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arXiv:quant-ph/9504016v1 24 Apr 1995

The Classical Limit of Quantum Theory

R.F. Werner

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Abstract. For a quantum observable Ah ? ? depending on a parameter h we de?ne the notion “Ah ? converges in the classical limit”. The limit is a function on phase space. Convergence is in norm in the sense that Ah ? -wise product of convergent ? → 0 is equivalent with Ah ? → 0. The h observables converges to the product of the limiting phase space functions. h ? ?1 times the commutator of suitable observables converges to the Poisson bracket of the limits. For a large class of convergent Hamiltonians the h ? -wise action of the corresponding dynamics converges to the classical Hamiltonian dynamics. The connections with earlier approaches, based on the WKB method, or on Wigner distribution functions, or on the limits of coherent states are reviewed.

Physics and Astronomy classi?cation scheme PACS (1994): 03.65.Sq, 03.65.Db

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FB Physik, Universit¨ at Osnabr¨ uck, 49069 Osnabr¨ uck, Germany Electronic mail: reinwer@dosuni1.rz.Uni-Osnabrueck.DE 1

1. Introduction

The problem of taking the limit of quantum mechanics as h ? → 0 is as old as quantum mechanics itself. Indeed, under the name “correspondence principle” it was one of the important guidelines for the construction of the theory itself. Naturally, there is a vast literature on the subject, and it requires some justi?cation to add yet another paper to it. I will therefore begin by stating the aims of the present paper more carefully than usual, and proceed to review some of the existing approaches to the classical limit with regard to these aims. This will be done in a separate subsection of the introduction. In Section 2 and Section 3 we describe the basic notions of our approach. It is based on a set of ? . This framework “comparison maps” jh ?h ? ′ which relate observables at di?erent values of h was originally designed for applications in statistical mechanics [We3], and has many further conceivable applications. In Section 2 it is shown that this furnishes a language in which the convergence of sequences of observables, and the theorems of the desired type can be adequately expressed. The de?nition of the comparison maps jh ?h ? ′ requires some additional structure from phase space quantum mechanics, and is undertaken in Section 3. Section 4 gives an extensive list of examples and applications. We hope that this section especially will help to convince the reader that the present approach to the classical limit is a natural, if not canonical one. Section 5 contains the more technical aspects, including, of course, the proofs of the main results. Some of these technical points, notably the proofs of the theorems about convergence of commutators to Poisson brackets, and the convergence of dynamics were beyond the scope of a single journal article, and will therefore be treated in a separate publication [We5]. The concluding Section 6 contains previews of such further extensions, and also some remarks about how some simplifying assumptions (like the boundedness of Hamiltonians) can be relaxed. 1.1. Motivation and review of the literature There are basically two reasons for studying the classical limit. The ?rst is concerned with the architecture of theoretical physics, and demands the reconstruction of classical mechanics in terms of its supposedly more comprehensive successor. This “correspondence principle” was part of the supporting evidence for the new quantum theory. Now that this is hardly needed anymore, some theorists feel that there is no more reason to study the classical limit. Some physicists also seem to feel uneasy about the sacrilege of changing the value of the Fundamental Constant h ? = 1.0545887 ? 10?34 kg m2 /s (or h ? = 1 in more practical units). Are we free to do this without talking about a di?erent possible world of no relevance to our own? This leads to the second motivation for discussing the classical limit: it is seen mainly as a practical tool for the simpli?ed approximate evaluation of quantum mechanical predictions. In this interpretation a limit theorem says that the classical treatment is accurate (within certain bounds) as long as the relevant observables change su?ciently slowly relative to the phase space scale ?xed by h ? . The introduction of a changeable parameter h ? is then merely a convenient shorthand for this comparison. What makes it especially convenient is that the comparison parameter ? h will show up in all those places, where we are used to seeing the constant ? h in the textbooks. 2

For the mathematical formulation of the classical limit both readings amount to the same thing. The following are some of the features, which one might ask of a satisfactory explanation, and which the present paper aims to implement. (a) The limit should be de?ned for the whole theory, not of certain isolated aspects. That is, we should de?ne the limits of general states, observables, and expectation values, and these should go to their classical counterparts. (b) The de?nition should be conceptually simple and general. That is, it should be appropriate for inclusion in a basic course on quantum mechanics. It should not depend on the choice of a special (e.g., quadratic or classically integrable) Hamiltonian, or special (e.g., coherent) states. (c) It should be a rigorous version of accepted folklore on the subject. For example, the limit of ?? h2 /(2m)? + V (x) should be the Hamiltonian function p2 /2m + V (q ), and some intuition should be given, for what kinds of observables the classical approximation is sensible. (d) The limit should be in the strongest topology possible. We want the statement of the limit to be a equivalent to an asymptotic estimate of operator norms for observables and trace norms for states. These norms carry special signi?cance in the statistical interpretation of quantum theory, since they correspond to uniform estimates on probabilities. (e) In the limit, the product of bounded operators should become the product of functions on phase space. (f) In the limit, “i/? h times a commutator” should become the Poisson bracket of the limits. (g) The quantum mechanical time evolution should converge (uniformly in ?nite time intervals) to the classical Hamiltonian evolution. (h) Equilibrium states (canonical Gibbs states) and partition functions of quantum theory should converge to their classical counterparts.

On the other hand, we can distinguish in the literature the following approaches to the classical limit, each of which naturally has a considerable overlap of results and applications with the approach we are going to present. This list is necessarily incomplete, and no attempt has been made to evaluate the historical development of the subject, or to decide any priority claims. Nor can we adequately portray the merits of the di?erent schools since our perspective is limited to the comparison with the approach of the present paper. (A) The WKB method. [Mas,Sch,Hel,Fr¨ o,DH,BS] One virtue of this well-known approach is that it is so close to Schr¨ odinger’s beautiful series of papers establishing his wave mechanics. It fails mainly on item (a): the Schr¨ odinger equation is only one aspect of quantum mechanics, and its short wave asymptotics is only one aspect of the classical limit. For example, it seems hopeless to try to understand the opera3

tor properties (e) and (f) in WKB terms. The WKB wave functions do correspond to (a subclass of ) convergent states in our approach (see Section 4.8). Their limits are measures supported by Lagrangian manifolds in phase space, hence they have a curious intermediate position between point measures and general measures. (B) Wigner functions. [Wig,BB,Bru,BCSS,Ara] It is often claimed that quantum mechanics has an equivalent reformulation in terms of Wigner’s phase space distribution functions. The classical limit could then be stated very simply in terms of these functions. However, the premise is only partly correct. Since the Wigner function of a state need not be integrable, it often represents a “probability” density, in which an in?nite positive probability is cancelled by an in?nite negative probability to give formally the normalization to unity. This is highly unsatisfactory from the conceptual point of view. Technically it means that operator norms (see (d) above) cannot be estimated without arti?cial smoothness assumptions [Dau]. It is well-known that by averaging Wigner functions with a suitable Gaussian [Bop,Car] these di?culties disappear [Dav,Hol,We1]. Moreover, the Gaussians can be chosen such that in the classical limit this smearing out becomes negligible anyhow. In their averaged form Wigner functions play an important role in our approach. For a discussion of states that have positive Wigner functions “all the way to the classical limit” see Section 4.10. (C) Pseudodi?erential and Fourier integral operators. [Rob,Vor,Omn] Such operators have a rich mathematical theory, whose applications are by no means con?ned to the classical limit. However, much of the rigorous work on the classical limit has been done under this heading. The “symbol” of a pseudodi?erential operator is just its Wigner function, so much of what has been said under (B) applies. The main weakness is again the lack of control on operator norms, and hence of probability estimates, unless additional smoothness assumptions are introduced. Where such assumptions hold, the results ?t well into the framework of the present paper, too. (D) Feynman integrals. The basic observation here is that the phase of the Feynman integrand is stationary precisely for the classical paths, which therefore give the main contribution to the propagator. To the extent that the Feynman integral and the method of stationary phase in in?nite dimensional spaces can be given a mathematical meaning, this observation can be made rigorous [Tru,AHK], and reproduces WKB wave functions. The shortcomings of this approach are therefore similar to the WKB approach. It is maybe interesting to note that the propagator itself does not have a classical limit in our approach, whereas the time evolution it implements on observables does (see Section 4.5). (E) Limits of coherent states. In the papers [Hep,Hag] it is shown that in the limit h ? →0 the time evolution of a coherent state, which is initially concentrated near a given point in phase space, is well approximated by another coherent state, concentrated at the classically evolved point. This statement is essentially what one gets in the version of the present approach based on norm convergences of states [We6] rather than norm

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convergence of observables. What is missed in this approach are therefore the operator properties (e) and (f). (F) Limit of partition functions. [Lie,Sim,LS,WS] This aspect of the classical limit is conceptually straightforward, because it only requires the convergence of some numbers. Of course, it covers only a small fraction of the desirable features listed above. Nevertheless some of the techniques developed for this problem, like upper and lower symbols, or certain operators connecting spin systems of di?erent spin [LS] are close to the approach of this paper. (G) Deformation quantization. [Ri1,Ri2,Lan]. In this approach the emphasis is indeed on the structure of products and Poisson brackets, and it is in many ways close to ours. With each classical phase space function (typically the Fourier transform of a ?nite measure) one associates a speci?c family of ? h-dependent operators, belonging to an algebra in which the product is de?ned by some variant of the ? h-dependent Moyal formula. It is clear that such families are also convergent in our sense (see Section 4.3). Nevertheless, the very restricted ? h-dependence of such families is unnatural from the point of view of the classical limit (or “dequantization” [Em1]), natural as it may be for “quantization”. For another approach to quantization, based on a very restricted class of Hamiltonians, see [BV].

2. De?nition and Main Results

Consider a typical Hamiltonian operator Hh ? =? ?2 h ? + V ( x) 2m (2.1)

from a textbook on quantum mechanics. Our aim is to de?ne the limit of operators like Hh ? as ? h → 0. Since the na¨ ?ve approach of setting h ? = 0 in the above expression is obviously not what is intended, we have to be more careful with the de?nition of such limits. Rather than the algebraic expression (2.1), it must be the relation of Hh ? to other observables in the theory which has to be taken to the limit. So let us denote by Ah ? the algebra of observables “at some value of ? h > 0”. This will always be the set of bounded operators on a Hilbert space (or a suitable subalgebra), and hence in some sense independent of h ?. However, the notational distinction between these algebras may help keeping track of the various objects. Note that we will always consider bounded observables. Thus it is not ?1 the operator (2.1) we will take to the limit but, for example, its resolvent (Hh or ? ? z) the time evolution it generates. For an h ? -dependent observable Ah ? ∈ Ah ? we now want to de?ne “limh ? →0 Ah ? ”. Of course, since we have not yet put any constraint on the allowed h ? -dependence of Ah ? , this limit (whatever its de?nition) may fail to exist. The crucial notion we must de?ne is therefore “Ah h → 0”. Loosely speaking we must express the property that, ? converges as ? 5

for h ? and h ? ′ small enough, Ah ? and Ah ? ′ become “similar”. This shifts the problem to the de?nition of some connection between the spaces Ah ? and Ah ? ′ which would permit such a comparison. The basic idea of our approach is to use certain linear maps jh ? ? ′ → Ah ?h ? ′ : Ah , (2.2)

and then to compare elements in the norm of Ah ? . Once the operators jh ?h ? ′ are de?ned there will be no more arbitrariness in the de?nition of the classical limit. In order to illustrate this point, and to give a quick insight into the kind of limits we will describe, we will proceed as follows: in this section we will assume that the spaces Ah ? , and the maps jh ?h ? ′ have been de?ned. Our aim is to show how this su?ces to set up a language, in which we can describe a limit with the desirable features listed in the introduction. In particular, we will state the main theorems of our approach in this subsection. The actual de?nition of jh ?h ? ′ will be given later, in the next section, after the necessary preliminaries on phase space quantum mechanics have been provided. In Section 4 we will then be able to give examples of convergent sequences of operators and states, by which the reader will be able to judge whether we have indeed found a rigorous statement of the usual folklore and intuitions on the classical limit. Most proofs will be given in Section 5, but those relating to the dynamics had to be relegated to a sequel paper [We5]. The central notion of this paper is the following notion of convergence, which we can de?ne in terms of jh ?h ?′ .

– -sequence we mean a family of observables Ah 1 De?nition. By an h ? ∈ Ah ? , de?ned for all su?ciently small ? h. We say that an ? h-sequence Ah is j -convergent , if ? ′ ′ lim =0 . A lim A ? j h ? h ? h ?h ? ′

? →0 h ? →0 h

The set of j -convergent ? h-sequence will be denoted by C (A, j ). Two ? h-sequences Ah ? and Bh are said to have the same limit, if ? lim Ah ? ? Bh ? =0 .

h ? →0

Thus the limit of Ah h-sequences, and ? is de?ned as an equivalence class of j -convergent ? we will denote it by j-limh A , or sometimes just A . The space of all limits of j -convergent ? h ? 0 h-sequences will be denoted by A0 . ?

The abstract de?nition of j-limh ? Ah ? as an equivalence class is the best we can do without giving a concrete de?nition of jh ?h ?′ , ?h ? ′ . It will be evident from our de?nition of jh however, that the limits can be identi?ed with functions on phase space (see De?nition 6 and Proposition 7). The convergence of operator products to products of functions can then be stated as follows: 6

2 Product Theorem. Let Ah h-sequences, and de?ne, for each ? h, ? , Bh ? be j -convergent ? Ch ? = Ah ? Bh ? ∈ Ah ? . Then C is j -convergent, and j-lim(Ah ? Bh ? ) = (j-lim Ah ? )(j-lim Bh ?) ,

h ? h ? h ?

where the product on the right hand side is the product in the commutative algebra A0 .

are j -convergent to zero. Since the product in A0 is abelian, commutators Ah ? , Bh ? The interesting term for commutators is thus the next order in h ? . It is clear, however, that h?1 Ah ? , B cannot be j -convergent for arbitrary j -convergent Ah ? h ? ? and Bh ? : any sequences Ah ? , Bh ? with norm going to zero are j -convergent, but this does not even su?ce to force the scaled commutators to stay bounded. Hence we need better control of the h ? -sequences than mere j -convergence. A hint of the kind of condition needed here is given by the theorem below: the Poisson bracket to which these commutators converge is only de?ned for di?erentiable limit functions. Hence we need di?erentiability properties also for the 2 sequences Ah ? and Bh ? . The appropriate space of sequences, denoted by C (A, j ), will be 2 de?ned and discussed in [We5]. Brie?y, C (A, j ) consists of those sequences Ah ? such that ? αh ( A ) has Taylor expansions to second order in ε with derivatives in C ( A , j ) and an error h ? εξ estimate which is uniform for su?ciently small h ? . This space is norm dense in C (A, j ). The following theorem is also shown in [We5]. is j -convergent, and 3 Bracket Theorem. Let A, B ∈ C 2 (A, j ). Then ? h?1 Ah ? , Bh ? i Ah = j-lim Ah , j-lim ? , Bh ? ? , j-lim Bh ? h ? h ? h ? ? h where the product on the right hand side is the Poisson bracket of C 2 -functions on phase space.

Commutators and Poisson brackets determine the equations of motion for quantum and classical systems, respectively. Hence the above theorem says that the quantum equations of motion converge to the classical ones. Of course, one also wants to know that the solutions of the respective equations converge. This is the content of the following Theorem. Again the proof is given in [We5]. Note that the Theorem only makes a statement for ?nite times, i.e., it is not strong enough to allow the interchange the limits h ? → 0, and the ergodic time average, or some other version of the limit t → ∞. This would be very interesting for applications to “quantum chaos” (see [DGI] for a result in this direction).

2 ? 4 Evolution Theorem. Let Hh h. De?ne the ? ∈ C (A, j ) such that Hh ? = Hh ? for every ? time evolution for each ? h by t itHh h A e?itHh h , ? /? ? /? γh (2.3) ? (A) = e t t for A ∈ Ah R. Let Ah h. ? , and t ∈ I ? be j -convergent, and de?ne Ah ? ), for every ? ? = γh ? (Ah t Then Ah ? is also j -convergent, and t t j-lim γh , ? ) = γ0 j-lim Ah ? ? (Ah h ? h ?

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t where γ0 is the Hamiltonian time evolution on phase space generated by the Hamiltonian function H0 = j-limh ? Hh ?.

Finally, we would like to de?ne the convergence of states. The states for each ? h are, by de?nition, positive, normalized linear functionals on Ah ? . Since Ah ? is an algebra of operators on Hilbert space this includes all states given by density matrices, the so-called normal states. Non-normal states appear naturally in the description of limiting situations such as states with sharp position and in?nite momentum. They are also included in the present setup.

5 De?nition. For each ? h, let ωh h-sequence ? : Ah ? → C be a state. We say that the ? ω is j*-convergent, if for every j -convergent ? h-sequence Ah ? ∈ Ah ? of observables, the sequence of numbers ωh ( A ) has a limit as h ? → 0 . The limit of the sequence is the state ? h ? ? ω0 = j -limh ? ωh ? : A0 → C, de?ned by ω0 j-lim Ah ? = lim ωh ? (Ah ?) .

h ? h ? →0

ω0 will be called a cluster point of the sequence ωh hn , n ∈ N ? , if there is a subsequence ? such that the above equation holds for limits along this subsequence.

Since A0 = j-limh ? Ah ? is a function on phase space, the limit functionals ω0 are measures on phase space, or, more precisely, measures on a compacti?cation of phase space. We will see that every state on A0 occurs as the limit of suitable h ? -sequences of states. De?nition 5 gives the analogue of weak*-convergence of states on a ?xed algebra. In particular, every sequence ωh ? has cluster points. Norm limits of states will be considered in another paper [We6].

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3. De?nition of jh –h –′

Without the concrete de?nition of Ah ? and jh ?h ? ′ the statements made in the last section are void. In this section we will provide these de?nitions, and describe some further properties of the limits, which can be stated only in this more concrete context. The systems we treat will be non-relativistic with d < ∞ degrees of freedom. Let us denote by X = I Rd the con?guration space of the system. Then its Hilbert space is H = L2 (X, dx) (3.1)

In H we have a representation of the translations in con?guration space and momentum space, given by the unitary Weyl operators

? Wh (x, p)ψ (y ) = exp ?

i i p·x+ p·y 2? h h ?

ψ ( y ? x) .

(3.2)

This is a translation by the momentum p ∈ I Rd and the position x ∈ I Rd . Taken together these two determine a point in phase space Ξ, usually denoted by ξ = (x, p). The basic commutation relations for the Weyl operators then read σ (ξ, η ) W h ? ? ? h Wh (ξ )W h (η ) = e 2? (ξ + η ) , where σ (x, p ; x , p ) = p · x ? p · x

′ ′ ′ ′ i

(3.3) (3.4)

is the usual symplectic form on phase space. The phase space translations act on quantum observables, represented by bounded operators A ∈ B(H), (resp. classical observables, represented by bounded measurable functions f ∈ L∞ (Ξ)) via

? αh ξ (A)

α0 ξ (f )(η ) = f (η ? ξ ) .

? ? = Wh (ξ ) A W h (?ξ )

(3.5)

? h ? h ? In either case, i.e., for h ? ≥ 0, we get αh ξ+η = αξ αη . The Weyl operators are eigenvectors of the translations, i.e., i ? ? h ? h ? σ (ξ, η ) W h (η ) . (3.6) αh W ( η ) = e ξ

The comparison maps jh ?h ? ′ : B (H) → B (H) will be taken to be positive in the sense ′ I) = 1 I. These properties are simply that A ≥ 0 =? jh ?h ? ′ (1 ?h ? (A) ≥ 0, and unital, i.e., jh required by the statistical interpretation of quantum mechanics. The essential condition is the one linking the comparison to the phase space structure: we will demand that

h ? h ? jh ?h ?′ ?h ? ′ ? αξ = αξ ? jh

′

.

(3.7)

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?, ? h′ is convex and, Note that the set of operators jh ?h ? ′ satisfying these conditions for ?xed h with any operator jh ?h ? ′ , also contains the operator ?h ?h ?′ =

? ρ(dξ ) αh ?h ?′ ξ ? jh

,

where ρ is any probability measure on phase space. Obviously, in order to get a sensible limit we must require that the origin of phase space is not shifted around in some arbitrary way (so only ρ centered near the origin will be allowed in the above formula), and that no large scale smearing out (with ρ of very large variance) is contained in jh ?h ? ′ . We won’t go into making these requirements precise in this paper (see, however, [We4]). The main point is that all systems of comparison maps satisfying these requirements de?ne the same class of j -convergent ? h-sequences via De?nition 1. Since our whole theory is not based h-sequences, we on the detailed behaviour of jh ?h ? ′ , but only on the class of j -convergent ? are free in this paper to make a somewhat arbitrary but explicit choice of jh ?h ? ′ for the sake of simple presentation. The equivalence to other choices, including an essentially unique “optimal” one will be shown in [We4]. Our choice of comparison maps will have the special property that it maps quantum to quantum observables (at di?erent value of h) via a classical intermediate step. It is clear that something like this must be possible ? from the idea that the comparison described by the jh ?h ? ′ should be at least asymptotically transitive. Positive maps taking quantum observables to classical ones and conversely are wellknown [Bop,Sim,Dav,Tak,We1]. These maps depend on the choice of a normal state, which is usually taken to be coherent, i.e., the ground state of some harmonic oscillator. Let ?x2 ?d/4 (3.8) χh h) exp ? ( x) = ( π ? 2? h be the ground state vector of the standard oscillator Hamiltonian

osc Hh = ?

1 2

(Pi2 + Q2 i) ,

i

(3.9)

with Pi = (? h/i)?/?xi . By Γh ? = |χh ? χh ? | we will denote the corresponding one-dimensional ∞ projection. Then we set, for f ∈ L (Ξ), and A ∈ B(H),

h ? h ? j0? h (A)(x, p) = χh ? |W (?x, ?p)A W (x, p)|χh ? dx dp ? h ? f (x, p) W h (x, p)|χh jh = ? χh ? |W (?x, ?p) ? 0 (f ) (2π ? h)d

(3.10.a) . (3.10.b)

In terms of Γh ? we can write this as

h ? j0? h (A)(x, p) = tr A αx,p (Γh ?) dx dp ? f (x, p) αh jh = ?) . ? 0 (f ) x,p (Γh d (2π ? h)

(3.10.a′ ) (3.10.b′ )

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The integrals in (3.10.b) or (3.10.b′ ) are to be interpreted as weak integrals, i.e., we have to take matrix elements of the integral, and compute it as a family of scalar integrals, which converge by virtue of the “square integrability of the Weyl operators” (see [We1]). One readily veri?es that j0? h and jh ? 0 both take positive into positive elements, and preserve the respective unit elements. Moreover, these maps transform the phase space translations according to 0 h ? h ? 0 jh and j0? (3.11) ? 0 ? αξ = αξ ? jh ?0 , h ? αξ = αξ ? j0? h .

? ? Since ξ → W h (ξ ) is strongly continuous, ξ → αh ? ) is continuous in trace norm, ξ (Γh which implies that j0? h A is a uniformly continuous function for any A ∈ B (H). Uniform continuity of a function f can be expressed as α0 ξ f ? f → 0 for ξ → 0, where we have ∞ used the supremum norm of functions in L (Ξ). The same continuity argument applies to jh ? 0 and, indeed, all operators of the form A = jh ? 0 f are uniformly continuous in the h ? sense that αξ (A) ? A → 0. With these preliminaries we can now de?ne jh ?h ? ′ , and also describe the ranges of these maps.

6 De?nition. For ? h, ? h′ > 0, we set (3.12) jh ? 0 ? j0? h′ : B (H) → B (H) , ?h ? ′ = jh ∞ ∞ where the maps jh ? 0 : L (Ξ) → B (H) and j0? h : B (H) → L (Ξ) are de?ned by equations (3.10.b′ ) and (3.10.a′ ). Together with the convention j00 = id, the maps jh ?h ? ′ are thus ′ de?ned for ? h, ? h ≥ 0. From the above discussion it follows that, unless ? h=h ? ′ = 0, the range of jh ? , where ?h ? ′ is contained in Ah Ah ? = A ∈ B (H )

ξ→0 ? lim αh ξ (A) ? A = 0 ξ→0

(3.13.a) . (3.13.b)

A0 = f ∈ L∞ (Ξ) lim α0 ξ (f ) ? f = 0

The space of observables “at the value ? h” (see the beginning of Section 2) can be taken as all of B(H), independently of h ? . However, since after one application of a comparison ′ only continuous elements play a role, we will usually take Ah map jh ? from (3.13.a) as ?h ? the space of observables. Note that this space is also the same for all h ? . Other possible choices are brie?y indicated in Section 4.3. We have now used the symbol A0 for two di?erent spaces, and we have to justify this by showing that the space A0 of uniformly continuous functions on Ξ as de?ned in (3.13.b) is indeed a concrete representation of the abstract limit space A0 appearing in De?nition 1. This will also justify our referring to the limits j-limh ? Ah ? as functions on phase space in the previous section. 7 Proposition. Let Ah h-sequence. Then j0? ? be a j -convergent ? h Ah ? is a norm convergent sequence of functions in the space A0 , as de?ned in De?nition 6. The identi?cation j-lim Ah ? ≡ lim j0? h Ah ?

h ? h ?

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de?nes an isometric isomorphism between A0 , and the abstract limit space of De?nition 1.

It is suggestive at this point to try an alternative de?nition of “convergence as ? h → 0”: the map j0? already takes operators to functions, i.e., quantum to classical observables, h and the convergence of these functions is at least implied by the de?nition we have given. Hence we might try to take the uniform convergence of j0? h Ah ? as a de?nition. We will see in Section 4.5, however, that with this de?nition the Product Theorem 2 would fail, so with this restricted de?nition we would miss an important desirable feature of the classical limit. The example in Section 4.5 is an operator which in a sense oscillates more and more rapidly as h ? → 0. If we exclude this sort of oscillation by an “equicontinuity” condition, i.e., if we make the uniform continuity condition in Ah ? , the convergence of ? also uniform in h j0? h Ah ? indeed becomes equivalent to convergence in the sense of De?nition 1 (seeTheorem 8 below). In order to state this precisely, we de?ne the modulus of continuity of X ∈ Ah ? ≥ 0, ?, h as the function λ → mh ? (X, λ), with mh ? (X, λ) := sup

? αh ξ (X ) ? X

ξ2 ≤ λ

,

(3.14)

where the “square” of a phase space translation ξ = (x, p) is de?ned by ξ 2 = x2 + p2 . This involves some arbitrariness since positions and momenta have di?erent physical dimensions. Any choice of the form λq 2 + λ?1 p2 would have done just as well, except that the estimates 2 involving jh ?h ? ′ look a bit simpler when the Euclidean norm“ ξ ” in phase space matches the oscillator Hamiltonian (3.9), whose ground state χh ? enters the de?nition of jh ?h ?′ . Uniform continuity of X ∈ Ah ? is equivalent to limλ→0 mh ? (X, λ) = 0. Moreover, the properties (3.7) and (3.11), together with the norm estimate jh ?h ? ′ X ≤ X imply mh ? (jh ? ′ (X, λ) ?h ? ′ (X ), λ) ≤ mh for h ?, ? h′ ≥ 0. (3.15)

(Note that the cases ? h = 0 and h ? ′ = 0 are included). Now, for a j -convergent ? h-sequence, ′ ′ ), which has h ? -modulus of continuity at ( A Ah is well approximated for small h ? by j ? h ? h ?h ? ? , thus excluding rapid oscillations most mh ? ′ , λ). This bound holds uniformly for small h ? ′ (Ah of Ah ? . This is the basic idea of the following characterization of j -convergent ? for small h sequences. It will be our basic tool for verifying j -convergence of the various sequences of observables in the examples of the next section. It also gives a quantitative meaning to the intuition that “nearly classical” observables are those that change little on a classical phase space scale, i.e., have small modulus of continuity. Whenever all relevant observables in some given physical situation satisfy this criterion, the classical limit is a good approximation, and quantitative bounds of this type can also be given, by following the proofs. This intuition can also be used [WW] to give a very direct (although “nonstandard”) de?nition of the classical limit, which is essentially equivalent to the one given in this paper. 12

8 Theorem. A sequence of observables Ah ? ∈ Ah ? is j -convergent, if and only if the following two conditions hold: (b) Ah h(ε), λ(ε) ? is equicontinuous in the following sense: for any ε > 0, we can ?nd ? such that, for ? h≤? h(ε), and λ ≤ λ(ε), we have mh ? (Ah ? , λ) ≤ ε. The idea of introducing the maps jh ? and Ah ?′ ?h ? ′ was to get a precise meaning of “Ah are similar”. Of course, this relation should be approximately transitive. This is expressed by the following estimate. Its concrete form depends on the choice of the coherent state (3.8) in the de?nition (3.10), and on (3.14). Note that each of the three parameters h ? in the theorem may take the value zero. 9 Theorem. Let ? h, ? h′ , ? h′′ ≥ 0, and let X ∈ B(H). Then

∞

(a) j0? h → 0. h (Ah ? ) ∈ A0 converges uniformly as ?

(jh ? ′h ? ′′ )X ≤ ?h ? ′ jh ?h ? ′′ ? jh X ? jh ? 0 j0? hX ≤ where ?d (dθ ) =

h′ θ ) ?d (dθ ) mh ? ′′ (X, 2?

0 ∞

(3.16) (3.17)

?d (dθ ) mh hθ ) , ? (X, 2?

0

θ d?1 e?θ dθ . (d ? 1)! h′ → 0, uniformly in ? h. In particular, if X ∈ Ah ? ′′ , the norm (3.16) goes to zero as ? An important Corollary of Theorem 9 is the following construction of j -convergent sequences and j *-convergent states. The sequences described in (1) are called “basic sequences” in the theory of “generalized inductive limits” [We3,GW,DW]. Their convergence is equivalent to the asymptotic transitivity jh ? ′h ? ′′ of the comparison. ?h ? ′ jh ?h ? ′′ ≈ jh 10 Corollary. (1) Fix h ? ′ ≥ 0 and X ∈ Ah ? = jh ?h ? ′ X is j -convergent, and ? ′ . Then Xh . j-lim jh h′ X ?h ? ′ X = j0?

h ?

(2) Let ω : A0 → C be a state, and de?ne, for every ? h > 0 a state ωh ? : Ah ? → C by ? ωh ? (X ) = ω (j0? h (X )). Then ωh ? is j *-convergent, and j -limh ? ωh ? = ω. (3) An ? h-sequence ωh ? of states on Ah ? is j *-convergent if and only if the sequence ωh ? ? jh ?0 is weak*-convergent in the state space of A0 .

Usually we are interested in normal states on Ah ? , i.e., states of the form ωh ? (A) = tr Dh ? A, where Dh ? is a density matrix. This excludes, for example, states with sharp position and in?nite momentum. (These can be obtained as the Hahn-Banach extensions 13

of a pure state on the algebra of uniformly continuous functions of position alone, and assign zero probability to any ?nite momentum interval). Similarly, on the classical side we often consider states of the form ω0 (f ) = ?(dξ )f (ξ ), where ? is a probability measure on phase space. Note that this is a strong assumption on the state: there are many states on A0 which live “at in?nity”, i.e., on the compacti?cation points [We2] of the spectrum space of A0 . However, for those states for which position and momentum are both ?nite with probability 1, we get the following somewhat simpli?ed criterion for convergence. It is analogous to the convergence theorems for characteristic functions in probability theory (see, e.g., [Chu]). Recall that C0 (Ξ) denotes the complex valued functions on Ξ vanishing at in?nity.

11 Proposition. Let ωh h-sequence of normal states. Then the following conditions ? be an ? are equivalent: (1) j ?-limh ? ωh ? = ω0 exists, and is a measure on phase space. (2) For every f ∈ C0 (Ξ), the limit limh ? ωh ? (jh ? 0 f ) = ω0 (f ) exists, and ω0 is normalized, i.e., sup {ω0 (f ) f ∈ C0 (Ξ), f ≤ 1} = 1.

h ? (3) For all ξ ∈ Ξ, the limit limh hξ )) = ω0 (ξ ) exists, and ξ → ω0 (ξ ) is a continuous ? ωh ? (W (? function.

14

4. Examples and Miscellaneous Results

1. Functions of position or momentum Let f : I Rd → I R be bounded and uniformly continuous, and let Fh ? be the multiplication operator (Fh ψ )( x ) = f ( x ) ψ ( x ). Then F satis?es the equicontinuity condition in Theo? h ? rem h (F ) is the convolution of f with a Gaussian of variance proportional √8. Moreover, j0? h. Hence, by the uniform continuity of f , to ? j-lim Fh ? (x, p) = f (x) .

h ?

(4.1)

Similarly, let Fh ? = f (P ), where f is evaluated in the functional calculus of the d commuting ? ? self-adjoint operators Pk = h i ?xk . (This is the same as taking the Fourier transform, multiplying with f (p), and transforming back). Then j-lim Fh ? (x, p) = f (p) .

h ?

(4.2)

2. Weyl operators The Weyl operators (3.2) play a fundamental role. They oscillate too rapidly to be convergent (see Section 4.5), but with a suitable rescaling of the arguments they do converge. For ?xed x, p ∈ I Rd , we set

ih ? h ? Eh hx, ? hp) = e? 2 x · p ? (x, p) = W (?

eip · Q e?ix · P

.

(4.3)

By the Product Theorem and the previous example, this converges to the phase space function E0 (x, p), de?ned as E0 (x, p)(x, p) = exp i(p · x ? x · p) = eiσ(x,p;x,p) E0 (η )(ξ ) = eiσ(η,ξ) . , (4.4)

or

The notational distinction between the two sets of Weyl operators re?ects a di?erence ? in interpretation: while the basic Weyl operators W h (ξ ) implement a symmetry transformation, expectations of Eh ? (ξ ) determine the probability distribution of position and momentum observables. This is precisely analogous to the dual role of selfadjoint operators in quantum mechanics as generators of one-parameter groups on the one hand, and as observables on the other. These also di?er by a factor h ? , e.g., the generator of the time evolution is not the observable H , but H/? h. Of course, this distinction is usually irrelevant (? h = 1!), but is crucial in the classical limit (see also Section 4.5 below). 3. Integrals of Weyl operators Let ? be a ?nite (possibly signed) measure on I R2d , and de?ne Fh ? (?) = ?(dη )Eh ? (η ) . 15 (4.5)

By the previous example this is an integral of j -convergent sequences with ? h-independent weights. It is easy to check using the Dominated Convergence Theorem that such sequences are also j -convergent. Moreover, the limit is the integral of the limits. In the present case we get the Fourier transform of the measure ? (with a symplectic twist, because Ξ and its dual vector space are identi?ed via σ ): F0 (?)(ξ ) = j-lim(Fh ? (?) (ξ ) =

h ?

?(dη ) eiσ(η,ξ)

.

(4.6)

There are two interesting special cases: If ? happens to be absolutely continuous with respect to Lebesgue measure, the “quantum” Riemann-Lebesgue Lemma [We1] asserts that Fh ? , and F0 (?) is a continuous function vanishing ? (?) is a compact operator for all h at in?nity. On the other hand, if ? is a sum of point measures, Fh ? (?) is an element of the CCR-algebra, i.e., the C*-algebra generated by the Weyl operators, and the limit function F0 (?) is almost periodic. These correspondences are a special case of a correspondence theorem [We1,We2] for general phase space translation invariant spaces of operators and functions, respectively. This general result can be used to set up limit theorems for a variety of subspaces of Ah ?. The sequences Fh ? (?) with absolutely continuous ? of compact support have been made the basis of a discussion of the classical limit by Emch [Em1,Em2]. In his approach each classical observable F0 thus has a unique h ? -sequence of quantum observables Fh ? associated with it, which is also typical for “deformation quantization” approaches [Ri1,Ri2,Ri3]. In our approach this constraint becomes unnecessary, both from a technical and from a conceptual point of view. Emch’s main emphasis is on de?ning the (weak) convergence of states with respect to this particular set of sequences. The intersection between his “classical states”, and our j *-convergent states is described precisely by Proposition 11. 4. Resolvents of unbounded operators By de?nition, j -convergent sequences are uniformly bounded in norm, which excludes the treatment of all standard quantum mechanical Hamiltonians. As a substitute, however, we can consider the resolvents of such operators. The following Theorem summarizes a few basic facts of this approach to unbounded operators. 12 Theorem. Let Hh h-sequence of (possibly unbounded) self-adjoint operators. ? be an ? ?1 We call Hh is j -convergent for ? j-convergent in resolvent sense, if Rh ? (z ) = (Hh ? ? z) some z ∈ C with ?mz = 0. Then (1) Rh ? (z ) is j -convergent for all z with ?mz = 0.

? (2) If Vh ? is a j -convergent sequence with Vh ? = Vh ? is j -convergent in resolvent ? , and Hh sense, then Hh + V is j -convergent in resolvent sense. ? h ?

Proof : (1) By the resolvent equation we have

∞

Rh ? (z ) =

n=0

′

n+1 (z ′ ? z )n Rh ? (z )

,

16

provided that (z ′ ? z )Rh < 1, which by self-adjointness of Hh ? (z ) ? is guaranteed by ′ |z ? z | < |?mz |. Each term in this sum is j -convergent by the Product Theorem, and convergence is uniform in h ? . This su?ces to establish j -convergence of the sum. Iterating ′ this argument, we ?nd j -convergence of Rh ? (z ) for all z in the same half plane as the originally given z . Since Hh ? is assumed to be self-adjoint, we also get j -convergence of ? Rh ( z ) = R ( z ) . ? h ? (2) We can argue exactly as in (1), using the series

∞ k

(Hh ? + Vh ? ? z)

?1

= (Hh ? ? z)

?1 k=0

?1 Vh ? (Hh ? ? z) ?1

,

which converges uniformly in h ? , provided Vh ≤ ε < 1 for small h ? . This will be ? |?mz | the case if |?mz | > V0 . For other values of z the convergence follows by (1).

An immediate application is to Schr¨ odinger operators: the kinetic energy Hh ? = ?? h2 /(2m)? is j -convergent in resolvent sense by Section 4.1, and if V is a ?xed uniformly continuous bounded potential, we conclude, for ?mz = 0: ?? h2 j-lim ? + V ( x) ? z 1 I h ? 2m p2 ?1 + V ( x) ? z 1 I R0 (z ) (x, p) = 2m

?1

= R 0 (z ) (4.7) .

with

At ?rst sight, it seems that the class of potentials for which this result holds is much larger. Indeed, the same technique is used to construct the Hamiltonian for relatively bounded perturbations [Kat], i.e., perturbations V for which V (H ? z )?1 < 1 for large z . The Coulomb potential is bounded relative to the Laplacian in this sense. However, in the above application the Laplacian is scaled down with a factor h ? 2 , so this relative boundedness of V with respect to H cannot be used uniformly in h ? , and this destroys the proof. It is easy to see that not only this particular method fails for the attractive Coulomb potential, but the statement itself is false: suppose that the potential V is not bounded below, and let R(x, p) = (p2 + V (x) ? z )?1 be the classical resolvent function at z ∈ C. If the resolvents of the corresponding Schr¨ odinger operators were j -convergent, this function would have to be uniformly continuous. This is impossible: Let xn be a sequence such that V (xn ) → ?∞, and let pn be a sequence such that p2 n = ?V (xn ). Then R(xn , pn + ε) ? R(xn , pn ) = (2pn ε + ε2 ? z )?1 + z ?1 .

?1

For ?xed ε the ?rst term goes to zero, i.e., supx,p |R(x, p + ε) ? R(x, p)| ≥ |z | , and hence R is not uniformly continuous. It should be noted, however, that this negative result only 17

concerns norm convergence. Singular objects like the Coulomb resolvent may still be weakly convergent in the sense dual to the norm convergence of states [We6]. 5. Implementing unitaries never converge The time evolution, and all other symmetry transformations on B(H) are implemented by ? unitaries Uh ? as Ah ? → Uh ? Ah ? Uh ? is j -convergent. Then we conclude with ? . Suppose that Uh 2 ? the Product Theorem that j-limh ? Uh ? Ah ? Uh ? Ah ? )|j-limh ? Uh ? | = j-limh ? Ah ? . In other ? = (j-limh words, the symmetry transformation becomes trivial in the classical limit. On the other hand, the time evolution and many other canonical transformations act non-trivially in the limit by the Evolution Theorem 4. Hence in all these cases the implementing unitaries cannot converge. An instructive special case is the phase space translation by η = 0. This clearly acts h ? non-trivially in the limit, and is implemented by Xh ? = W (η ). We have

i ?1 2 j0? h (Xh ? )(ξ ) = exp h ? σ (η, ξ ) · exp 4? hη

.

(4.8)

This converges to zero, uniformly in ξ . Hence the criterion (a) of Theorem 8 is satis?ed, and would indicate the limit X0 = 0. But, of course, (b) is violated for this “rapidly oscillating operator”: we get mh ? (Xh ? , λ) = sup eiα ? 1 α2 ≤

λ (η 2 h ?2

.

(4.9)

For ?xed λ = 0 this expression is equal to 2 for all su?ciently small h ? . It is clear from this example, that a notion of convergence based on Theorem 8.(a) alone would not satisfy the Product Theorem, and is hence too weak for many applications (compare Section 4.4, Section 4.7, and Section 4.9). 6. Point measures The operators Γh ? = |χh ? χh ? |, which we have used in the de?nition of j0? h and j0? h are 2 not j -convergent: (j0? (Γ ) ( ξ ) = exp ? ξ / (2? h ) converges pointwise as ? h → 0, but not h h ? uniformly, (and not to a continuous function). On the other hand, we can also interpret the operators Γh ? -sequence of states ωh ? as the density matrices of an h ? . This sequence is j *-convergent: for A ∈ C (A, j ) we have lim ωh ? (Ah ? ) = lim tr Γh ? Ah ? = lim j0? h (Ah ? ) (0) = A0 (0) .

h ? h ? h ?

(4.10)

Hence these states converge to the point measure at the origin. More generally, we get from Proposition 11 the following statement: a sequence of normal states ωh ? converges to the point measure at the origin i? ωh E ( ξ ) → 1 for every ξ ∈ Ξ. ? h ? In case each ωh ? has ?nite second moments we can give a simple and intuitive suf?cient criterion for convergence to this point measure. Consider the standard oscillator osc (3.9). Then we claim the inequality Hamiltonian Hh ? 1 ? osc (Eh I ? ξ 2 Hh ? (ξ ) + Eh ? (ξ ) ) ≥ 1 ? 2 18 , (4.11)

interpreted as an inequality between quadratic forms. To prove this, note that the inequality is unchanged under any symplectic linear transformation leaving the metric ξ 2 , and osc hence Hh invariant. We may thus transform to a standard form in which only one compo? nent, say the p1 -component of ξ is non-zero. Then, according to (4.4), Eh ? (ξ ) = exp(ip1 Q1 ), 2 and in the functional calculus of Q1 , we ?nd ?e Eh I ? p2 ? (ξ ) = cos(p1 Q1 ) ≥ 1 1 Q1 /2 ≥ 2 osc 1 I ? ξ Hh ? . Evaluating now the inequality (4.11) on a sequence of states, we ?nd that, if

osc ωh ? (Hh ? ) ?→ 0

as ? h→0 ,

? then ωh ? (Eh ? (ξ )) → 1 for all ξ , and hence j -limh ? ωh ? is the point measure at 0 by the above arguments. It is shown in [We4] that any such sequence ωh ? could have been used in the ′ instead of Γ , without changing the class of convergent sequences. de?nition of jh h ? ?h ?

7. Eigenstates Let Hh ? be a sequence of self-adjoint operators which are j -convergent in resolvent sense. Let λh ? be a sequence of real numbers, converging to λ0 , and let ψh ? be an eigenvector with Hh ? ψh ? = λh ? ψh ? , (4.12)

for each h ? > 0. Let ωh ? (X ) = ψh ? , Xψh ? be the corresponding state on Ah ? . Consider a cluster point ω? of this sequence of states, i.e., the limit along a subsequence ? hn . Then by the Product Theorem we have ω? R0 (z ) ? (λ0 ? z )?1

2

= lim ωh ?n

h ? n →0

?1 Rh ? (z ) ? (λh ? ? z)

2

=0 ,

because ωh ? n is a sequence of eigenstates. It follows that ω? , considered as a measure on (a compacti?cation of ) phase space is supported by the level set {ξ H0 (ξ ) = λ0 } .

In the one-dimensional case, and when the dynamics associated with Hh ? is also j convergent, we can say more: then ω0 has to be invariant under the phase ?ow generated by H0 . Hence it has to be equal to the micro-canonical ensemble at energy λ for the classical Hamiltonian H0 . In particular, all cluster points of ωh ? coincide, and we have convergence. 8. WKB states The basic states for the WKB-method are vectors of the form

iS (x)/h ? ?h ? (x) = ?(x)e

,

(4.13)

with a ?xed vector ? ∈ L2 (Ξ), and the “action” S : I Rd → I R. The distribution of 2 “position” in these vectors is |?(x)| , independently of h ? , and the rapidly oscillating phase determines the momentum. Asymptotic estimates of expectation values in such states 19

are traditionally evaluated using the stationary phase method [Mas]. Since this typically involves some partial integration, the technical conditions in such results usually demand some smoothness of ? and S . In our context we can get by with the minimal assumptions needed to even state the asymptotic formula. 13 Theorem. Let ? ∈ L2 (I Rd ) with ? = 1, and let S : I Rd → I R almost everywhere di?erentiable. Set ωh ? (A) = ?h ? , A?h ? , with ?h ? from (4.13). Then ωh ? is j *-convergent with limit ω0 given by ω0 ( f ) = dx |?(x)| f (x, dS (x)) .

2

Proof : The states ωh ? are normal, and ω0 is a probability measure on phase space. Hence we may apply Proposition 11. In the expression 1 h ? hx)) ?(y ? ? h x) ωh dy ?(y ) exp i x · p + p · y ? (S (y ) ? S (y ? ? ? (Eh ? (x, p)) = 2 h ? ? we may replace ?(y ? ? hx) by ?(y ): the error is bounded by W h (? hx, 0)? ? ? , which 2 2 goes to zero by strong continuity of the translations on L . Since |?(y )| is integrable, and independent of h ? , we may carry out the limit under the integral by the Dominated Convergence Theorem. This gives lim ωh ? (Eh ? (x, p)) =

h ?

dy |?(y )| exp i p · y ? x · dS (y )

2

.

The exponential can be written as E0 (x, p)(y, dS (y )), which shows that ωh ? (Eh ? (x, p)) → ω0 (E0 (x, p)) with the ω0 given in the Theorem.

When S is reasonably smooth, the set LS = (x, dS (x)) x ∈ Ξ , which contains the support of the measure ω0 is a Lagrange manifold in phase space, i.e., a manifold on which the symplectic form vanishes. This property remains stable under time evolution, whereas the uniqueness of the projection (x, dS (x)) → x from L onto the con?guration space is obviously not stable. The points where this projection becomes singular are called caustics, and play an important role in the time dependent WKB-method [Mas]. At such points, and at the turning points of a bound state problem, it may become more pro?table to play the same game with wave functions ? in momentum representation, and a p-dependent action S . The limits of such states can be treated exactly as above, so we will not do it explicitly. 9. Interference terms, and pure states converging to mixed states The WKB-states ωh ? . Yet their limit ? of the previous example are pure for every non-zero h is not a point measure, i.e., the limit is a mixed state. Is the funny support of the limit measure (the Lagrange manifold) perhaps a consequence of this purity? Are the limits of pure states always singular with respect to Lebesgue measure, as Section 4.7 also suggests? 20

We will see in this example that, to the contrary, any measure on phase space can be the limit of a sequence of pure states. The basic observation is that the classical limit annihilates certain “interference terms”. The following Proposition describes a general situation in which this happens. Recall that two states ω, ω ′ on a C*-algebra are called orthogonal, if ω ? ω ′ = 2, or, equivalently, if, for every ε > 0, there is an element 0 ≤ Fε ≤ 1 I in the algebra such that ω (Fε ) ≤ ε, and ′ ω (Fε ) ≥ 1 ? ε. On the abelian algebra A0 two states (measures) are orthogonal if they have disjoint supports, but also if one is, say, a sum of point measures, and the other is absolutely continuous with respect to Lebesgue measure on phase space. 14 Proposition. Let ?h h-sequences of unit vectors such that the states ?h ? , ψh ? be ? ? , ·?h ? and ψh h? , ·ψh ? are j *-convergent with orthogonal limits. Then, for any j -convergent ? sequence Ah , we have ? lim ?h ? , Ah ? ψh ? =0 .

h ?

. ≤ Ah Fε,h (1 I ? Fε,h Ah ? ? ?h ? + Ah ? ? )ψh ? + ? , Fε,h ? The ?rst two terms converge to limits less than ε by the choice of Fε , and the last term goes to zero by the Product Theorem.

Proof : Let ω? , ωψ be the limit states of the sequences in the Proposition. Pick Fε such that ω? (Fε ) ≤ ε, and ωψ (1 I ? Fε ) ≤ ε, and let Fε,h h-sequence with ? be a j -convergent ? j-limh ? Fε,h ? = Fε . Then | ?h I ? Fε,h ? , Fε,h ? ψh ? ? , Ah ? ψh ? | ≤ | ?h ? , Fε,h ? Ah ? ψh ? | + | ?h ? , Ah ? (1 ? )ψh ? | + ?h ? , Ah

15 Theorem. Let ω0 be a state on A0 , represented by a probability measure on phase space. Then there is an ? h-sequence ?h ? of unit vectors such that ω0 is the limit of the j *-convergent ? h-sequence ωh = ? , · ? ? h ? h ? of pure states. Proof : By Proposition 11.(2) we have to construct ?h ? such that ′ ωh ( ?) ? , jh ? 0 (f )?h ? ?→ ω0 (f ) , ? (f ) = ?h for all f ∈ C0 (Ξ). Let fn ∈ C0 (Ξ) be a norm dense sequence. Since the states are uniformly ′ bounded, it su?ces to show ωh ? (fn ) → ω0 (fn ) for all n. We will do this by constructing a sequence of vectors ?h ? (N ), N ∈ I N, such that h ? (N ) → 0 as N → ∞, ? , and a sequence h and ?N | ?h , ? , jh ? 0 (fn )?h ? ? ω0 ( f n ) | ≤ 2 for n ≤ N , and h ?≤? h(N ). We ?rst pick a state ω ˙ N , which is a sum of ?nitely many point measures (supported on di?erent points) such that |ω0 (fn ) ? ω ˙ N (fn )| ≤ 2?(N +1) . We know from Section 4.6 that we can ?nd pure states converging to any point measure, and combining these using Proposition 14 we ?nd vectors ?h ˙ N ( fn ) | ≤ ? such that | ?h ? , jh ? 0 (fn )?h ? ?ω 21

2?(N +1) , for su?ciently small h ? . These are the vectors that have the desired approximation property.

10. Wigner functions The Wigner function [Wig], or “quasi-probability density” of a state ω can be written as

? (Wh h)d ω αh ? ω )(ξ ) = (2/? ξ (Π)

,

(4.14)

where (Π?)(x) = ?(?x) is the parity operator [Gro]. Here we have chosen the normalization such that formally, or with suitable regularization, (2π )?d dx dp (Wh ? ω )(x, p) = 1. Of course, Wh ? ω is rarely positive [Hud,BW], and in general not even integrable. Ignoring such technical quibbles, however, as most of the literature on Wigner functions does, we get a “simpli?ed” formulation of the classical limit, and also an interesting class of convergent states. The modi?ed de?nition of the classical limit is based on an alternative de?nition of j0? h and jh ? 0 , namely as the usual Wigner-Weyl quantization and dequantization maps. These can be de?ned using the adjoint of (4.14): dx dp W Wh ? ω (x, p) j 0? h A (x, p) = ω (A) (2π )d

W jh ?0 W jh ?h ?′

= jW 0? h

?1

(4.15) .

W W = jh ? 0 j 0? h′

From the observation that Wh ? maps the state ω to the measure on phase space with the same Fourier transform [We1] (or Weyl transform) we get

W jh ? (ξ ) . ? ′ (ξ ) = Eh ?h ? ′ Eh

(4.16)

Because Wh ? ω is in general not integrable, the transformations (4.15) are all ill-de?ned as they stand, and unbounded for the norms of B(H) and L∞ (Ξ) [Dau]. There are several ways to give them a meaning on some restricted domain. For example, all transformations make sense on the Hilbert-Schmidt class and L2 (Ξ), because Wh ? is unitary up to a factor: dx dp ′ ? Wh ? ?d tr Dω Dω′ ? ω (x, p) Wh ? ω (x, p) = h (2π )d , (4.17)

where Dω and Dω′ are the density matrices of ω and ω ′ . Further customary domains of such transformations involve additional smoothness assumptions [Rob]. Whether one wants to burden the de?nition of the classical limit with such constraints is a matter of taste. That they are not necessary is demonstrated by the present paper, or so the author hopes. 22

The transformation Wh ? can also be used directly to de?ne a sequence of states ωh ? by 1 ?xing a density ρ ∈ L (Ξ), and demanding Wh ? ωh ? =ρ . (4.18)

We have to assume that ρ is chosen so that ωh ? is given by a trace class operator Dh ? . Let us denote the the classical state with density ρ by ω0 . Is this the classical limit of the sequence ωh ? , i.e., do we have j ?-lim ωh (4.19) ? = ω0 ?

h ?

In order to decide this, let us ?rst consider a j -convergent sequence of quantum observables of the special form Ah ? = jh ? 0 A0 . Then ωh ? (Ah ? ) = tr(Dh ? Ah ?) dx dp h ? A0 (x, p) tr(Dh = ? αx,p (Γh ? )) d (2π ? h) dx dp dx′ dp′ ′ ′ ′ ′ = A0 (x, p)Kh ? (x ? x , p ? p )ρ (x , p ) , d d (2π ) (2π ) where we have evaluated the trace using (4.17), and have used the Wigner function Kh h)d exp ?(x2 + p2 )/? h of the coherent projection Γh ? (x, p) = (2/? ? . This kernel goes to a δ -function as h ? → 0, and since A0 is uniformly continuous, ωh ? (Ah ? ) converges to ?d (2π ) dx dp A0 (x, p)ρ(x, p) = ω0 (A0 ). So it appears that (4.19) holds. What makes this computation work is the fact that the convolution of two Wigner functions of trace class operators (here Dh ? and Γh ? ) is always integrable. Thus the bad properties of ωh ? are averaged out. The argument fails, however, when Ah ? is not of the special form Ah = j A . It is true that the A of this form are norm dense ( Ah ? h ?0 0 h ? ? ? jh ? 0 A0 → 0 for any j -convergent sequence). However, this kind of approximation for a general Ah ? is only su?cient to show the convergence of ωh ( A ), when D is uniformly bounded in ? h ? h ? trace norm as ? h → 0. Only in this case the conclusion (4.19) is valid. It is easy to ?nd densities ρ, however, such that Dh ? (any density ρ, which is ? is not even trace class for any h unbounded, or discontinuous, or does not go to zero at in?nity will do). For such densities the sequence ωh ? (Ah ? ) may diverge, even if Ah ? → 0. In particular the Ansatz (4.18) with such ρ never yields a j *-convergent sequence ωh ?. If we value the statistical interpretation of quantum mechanics, we should demand not only that Dh ? has uniformly bounded trace norm, but also that Dh ? (and hence ωh ? ) is positive for all h ? , or at least for a sequence ? hn along which we want to take the classical limit. In the terminology of Narcowich [Nar] this means that the “Wigner spectrum” of the Fourier transform of ρ contains the sequence ? hn . This is a severe constraint on the classical densities ρ [BW].

23

5. Proofs

In this section we prove the results stated in Section 2 and Section 3, apart from the Theorems 3 and 4 about Poisson brackets and the dynamics [We5]. We ?rst state a ?, ? h′ ≥ 0 and their compositions Lemma that allows us to handle the maps jh ?h ? ′ with h more easily. The basic observation is that since all these maps are normal, they are completely determined by their action on Weyl operators. Moreover, since Weyl operators are eigenvectors of the phase space translations (3.6), and these are intertwined by jh ?h ?′ , Weyl operators must be mapped into Weyl operators— up to a scalar factor. This scalar W factor is what distinguishes j h ?h ? ′ , and some such a factor is necessary ?h ? ′ after (4.16) from jh to make jh ?h ? ′ positive (see [We4] for a complete discussion).

16 Lemma. For ? h ≥ 0, let Eh ? (x, p) be de?ned as in (4.3), and let jh ?h ? ′ be as de?ned in equations (3.10) and (3.12). Then, for ? h, ? h′ ≥ 0, and A ∈ B(H), we have ?(? h+h ? ′) 2 ′ ′ jh (x + p2 ) , (5.1) ? (x, p) exp ? (x, p) = Eh ?h ? Eh 4 and jh ?h ? ′ jh ? ′h ? (A) = ?(x2 + p2 ) h dx dp exp α? x,p (A) , (2π (? h+h ? ′ ))d 2(? h+h ? ′) (5.2)

Proof : A Gaussian integration using (3.8) and (3.2) gives ?1 2 h ? χh (x + p2 ) . (5.3) ? , W (x, p)χh ? = exp 4? h From this and the Weyl relations (3.3) we get the equation (5.1) for the special case ? h = 0. The case ? h′ = 0 is veri?ed by the following computation: dx′ dp′ ? exp i(x′ · p ? p′ · x) αh jh ?) ? 0 E0 (x, p) = x′ ,p′ (Γh (2π ? h)d dx′ dp′ h ? ? α? ′ ′ W h (? hx, ? hp)? Γh = Wh (? hx, ? hp) ? (2π ? h)d x ,p

h ? = Eh hx, ? hp)? Γh ? (x, p) tr W (? ? ?? h 2 (x + p2 ) . = Eh ? (x, p) exp 4 For general h ?, ? h′ we get (5.1) by composition. In the same way we ?nd jh ? (x, p) = ?h ? ′ jh ? ′h ? Eh ?(? h+? h′ ) 2 2 (x + p ). When A is a Weyl operator, (5.2) can be veri?ed by Eh ? (x, p) exp 2 computing the Gaussian integral. For other operators A, (5.2) follows, because jh ? ′h ? is ?h ? ′ jh ultraweakly continuous, and the Weyl operators span an irreducible algebra of operators, which is hence ultraweakly dense in B(H).

24

Proof of Theorem 9: From Lemma 16 and the de?nition (3.14) of the modulus of continuity we get dx′ dp′ ?1 f ? j0? (x ? x′ )2 + (p ? p′ )2 f (x, p) ? f (x′ , p′ ) exp h jh ? 0 f = sup d (2π ? h) 2? h x,p ?1 2 dx dp x + p2 m0 (f, (x2 + p2 )) exp ≤ d (2π ? h) 2? h ∞ θ d?1 ?θ = dθ e m0 (f, 2? hθ ) . (d ? 1)! 0 The estimate (3.16) now follows from , (jh ? 0 (id ?j0? ? ′ 0 )j0? h′′ X h′ jh ? ′h ? ′′ )X = jh ?h ? ′ jh ?h ? ′′ ? jh jh ≤ X , and m0 (j0? ?0X h′′ X, λ) ≤ mh ? ′′ (X, λ). The proof of the estimate (3.17) is completely analogous.

Proof of Theorem 8: Assume that Ah ? is j -convergent. We ?rst show the equicontinuity condition (b). By De?nition 1 we can pick h ?′ such that limh ≤ ε/8. Next we pick ? h(ε) such that Ah ≤ ε/4 for ? Ah ? ? jh ? ? jh ?h ? ′ Ah ?′ ?h ? ′ Ah ?′ ′ ′ ′ ′ , λ ) ≤ ε/ 2 for λ ≤ λ(ε). ( A , we can ?nd λ ( ε ) such that m ∈ A h≤? ? h(ε). Since Ah h ? h ? h ? ? Hence, for h ?≤? h(ε), and λ ≤ λ(ε), mh ? ? jh ? (Ah ? , λ) ≤ mh ? (jh ?′ ?h ? ′ Ah ? ′ , λ) + 2 Ah ?h ? ′ Ah ≤ mh ? ′ (Ah ? ′ , λ) + ε/2 ≤ ε . To see condition (a), the uniform convergence of j0? h Ah ? , we estimate j0? h jh h (Ah ? ? jh h Ah ? ? j0? ?′ h′ )Ah ?h ? ′ ? j0? ? ′ ) + (j0? ?h ? ′ Ah ? ′ ≤ j0? h′ Ah

. ≤ Ah h jh ? 0 )j0? ? ? jh ?′ h′ Ah ? ′ + (id ?j0? ?h ? ′ Ah ? → 0. Hence, using Since j0? ? ′ is uniformly continuous, the second term goes to zero as h h′ Ah ′ the j -convergence of Ah ? for the ?rst term, we get limh ? j0? h Ah ? ? j0? ? ′ = 0, which ? limh h′ Ah implies that j0? h Ah ? is norm-Cauchy in A0 , and hence converges. Conversely, assume that (a) and (b) are satis?ed. Then Ah ? ? jh ? 0 j0? h Ah ? + jh ? 0 j0? h Ah ? ? jh ? 0 j0? ? ? jh ?′ h′ Ah ? ′ ≤ Ah ?h ? ′ Ah ≤ ?d (dθ )mh hθ ) ? (Ah ? , 2? + j0? h Ah ? ? j0? ?′ h′ Ah The integrand in the ?rst term goes to zero as h ? → 0, for every θ due to condition (b), so the ?rst term vanishes in this limit by dominated convergence. Hence by condition (a), limh ? Ah ? ? jh ?h ? ′ Ah ? ′ = 0. ? ′ limh

Proof of Proposition 7: Let us denote, for the sake of this proof, the abstract limit space of De?nition 1 by A∞ , and the space of uniformly continuous functions from (3.13.b) by A0 . Then the equation j0∞ j-lim Ah , ? = lim j0? h Ah ?

h ? h ?

25

for j -convergentAh ? , de?nes an operator j0∞ : A∞ → A0 , because j-limh ? Ah ? = 0 is de?ned as limh ? Ah ? = 0, and hence implies limh ? j0? h Ah ? = 0. j0∞ is surjective, because j-limh ? jh ?0f = f for f ∈ A0 , and is injective by the estimate (3.17). Since both j0? and j are contractive h h ?0 the same arguments also show that j0∞ is isometric.

Proof of Corollary 10: j -convergence in (1) follows immediately from the Theorem, and the value of the limit follows from the identi?cation of the limit space. For (2) it su?ces to evaluate ωh ? on j -convergent sequences of the form (1), for which the convergence again follows from (3.16) with h ? = 0. (3) follows from the observation that lim Ah (5.4) ? ? jh ? 0 A0 = 0 ,

h ?

for any A ∈ C (A, j ).

h ? Proof of Proposition 11: (1)?(3): By Section 4.2, Eh hξ ) is j -convergent, ? (ξ ) = W (? hence the existence of the limit is clear, which is then equal to ω0 (E0 (ξ )). This is the Fourier transform of the measure ω0 , which is continuous by Bochner’s Theorem [Kat].

for all choices of ξ1 , . . . , ξN ∈ Ξ. In the limit h ? → 0 this becomes the positive de?niteness hypothesis in Bochner’s theorem, which together with the postulated continuity implies that ω0 (ξ ) is the Fourier transform of a positive measure ω0 on Ξ, i.e., ω0 (ξ ) = ω0 E0 (ξ ) . The normalization of this measure follows by setting ξ = 0. It su?ces to show convergence for f = F0 in a norm dense subset of C0 (Ξ). For this we take the Fourier transforms of L1 -functions in the sense of Section 4.3. Explicitly, we let f = dξ ρ(ξ )E0 (ξ ) with ?xed ρ ∈ L1 (Ξ). Then jh ?0 = and ωh ? (jh ?0f ) =

?ξ dξ ρ(ξ )e?h ?ξ dξ ρ(ξ )e?h

2

(3)?(2): Positivity of ωh ? is equivalent [We1,BW] to the positive de?niteness of all mah ? trices Mν? , ν, ? = 1, . . . , N de?ned by ? h σ (ξ ν , ξ ? ) , Mh = ω E (ξ ? ξ ) ei?

ν? h ? h ? ν ?

/4

Eh ? (ξ ) , ωh ? Eh ? (ξ )

2

/4

′ (2)?(1): By Corollary 10.(3) we have to show that the convergence ωh ? (jh ? 0 f ) ≡ ωh ? (f ) → ω0 (f ) extends from f ∈ C0 (Ξ) to all f ∈ A0 . By the normalization condition in (2), we ′ can ?nd fε ∈ C0 (Ξ) such that 0 ≤ fε ≤ 1, and ω0 (f ) ≥ 1 ? ε. Hence ωh ? (fε ) ≥ 1 ? 2ε for h≤? ? h(ε). But then, for arbitrary f ∈ A0 , ′ ′ ′ |ωh ? (f ) ? ω0 (f )| ≤ |ωh ? (f (1 ? fε )| + |ωh ? (f fε ) ? ω0 (f fε )| + |ω0 (f (1 ? fε )| . Then, for h ?≤? h(ε), the ?rst and last term are bounded by 2ε f and ε f , respectively, and the middle term goes to zero, because f fε ∈ C0 (Ξ).

holds for all h ? ≥ 0, and the claim follows by dominated convergence.

26

Proof of the Product Theorem 2: Let Ah ? -sequences. We have ? , Bh ? be convergent h to show the convergence of Ch ? = Ah ? Bh ? . Two observations help to simplify the proof: ′ ′ ?rstly we may replace Ah by A such that Ah ? ? ? Ah ? . Note h ? ? → 0, and similarly for Bh that this modi?cation will also not change j-limh ? Ch ? . Hence we may take Ah ? = jh ? 0 A0 , and Bh ? = jh ? 0 B0 . Secondly, the estimate mh ? (Ah ? Bh ? , λ) ≤ mh ? (Ah ? , λ) Bh ? + Ah ? mh ? (Bh ? , λ) shows that Ch ? satis?es the equicontinuity condition in Theorem 8, since Ah ? and Bh ? do. Therefore, by that Theorem, it su?ces to show that, for A0 , B0 ∈ A0 , lim j0? h (jh ? 0 A)(jh ? 0 B ) ? A0 B 0 = 0 .

h ?

This norm is the supremum norm in the function algebra A0 , hence it su?ces to estimate it at any point, say the origin, in terms of data, which do not change under translation. Speci?cally, we will give a bound on j0? ( ?) h (jh ? 0 A0 )(jh ? 0 B0 ) (0) ? A0 (0)B0 (0) by a quantity depending only on moduli of continuity of A0 and B0 . Then (?) is bounded by j0? I)jh I) (0) h jh ? 0 (A0 ? A0 (0)1 ? 0 (B0 ? B0 (0)1 + |A0 (0)| B0 ? j0? h jh ? 0 B0 + A0 ? j0? h jh ? 0 A0 |(j0? h jh ? 0 B0 )(0)| , where the terms in the second line go to zero by virtue of (3.16). Hence in (?) we may suppose that A0 (0) = B0 (0) = 0 and, consequently, |A0 (ξ )| ≤ m0 (A0 , ξ 2 ) . Inserting the de?nitions (3.10.a′ ) and (3.10.b′ ) of j0? h and jh ? 0 , we obtain j0? h (jh ? 0 A0 )(jh ? 0 B0 ) (0) ≤ ≤ dx dp (2π ? h)d dx dp (2π ? h)d dx′ dp′ h ? h ? |A0 (x, p)| |B0 (x′ , p′ )| tr Γh ? αx,p (Γh ? )αx′ ,p′ (Γh ?) (2π ? h)d dx′ dp′ m0 (A0 , x2 + p2 ) m0 (B0 , x′2 + p′2 ) × d (2π ? h) ×

h ? h ? ? h ? ′ ′ h ? ′ ′ ? χh ? , W (x, p)χh ? χh ? , W (x, p) W (x , p )χh ? χh ? , W (x , p ) χh ?

≤

dx dp dx′ dp′ m0 (A0 , x2 + p2 ) m0 (B0 , x′2 + p′2 ) × d d (2π ? h) (2π h ?) ?1 2 ?1 ′2 × exp (x + p2 ) exp (x + p′2 ) 4? h 4? h ?d (dθ )m0 (A0 , 4? hθ ) ?d (dθ ′ )m0 (B0 , 4? hθ ′ ) .

≤ 22 d

Note that we are justi?ed in using the weak* integrals de?ning jh ? 0 because in both integrations in the second line the de?nition of jh is used under the trace with a trace ?0 class operator. In any case, since the integrals in the last line go to zero by dominated convergence, we ?nd that j0? h (jh ? 0 A0 )(jh ? 0 B0 ) (0) → 0. The estimate involves only the moduli of continuity of A and B , which concludes the proof.

27

6. Further extensions

(1) Alternative de?nitions of jh ?h ?′ It was claimed in Section 2 that the precise de?nition of jh ?h ? ′ is not essential, since asymptotically close systems of comparison maps yield the same class of j -convergent h-sequences. In [We4] the class of alternative choices of jh ? ?h ? ′ with this property is studied systematically. Surprisingly, there is even one choice for which the chain relation jh ?h ? ′′ = jh ?h ? ′ jh ? ′h ? ′′ is satis?ed exactly. Hence the classical limit can be understood as an ordinary inductive limit of ordered normed spaces. The “sharpest possible” comparison maps satisfying the chain relation are essentially unique (i.e., up to the choice of a complex structure on phase space). (2) Norm limits of states It is easy to de?ne comparison maps ?h ?h ? ′ for density matrices, which determine the notion of a norm convergent sequence of states in the classical limit: for example we may take ?h ? ′h ? from Section 3. The limit space then consists ?h ? ′ as the pre-adjoint of jh of all integrable functions on phase space [We6]. Also the evaluation of a norm convergent sequence of states on a norm convergent sequence of observables produces a convergent sequence of numbers, or, what is the same thing, norm convergence of either states or observables implies weak convergence. The notion of weak convergence of observables in the classical limit allows one to discuss, for example, the convergence of spectral projections. However, the Product Theorem is lost for this weak convergence. The limits of WKB states or eigenstates (see Section 4.7 and Section 4.8) do not exist in norm, since the limit measures are not absolutely continuous. On the other hand, under suitable conditions the equilibrium states belonging to a norm convergent sequence of Hamiltonians do converge in norm. (3) Dynamics The de?nition of the class C 2 (A, j ), as well as the proofs of Theorem 3 and Theorem 4 will be given in [We5]. As written, these theorems require bounded Hamiltonians, which comes from the technical requirement that the time evolution should be strongly continuous on Ah ? . Not even the time evolution of the free particle satis?es this. On the other hand, by restricting Ah ? to the space of compact operators with adjoined identity, certain unbounded Hamiltonians can be treated, as well. Note, however, that a version of the Evolution Theorem can only hold if the classical time evolution exists for all times, so some restrictions on Hh ? are always needed. A good way of handling unbounded Hamiltonians is also to study the dynamics in the norm limit of states (see (2), and [Hep,Hag]). In the deformation quantization approach, dynamics was recently discussed in [Ri3]. (4) Classical trajectories The Evolution Theorem does not explain how, in the classical limit, a description of the systems in terms of trajectories becomes possible. The statistics of trajectories 28

should be the limit of a sequence of continual measurement processes depending on h. One can set up such processes quite easily in the framework of E.B. Davies [Dav], ? and thus obtains an idealized description of a measuring device which is always in interaction with the system under consideration, and produces as output a sequence of random events, each of which is described by a Poisson distributed random time, and a random point in phase space. The rate ν of random events can be chosen arbitrarily, but it is clear that a larger rate will introduce a stronger perturbation of the free evolution. What happens in the classical limit now depends on how this rate ν is scaled as ? h → 0. If we take ν → ∞, but ν ? h → 0, we will get a classical process, which is concentrated on the classical orbits, and has the initial condition as the only random parameter. On the other hand, if we take ν ? h → C , some quantum perturbations of the free evolution survive the limit, and we get a di?usion in phase space with di?usion constant proportional to C (compare [FLM]), and with a drift given by the Hamiltonian vector ?eld. Joint work on these issues is in progress with Fabio Benatti. (5) Higher orders in ? h In the WKB method one is usually not only interested in the classical limit, but in the asymptotic expansion of the wave functions to all orders in h ? . In this paper we have only considered the limit itself, for the following reason: we wanted to emphasize that the notion of convergence is almost completely insensitive to special choices of identi?cation operators jh ?h ? ′ . Given these identi?cations one can also de?ne higher orders of the asymptotic expansion of an h ? -dependent operator. But these are now much less “canonical”, and there seems little point in computing such quantities which depend on a special choice of, say, coherent states, unless there is a speci?c reason for considering a particular choice. One possible “canonical choice” of identi?cations is given in [We4] (see (2) above). (6) in?nitesimal ? h In the framework of nonstandard analysis [AFHL] the limit h ? → 0 can be carried out simply by taking h ? literally in?nitesimal. The art, as usual in this theory, is to extract from the resulting structure the relevant “standard part”. For the classical limit the idea is essentially taken from Theorem 8: the relevant observables for the classical limit are those, which are strongly continuous for phase space translations “on the standard scale”. Up to corrections of in?nitesimal norm this observable algebra is precisely the algebra A0 obtained above [WW]. This formulation is perhaps even closer to physical intuition than the one presented here. However, for the proofs we mostly had to go back to the standard proofs given in this paper. (7) Spin systems Of course, one can also consider particles with spin, or other internal degrees of freedom. How the classical limit on these internal degrees of freedom is to be taken depends on the physical question under consideration. For example, in the kinetic theory of gases, one sometimes leaves these degrees of freedom untouched, obtaining a theory of classical particles with quantum excitations. But we can also ?x the 29

spin in angular momentum units, which means that the half-integer labelling the irreducible representation of SU2 must go to in?nity. This limit can be stated exactly along the lines of this paper, with analogous results. It is essentially equivalent to a mean-?eld limit [GW]. It can also be carried out for systems of many spins [RW], for more general compact Lie groups [Duf ], and for some quantum groups [GW]. For a nonstandard version, see [WW].

Acknowledgements

This paper has grown out of a series of lectures given at the Marc Kac Seminar in Amsterdam in Summer 1993. The topic of the lectures was non-commutative large deviation theory, and the classical limit was included at the request of some members of the seminar, taking me up on my claim that the techniques I was presenting had applications to this problem. I would like to thank the members of the seminar, and in particular the organizers, Hans Maassen and Frank den Hollander, for the stimulating atmosphere of the seminar.

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33

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