuc&.

2. Evolving Dm

We shall now concentrate on the case of a system with a classical limit initial data. The aimis to calculate, in the representation ofA ~22!, the value of themth-order spread of a wavefunction uc& satisfying~5!. To do this the main point is to calculate the norm of a general error ket:Êx1 , . . . ,xn

uEx1 , . . . ,xn&, wherex1 , . . . ,xn is an arbitrary sequence of fundamental observables:

xiP$q1 ,¯ ,qN ,p1 ,¯pN%,i 51¯n,nPN. The following theorem will do this.Theorem: If uc& is a classical limit initial data wave function then the norm of the error ket

uE( x1 ,..,xn ,c,x10 ,..,xn

0)& satisfies the following relation:

Êx1 , . . . ,xnuEx1 , . . . ,xn

&<d12 . . . dn

2 , ~31!

where (xi0 ,d i),i 51¯n is the classical initial data associated to the observablexi .

Proof: This result will be proved by induction:~i! For n51 andx1 an arbitrary fundamental observable Eq.~5! immediately implies~result

~3.1c!: Êx1uEx1

&<d12.

~ii ! For an arbitraryn we use~10! and write

Êx1 , . . . ,xnuEx1 , . . . ,xn

&5(s,xn

Êx1 , . . . ,xn21uxn ,s& ^xn ,suEx1 , . . . ,xn21

&u~xn2xn0!u2, ~32!



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wherexn are the eigenvalues of the operatorxn , with degeneracy indexs. We want to show thatfor all eigenvaluesxn¹I n , whereI n5@xn

02dn ,xn01dn#, we have

^xn ,suEx1 , . . . ,xn21&5^xn ,su)

i 51

n21

~ xi2xi0!uc&50, ;xn¹I n

. ~33!

To prove this let us expand the wave functionuc& using a second set of eigenstates ofxn ,

^xn ,su)i 51

n21

~ xi2xi0!uc&5 (

xn8PI n ,s8^xn ,su)

i 51

n21

~ xi2xi0!uxn8 ,s8&^xn8 ,s8uc&, ~34!

wherexn8 are eigenvalues ofxn with degeneracy indexs8. Notice that in the representation ofxn

the wave function is completely confined to the intervalI n . To prove the result~33! is thensufficient to show that

^xn ,su)i 51

j

xi uxn8 ,s8&50, ;xn¹I n ,xn8PI n

,; j ,n . ~35!

Let us then prove the former identity

^xn ,su)i 51

j

xi uxn8 ,s8&51

xn8^xn ,su)

i 51

j

xi xnuxn8 ,s8&51

xn8^xn ,suxn)

i 51

j

xi1F)i 51

j

xi ,xnG uxn8 ,s8&

~36!

and if ^xn ,su@P i 51j xi ,xn#uxn8 ,s8&50 then

^xn ,su)i 51

j

xi uxn8 ,s8&51

xn8^xn ,suxn)

i 51

j

xi uxn8 ,s8&5xn

xn8^xn ,su)

i 51

j

xi uxn8 ,s8&. ~37!

SincexnÞxn8 Eq. ~37! immediately implies~35! and thus the result~33! is valid. The problem isnow reduced to prove thatxn ,su@P i 51

j xi ,xn#uxn8 ,s8&50, which in turn, and using the sameprocedure, will be reduced to prove that^xn ,su@@P i 51

j xi ,xn#,xn#uxn8 ,s8&50, and so on until weobtain at the most aj -commutator which will always have the value zero~notice thatxi arefundamental operators!. Inserting the result~33! into ~32! we get

Êx1 , . . . ,xnuEx1 , . . . ,xn

&5 (s,xnPI n

Êx1 , . . . ,xn21uxn ,s&^xn ,suEx1 , . . . ,xn21

&u~xn2xn0!u2

<Êx1 , . . . ,xn21uEx1 , . . . ,xn21

&dn2 , ~38!

which proves the theorem. h

A straightforward corollary of the former result is the one obtained by using the Schwartzinequality:

uÊx1 ,...,xnuEy1 ,...,ym

&u<dx1¯dxn

dy1¯dym

, ~39!

wherex1 ,...,xn andy1 ,...,ym are two arbitrary sequences of fundamental observables.Let us return to the calculation of them-order spread of the wave functionuc&, satisfying~5!,

in the representation ofA. The relevant calculation is that ofÊAmuEA

m&. From Eqs.~28! and~39!,we get



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ÊAmuEA

m&5^cu~A2A0!2muc&<S (L51

1`1

L! (j 1 , . . . ,j L51

2N U ]LA0

]aj 1. . . ]aj L

Ud j 1. . . d j LD 2m

5dA02m ,

~40!

a result that is valid for allmPN. Them-order spread then reads

Dm~A,c,A0,p!5S ÊAmuEA

m&12p D 1/2m

<dA0

~12p!1/2m , ~41!

which, for all p,1 converges to the classical error margindA0 asm→`.From the results~3.1d! and~41! we can draw the conclusion that the classical limit initial data

wave functionuc&, in the representation of an arbitrary observableA, is completely confined to aninterval aroundA0 with the range of the classical error margin ofA0, the relation betweenA andA0 being the one given by~27!.

IV. UNQUANTIZATION

In the last section we proved that the wave functionuc&—satisfying~5!—in the representationof a given quantum operatorA, which in the end is to be identified withA(t) given in ~6!, iscompletely confined to an interval centered at the value of the classical observableA0 and with therange of the classical error interval ofA0. This implies that the output of a measurement ofA,performed with an experimental apparatus of any resolution, will certainly belong to the previouserror interval, which is exactly the classical prediction for the output of a measurement of theclassical observableA0.

However these are not our final results, yet. The reason is straightforward: LetA5A(t), whatremains to be proven is simply thatA05V0(A(t))5A(t), i.e., that the classical observableA0,obtained from the quantum observableA(t) using the mapV0 , coincides with the classicalobservableA(t) obtained by evolvingA(0)5V0(A(0)) using the classical theory.

Hence the relevant questions are: Is the mapV0 well defined? And will it map a quantumobservableA(t) to the classical observableA(t) given by ~7!?

Starting with the first question it is easy to see that the mapV0 is not well defined. In generalA can be displayed in several different functional forms~all of them satisfying the order require-ment!, each of which will be mapped byV0 to adifferent~however very similar! classical observ-able. A simple example will elucidate this point: letA51/2(xyz1 zyx) with x5q1p2 , y5 p1 andz5q1q2 whereq1 , p1 , q2 , p2 are the canonical variables of a two dimensional system. Alterna-tively, A might be written asA51/4(xyz1 zyx1 xzy1 yzx)11/4@ x,@ y,z##. The first form ofA ismapped byV0 to the classical observableA05xyz while the second form is mapped to theobservableA05xyz21/4\2q1 . Moreover for each different classical observable obtained we will,in general, also get a different associated error ket and error margin. This does not mean that theformer results~30!, ~41! concerning the error ket and the spread of the wave function are incorrect.These results have been proved to be valid~up to a correction term proportional to\2) for alldifferent orders we may choose for the operatorA, and consequently for all differentA0 obtainedfrom A, providing the order requirement is satisfied.

Notice however that this ambiguity would be problematic if the difference between twodifferent classical observables, obtained from a single quantum one, had meaningful values. How-ever, one can easily realize that ifA1

0 andA20 are two such [email protected]., A1

05V0(A) and alsoA2

05V0(A)] then A102A2

0 is proportional to a factor of, at the most\2. An imprecision of thismagnitude is not meaningful when compared to the errors associated to the classical observables.That is the predictionsA1

0 and A20 are well within the error interval of each otheruA1

02A20u



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!dA10orA

20. Hence, the two predictions are consistent with each other and we conclude thatA1

0 and

A20 are equally valid candidates for a classical description of the quantum observableA.

This takes us to the second question, which now can be restated as: will the classical observ-ableA(t), given by~7!, be one of the images ofV0(A(t))?

To answer this question let us start by presenting a second proposal for the unquantizationmap:

Definition: Unquantization V. Using the notation of the previous definition we define the newunquantizationV to be the map:

V:A~H!→A~T* M !, A5V~A!, ~42!

that satisfies the following requirement:V+∧51 where∧ is the Dirac quantization map.32,37Thatis V is the inverse of the Dirac quantization map∧. The properties ofV follow immediately fromthe properties of∧:

~a! V(qi)5qi , V( pi)5pi , i 51¯N, andV( I )51.

~b! V(1/i\@A,B#)5$V0(A),V0(B)% for all A and B.

For a general operatorA displayed in the required order~see definition ofV0):

~c! V is linear: if A5bB1C thenV(A)5bV(B)1V(C), bPC.

~d! The product rule is valid: ifA5¯1BC1¯ thenV(A)5V(¯)1V(B)V(C)1V(¯). h

We should point out that, since the Dirac quantization map∧ is not injective, the unquanti-zation mapV is also nonunivocous. The simple example above—A51/2(xyz1 zyx)—can also beused here to make this point clear. Being beset by the same type of order problems stillV displaysan important advantage overV0 : it is straightforward to recognize that whenV is applied to theoperatorA(t)—given by ~6!—yields the classical observableA(t):

V~A~ t !!5VF (n50

`1

n! S t

i\ D n

@¯@A,H#,H#¯G5 (n50

`tn

n!$¯$A,H%,H%¯%. ~43!

Let us now consider a general operatorX. We want to prove thatV(X) provides a set of classicalobservables that is included in the setV0(X). Clearly the action of the two maps on an operatorthat does not contain antisymmetric components is identical. Moreover the two commutationrelations 1/i\@ ,# and$,% have ‘‘compatible’’ algebraic structures, that is we can resolve 1/i\@A,B#and express the result in such an order that when we perform the substitution of the quantumobservables by the corresponding classical ones~i.e., when we unquantize 1/i\@A,B# using themapV0) we get exactly the same final result as if we just compute$A,B% ~i.e., as if we use themap V to unquantize 1/i\@A,B#). Thus, if X contains antisymmetric components the classicalobservableV(X) might be obtained by displayingX in an adequate order and calculatingV0(X).Hence, it is always possible to obtain the classical observablesV(X) using the mapV0 . This is theresult we were looking for. It means that expansion~28! and thus all subsequent results are validfor A05V(A).

Finally, we are able to state that the quantum mechanical predictions for an arbitrary dynami-cal system with a classical limit initial data wave function are that a measurement of an arbitraryfundamental observableA(t) at the timet will yield the value given by~43! with an error margingiven by ~41! ~with m→`). These are exactly the predictions of the classical mechanical treat-ment of the same system.



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V. CRITERIA OF CLASSICALITY

In the last section we proved that a dynamical system with a quantum initial data satisfying~5!will evolve exactlyaccordingly to the predictions of classical mechanics, that is the system is`-consistent for all times. The main interest of this result is formal since, in most cases we areunable to provide a wave function satisfying the classical limit criterion~5!.

To proceed we will now study dynamical systems with general physical initial data. The aimis to use the formalism developed in the previous sections to derive two criteria providing ameasure of the degree of classicality of an arbitrary quantum system with arbitrary initial data.

Let then uf& be the initial data wave function of aN dimensional dynamical system withcanonical variablesak , (k51¯2N). Let Ski

5(ak1,ak2

, . . . ,akn), kiP$1¯2N% be a sequence of

fundamental operators associated to then-terms sequenceki ( i 51¯n;nPN). The relevant quan-tities that we have to calculate will be then-order mixed error kets associated to the sequencesSki

:

uESki&5~ ak1

2ak1

0 !~ ak22ak2

0 !¯~ akn2akn

0 !uf&, ~44!

whereaki

0 is the classical initial value of the canonical variableaki, i.e., aki

0 5aki(t50).

A. First criterion

The first step is to obtain the time evolution of the canonical variables using the standardclassical formulation of the system

aj~ t !5 (m50

`tm

m!$ . . . .$aj ,H% . . . .,H%5F j~ak ,t !. ~45!

We then consider the sequencesSkisuch that

]F j

]Ski

5]nF j

]ak1]ak2

¯]akn

Þ0, ~46!

for at least onej 51¯2N. For each of these sequences we construct the associated error ket~44!.This way we obtain a set of error kets. The first order classicality criterion is given by thefollowing conditions:

ÊSkiuESki

&<~dSki!25~dk1

dk2¯dkn

!2, ~47!

where the inequalities should hold for all the sequences determined in~46!. In ~47! dkiis the initial

data classical error margin associated to the classical observableaki, i 51¯n. If the initial data

wave function satisfies the former conditions we say that the dynamical system isfirst orderclassical.

To proceed we construct the sequencesSki

M formed byMPN original sequencesSki, that is

Ski

M5Ski,Sk

i8, . . . ,Sk

i9, where each of theM former sequencesSki

might be any of the ones

determined in~46!. Once again, we calculate the values ofÊSkiMuES

kiM& and compare them with

(dSkiM)2. If the initial data satisfies:

ÊSkiMuES

kiM&<~dS

kiM!2, ~48!

for all possible sequencesSki

M we say that the dynamical system isM -order classical.

Let us make two remarks: the first one to say that the classification of a given initial data asM -order classical isdependenton the scales~error marginsdk) that characterize the classicaldescription. In particular fordk5` all dynamical systems will be -order classical~and will havea classical limit initial data! while for scales smaller than the Planck scale none will even be



169.230.243.252 On: Thu, 18 Dec 2014 02:48:03

first-order classical. The second remark is to point out that if a system isM -order classical then@result~3.1d!# its initial data wave function, in the representation of any of the observablesak(0),will have a minimum probabilityp in the interval I k5@ak(0)2dk /(12p)1/(2M ),ak(0)1dk /(12p)1/(2M )#, whereak(0) is the value of the classical observableak at the initial time,t50. Thatis the initial data isM -order consistent. This is because in~46! we always determine the 2N singlevalue sequencesSk1

5ak , (k51¯2N).Furthermore, if the classical and the quantum initial data satisfy the inequalities~48! then we

can substituteESkiMuES

kiM& by (dS

kiM)2 when computing~40! for A05aj (t). The result~41! can then

be easily obtained being valid up to the orderm5M . Hence, using the result~3.1d! we can statethat an arbitraryM -order classical system, as defined above, will evolve in such a way that in therepresentation ofa j (t) ~for all j 51•••2N and for all t) the initial data wave function has at leasta probability p confined to the intervalI j (t)5@aj (t)2d j (t)/(12p)1/(2M ),aj (t)1d j (t)/(12p)1/(2M )# with aj (t) andd j (t) being the classical time evolution of the canonical variableaj andassociated error margin. According to the second consistency criterion this means that the classicaland the quantum data describing the system at the timet areM -order consistent.

We conclude that ifM is the order of classicality of a given dynamical system then: first theclassical and the quantum data describing the initial time configuration of the system areM -orderconsistent and, second the degree of consistency is preserved through the time evolution.

Finally, notice that the higher the order of classicalityM of the dynamical system the moresimilar will be the range ofI j (t) and the classical error margin. WhenM goes to infinity we obtainthe classical limit description of the system.

B. Second criterion

A second classicality criterion can be easily devised. Once again the first step is to calculatethe time evolution of the canonical variablesaj using the classical formulation of the theory~45!.Again we use this result to obtain the sequences of fundamental operatorsSki

~46! and theassociated error kets~44!. Using these error kets we can write the second classicality criterion:

ÊSkiuESki

&<~dSki!2~12p0!5~dk1

dk2. . . ..dkn

!2~12p0!, ;Skiin ~46!. ~49!

If the classical and the quantum initial data satisfy the former inequalities for a givenp0 (0<p0,1) and for all sequences determined in~46!, then we say that the dynamical system isp0-order classical. A straightforward use of the result~3.1d! will prove that ap0-order classicalsystem is described by an initial data wave function that, in the representation of any of itsfundamental observablesak(t50), k51¯2N, has at least a probabilityp0 confined to the clas-sical error interval@ak(0)2dk ,ak(0)1dk#. That is the initial time configuration isp0-orderconsistent~according to the first consistency criterion!.

Now, let us concentrate on the dynamical evolution of the former initial data. If the inequali-ties ~49! are satisfied we can perform the substitution ofÊSki

uESki& by (dSki

)2(12p0) whencomputing~40! for m51. Notice that there will be an extra factor of (12p0) in the final expres-sion in ~40!. We can then proceed and obtain the result~41! for D1(a j (t),f,aj (t),p) @which stillcontains the extra factor (12p0)1/2]. More precisely we makep5p0 in ~41! and get for all j51¯2N:

D1~ a j~ t !,f,aj~ t !,p0!5S Êaj (t)uEaj (t)

&

12p0D 1/2

<1

~12p!1/2 (L51

1`1

L! (k1 , . . . ,kL51

2N U ]Laj~ t !

]ak1. . . ]akL

Udk1¯dkL

~12p0!1/25d j~ t !.

~50!



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The previous result together with~3.1d! implies that in the representation ofa j (t) ~for all j51¯2N) the initial data wave function has at least a probabilityp0 confined to the classical errorinterval @aj (t)2d j (t),aj (t)1d j (t)#, i.e., the classical and the quantum descriptions of the con-figuration of the system at the timet arep0-order consistent. This result is valid for all times. Weconclude that if a dynamical system isp0-order classical then the classical and the quantumdescriptions of the configuration of the system arep0-order consistent for all times. Once again,whenp0→1 we obtain the classical limit description of the system.

VI. EXAMPLE—HARMONIC OSCILLATOR

To illustrate the use of the first classicality criterion~Sec. V A!, let us obtain the classicalityconditions for the simple example of the harmonic oscillator. The classical Hamiltonian is givenby H5 1

2(q21p2), whereq andp are a pair of canonical variables and, to make it simple we made

w5m51. Solving the equations of motion we obtain the classical time evolution of the canonicalvariables and the corresponding error margins

q~ t !5q~0!cost1p~0!sint, dq~ t !5ucostudq~0!1usintudp~0!,~51!

p~ t !5q~0!sint1p~0!cost, dp~ t !5usintudq~0!1ucostudp~0!.

Let uf& be the initial data wave function for the quantum harmonic oscillator. Let us then deter-mine the conditions thatuf& should satisfy so that the quantum system allows for a consistentM -order classical description. Following the general prescription of Sec. V A the first step is todetermine the fundamental sequences~46!. They are the single value sequences

S15q and S25p. ~52!

For aM -order classical system the relevant sequences are arrays ofM fundamental sequences:

SM5~z1 , . . . ,zM !, zi5q~p, i 51¯M , ~53!

and the condition ofM -order classicality~48! reads

ÊSMuESM&<dSM2 , ;SM in ~53!

⇔^fu~ z12z1~0!!. . . ~ zM2zM~0!!~ zM2zM~0!! . . . ~ z12z1~0!!uf&<dz1

2 ~0! . . . dzM

2 ~0!.

~54!

Given the classical initial data$q(0),p(0),dq(0),dp(0)%, Eq.~54! constitute a system of inequali-ties to be satisfied by initial data wave functionuf&. For a first-order classical system the classi-cality conditions take their simpler form

^fu~ q2q~0!!2uf&<dq2~0! E ~q2q~0!!2uf~q!u2dq<dq

2~0!⇔

^fu~ p2p~0!!2uf&<dp2~0! E ~p2p~0!!2uf~p!u2dp<dp

2~0!.~55!

To obtain explicit solutions we may want to consider Gaussian wave packets@notice however that,in general, there are many solutions of~55! which are not Gaussians#:

fc~q0 ,p0 ,Dq,q!51

~2p~Dq!2!1/4expH 2~q2q0!2

4~Dq!2 1 ip0q/\J , ~56!

whereq0 , p0 andDq are parameters and the wave function was displayed in theq representation.If we make q05q(0) and p05p(0) and substitutefc(q(0),p(0),Dq,q) in ~55! we get theequivalent set of inequalities:



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Dq<dq~0! ∧\

21/2Dq<dp~0!. ~57!

As expected, ifdp(0)dq(0),\/(21/2) there is no coherent state~and actually no wave function!that might satisfy the former inequalities, while for larger values of the classical error marginsthere are many solutions of~55!, including the coherent states with a parameterDq satisfying~57!.We see that the classicality criteria provide a comparative notion of classicality but not an absoluteone. In fact the degree of classicality is always relative to the classical description supplied.

The main result of the formalism is that a wave function satisfying~55!, and in particular acoherent state satisfying~57! displays a time evolution first order consistent with the classicalpredictions~51!, which means that for allt:

Eq(t)2 dq(t)/(12P)1/2

q(t)1 dq(t)/(12P)1/2

uf~q,t !u2dq>P, ∧ Ep(t)2 dp(t)/(12P)1/2

p(t)1 dp(t)/(12P)1/2

uf~p,t !u2dp>P, ;0<P,1 ,

~58!

where q(t), p(t), dq(t), dp(t) are given by~51!, uf(t)& is the solution of the Schro¨dingerequation:

i\]/]tuf~ t !&51/2~ q21 p2!uf~ t !&, uf~ t50!&5uf&,

and P is an arbitrary probability. Take for instanceP50.99, Eq.~58! states that 99% of theprobability of the wave functionuf(t)&, in both the representations ofq and p, is confined to theclassical intervals@q(t)210dq(t),q(t)110dq(t)# and @p(t)210dp(t),p(t)110dp(t)#, respec-tively. This statement is valid for all times and, for the case of coherent states can be easily verifiedby numerical computation of the integrals~58!.

To see what happens when the degree of classicality is increased let us consider a 10th-orderclassical system. In this case the initial data wave function should satisfy the conditions~54! for allM510 sequences~53!. If we consider Gaussian type solutions~56! and substitute~56! in ~54! weget the following set of inequalities for the parameterDq:

~2M21!!

2M21~~M21!! !~Dq!2M<dq

2M~0! ∧~2M21!!

2M21~~M21!! ! S \

21/2DqD 2M

<dp2M~0!, ~59!

whereM510. Clearly, for the same classical initial data the set of solutions of~59! is just a subsetof the set of solutions of~57!.

The solutions of~54! with M510 and in particular the coherent states satisfying~59!, displaya time evolution 10th-order consistent with the classical predictions~Sec. V A!. In particular forP50.99 we have (12P)1/2050.79 and thus

Eq(t)21.25dq(t)

q(t)11.25dq(t)

uf~q,t !u2dq>0.99 ∧ Ep(t)21.25dp(t)

p(t)11.25dp(t)

uf~p,t !u2dp>0.99, ~60!

whereuf(t)& is the quantum time evolution of a general initial data wave function satisfying~54!for M510. The result is valid for all times and can be checked explicitly for an arbitrary coherentstate satisfying~59!.

One of the most interesting properties of coherent states is that its quantum time evolution is~in some sense! ‘‘consistent’’ with the classical predictions. Because of this coherent states mightbe seen as ‘‘classical states.’’ We see that the classicality criteria provide a systematic procedure toobtain ‘‘classical states’’ for an arbitrary dynamical system: by solving the classicality conditions~54! we obtain a large set of classical states~much larger than the coherent states set! for which thequantum time evolution is consistent with the classical predictions. Moreover, and most important,the imprecise notions of ‘‘consistency’’ and of ‘‘classical states’’ are made fully precise in thisformalism.



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VII. FURTHER EXAMPLE

To further illustrate the use of the criteria let us consider the two-dimensional system de-scribed by the Hamiltonian:

H5P2

2M1

p2

2m1kQp2, ~61!

where (Q,P) are the canonical variables of a particle of massM , (q,p) the ones of the particle ofmassm and k is a coupling constant. The classical time evolution is obtained by solving theHamiltonian equations of motion which yield

Q~ t !5Q~0!1P~0!

Mt2

k

2Mp~0!2t2,

P~ t !5P~0!2kp~0!2t,~62!

q~ t !5q~0!1H p~0!

m12kQ~0!p~0!J t1

k

MP~0!p~0!t22

k2

3Mp~0!3t3,

p~ t !5p~0!

with error margins~taking into account all orders in the initial time error margins!:

dQ~ t !5dQ~0!1U t

MUdP~0!1Ukp~0!t2

M Udp~0!1U kt2

2MUdp2~0!,

dP~ t !5dP~0!1u2kp~0!tudp~0!1uktudp2~0!,

~63!

dq~ t !5dq~0!1US 1

m12kQ~0! D t1

kP~0!t2

M2

k2p~0!2t3

M Udp~0!1u2kp~0!tudQ~0!

1Ukp~0!t2

M UdP~0!1uktudQ~0!dp~0!1U kt2

2MUdP~0!dp~0!1Uk2p~0!t3

M Udp2~0!1Uk2t3

M Udp3~0!,

dp~ t !5dp~0!.

For this dynamical system the fundamental sequences~46! are~the indexes 1, 2, 3, and 4 refer tothe canonical variablesq, p, Q, andP, respectively!

S15q, S25p, S35Q, S45P,~64!

S225~p,p!, S235~p,Q!, S245~p,P!, S2225~p,p,p!

and the first order classicality condition reads

ÊSkiuESki

&<dSki

2 , ;Skiin ~64!. ~65!

To make the discussion simpler let us consider solutions of the formuj&5uc&uf&, whereuc& is thequantum state of the particle of massm, anduf& is that of the particle of massM . In the (q,Q)representation we havej(q,Q)5c(q)f(Q). To proceed we notice that if the condition~65! issatisfied for the sequencesS1 , S2 , S3 andS4 then is also satisfied forS23 andS24. Furthermore,if it is satisfied forS222 is also forS22 and S2 . Hence, the system of eight inequalities~65! isreduced to the system of four inequalities:ÊSki

uESki&<dSki

2 , SkiP$S1 ,S3 ,S4 ,S222%. Moreover, the

former system splits into two independent systems:



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^cu~ q2q~0!!2uc&<dq2~0! ^fu~Q2Q~0!!2uf&<dQ

2 ~0!and

^cu~ p2p~0!!6uc&<dp6~0! ^fu~ P2P~0!!2uf&<dP

2 ~0!.~66!

For Gaussian type solutionsjc(Q0 ,P0 ,q0 ,p0 ,DQ,Dq,q,Q)5fc(Q0 ,P0 ,DQ,Q)cc(q0 ,p0 ,Dq,q) it reduces to the form

Dq<dq~0! DQ<dQ~0!

and

~15!1/6\

21/2Dq<dp~0!

\

21/2DQ<dP~0!,

~67!

where we used Eq.~59! to obtain the second inequality of the first system. Any Gaussian wavefunction satisfying the former conditions displays a time evolution~solution of the Schro¨dingerequation:i\]/]tuj(t)&5Huj(t)&, uj(0)&5uj&) first order consistent with the classical predictions~62! and~63!. This means that for any of the canonical variablesZ5q,p,Q~P and for all times:

EwE

Z(t)2dZ(t)/(12P)1/2

Z(t)1dZ(t)/(12P)1/2

u^j~ t !uz,w&u2dzdw>P, ;0<P,1 , ~68!

where uz,w& is the general eigenstate of the observableZ with associated eigenvaluez anddegeneracy indexw, P is a probability and (Z(t),dZ(t)) are the classical time evolution of thecanonical variableZ and its error margin which are given by Eqs.~62! and ~63!, respectively.

Just like in the example of the harmonic oscillator we can increase the order of classicality ofthe system by requiring the initial data wave function to satisfy higher order classicality condi-tions. Likewise, this will increase the degree of consistency between the classical and the quantumpredictions.

VIII. CONCLUSIONS

A general procedure leading to the construction of a measure of classicality was presented.The procedure can be summarized in two main steps.

~1! Definition of a consistency criterion. This is a kinematical criterion. It should provide asuitable measure of the degree of agreement between the classical and the quantum descriptions ofa single, specific time configuration of the system. Two intuitive definitions of consistency criteriawere presented and certainly other, possible more interesting criteria might be defined.

~2! Using the consistency criterion the problem of studying the overall consistency betweenthe classical and the quantum descriptions of a general dynamical system is reduced to a simplerand more precisely formulated problem: that of identifying the conditions that should be satisfiedso that the degree of consistency between the classical and the quantum initial data is preservedthrough time evolution. These conditions were explicitly obtained and used to construct a classi-cality criterion for each of the consistency criteria proposed at the beginning. We expect that thesame general procedure might be used to derive the classicality criterion associated to otherproposals of consistency criteria.

Ultimately, the two measures of classicality obtained are arbitrary in nature since they are adirect consequence of the definition of a consistency criterion. Therefore, they should be tested inphysical examples to see if they yield sensible results. The criteria of this paper were applied totwo simple dynamical systems and yield expected results. Namely, Gaussian wave functions wereobtained as ‘‘classical-like states’’~solutions of the classicality conditions!.

Further applications of the criteria have been developed in Refs. 34, 35, and 36. Other pos-sible future applications which might be of considerable interest are in the field of nonlineardynamical systems, where the connection between classical and quantum mechanics becomesmore involved.

To finish let us make a few remarks.



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~i! The degree of classicality of a given quantum system is always relative to the classicaldescription supplied~more precisely, to the classical initial data supplied!. Therefore, the criteriado not provide an absolute classification of classicality but only a comparative one.

~ii ! The fully classical behavior~i.e., the complete consistency with the classical predictions!is obtained in the limit case in which the quantum system fully satisfies one of the classicalitycriteria to all orders or equivalently, if the quantum and the classical initial data are fully consis-tent. Notice that consistency and classicality, in this case and in general only in this case, areequivalent notions: the results of Sec. III B 2 imply this statement directly.

~iii ! The classicality conditions vary from one dynamical system to another. More precisely,once the classical initial data is given theM -order classicality conditions on the quantum initialdata are more restrictive for some systems~for instance, the second example! than for others~forinstance, the first example!. This is not surprising, as neither is the fact that for some dynamicalsystems, the classicality conditions are so restrictive that there is no wave function that, for typicalvalues of the classical data, might satisfy even the first order classicality conditions~consider forinstance the system of a particle in a potential step!.

~iv! The classicality criteria provide only a partial answer to the problem of measuring thedegree of classicality of a general dynamical system. We were able to prove that a generaldynamical system will evolve in agreement with the classical predictions up to some degree if itsinitial time configuration satisfies the classicality conditions up to some extent. However, these aresufficient but not necessary conditions: the system may very well not satisfy even the first orderclassicality criterion and still present a fairly classical-like evolution.

ACKNOWLEDGMENTS

I would like to thank Jorge Pullin for many suggestions and insights made through the presentwork. I would also like to thank Gordon Fleming, Don Marolf, Joaõ Prata, Troy Schilling, and LeeSmolin for several discussions and insights in the subject. This work was supported by fundsprovided by Junta Nacional de Investigaçao Cientıfica e Tecnolo´gica–Lisbon–Portugal, grant No.B.D./2691/93 and by Grants Nos. NSF-PHY 94-06269, NSF-PHY-93-96246, the Eberly Researchfund at Penn State and the Alfred P. Sloan foundation.

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