Non-integrability and mixing in quantum systems: On the way to quantum chaos

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Studies in History and Philosophy of Modern Physics 38 (2007) 482–513 www.elsevier.com/locate/shpsb

Non-integrability and mixing in quantum systems: On the way to quantum chaos Mario Castagninoa, Olimpia Lombardib, a

CONICET-IAFE, Universidad Nacional de Buenos Aires, Casilla de Correos 67, Sucursal 28, 1428 Buenos Aires, Argentina b CONICET-Universidad de Buenos Aires, C. Larralde 3440, 61D, 1430, Buenos Aires, Argentina Received 18 November 2005; received in revised form 11 August 2006; accepted 25 August 2006

Abstract In spite of the increasing attention that quantum chaos has received from physicists in recent times, when the subject is considered from a conceptual viewpoint the usual opinion is that there is some kind of conflict between quantum mechanics and chaos. In this paper we follow the program of Belot and Earman, who propose to analyze the problem of quantum chaos as a particular case of the classical limit of quantum mechanics. In particular, we address the problem on the basis of our account of the classical limit, which in turn is grounded on the self-induced approach to decoherence. This strategy allows us to identify the conditions that a quantum system must satisfy to lead to nonintegrability and to mixing in the classical limit. r 2006 Elsevier Ltd. All rights reserved. Keywords: Quantum mechanics; Quantum chaos; Classical limit; Decoherence; Quantum mixing; Quantum nonintegrability

1. Introduction At present, the study of high instability has pervaded many areas of classical physics. It is known that the existence of chaos in classical mechanics has been rigorously proved only in a few highly idealized systems, and that the KAM theorem seems to impose a strong Corresponding author.

E-mail addresses: [email protected] (M. Castagnino), olimpiafi[email protected] (O. Lombardi). 1355-2198/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.shpsb.2006.07.002

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limitation on classical chaos in Hamiltonian systems. Nevertheless, the behavior of many classical systems exhibits features that can be interpreted as symptoms of chaos (irregular evolutions, approach to equilibrium, etc.). These empirical manifestations contrast with the fact that chaos in quantum systems seems to be the exception rather than the rule. Some authors even claim that there is no quantum chaos because the models resulting from quantization of chaotic classical systems usually do not exhibit chaotic behavior. These considerations have been used to consider the relative scarcity of quantum chaos as a severe threat to the correspondence principle. Extreme positions even conclude that the impossibility of quantum chaos casts doubt on the adequacy of quantum mechanics as a fundamental theory. However, on the other hand, the opinion that there is some kind of conflict between quantum mechanics and chaos sounds surprising in the light of the increasing attention that quantum chaos has received from the community of physicists during the last years. Perhaps the main reason for this confusing situation is the disagreement about what ‘quantum chaos’ means. Some authors try to define quantum chaos by analogy to the definition of chaos for classical systems. Others seek the usual indicators of chaos, such as extreme sensitivity to initial conditions, unpredictability or positive Kolmogorov entropy, in the quantum domain. Still others find the clue of quantum chaos in the non-separability of the Hamiltonians of composite quantum systems. In this paper, we shall attempt to identify the necessary conditions for the existence of quantum chaos. Our aim is to analyze the problem of quantum chaos as a particular case of the classical limit of quantum mechanics. The question is, then, how classical chaotic properties can emerge from the quantum description of a physical system. When considered in the light of the supposed scarcity of quantum chaos, it is precisely this question what has been used to put pressure on the correspondence principle. In particular, we shall address the problem of quantum chaos on the basis of our account of the classical limit of quantum mechanics, which in turn is based on the selfinduced approach to decoherence according to which a single closed system can decohere when its Hamiltonian has continuous spectrum. Our explanation of the classical limit shows that, if the quantum system is macroscopic enough, after self-induced decoherence it can be described as an ensemble of classical densities weighted by their corresponding probabilities. We shall consider the program proposed by Belot and Earman (1997) as the starting point of our discussion, because of the many points of contact between their position about quantum chaos and our approach to the classical limit. This strategy of argumentation will allow us to stress the relevance of the way in which the classical limit is explained for the analysis of the quantum chaos problem. In fact, we shall show that, in spite of the several agreements between both positions, our approach to quantum chaos differs from Belot–Earman’s proposal precisely in the conception of the classical limit. In particular, when considered in the context of our theoretical account, Belot–Earman’s definition of quantum mixing may lead to an integrable—and, then, non-chaotic—classical system in the appropriate limit. On the basis of this result, we shall finally present a formalism developed for treating quantum systems whose classical limits are non-integrable, and this will allow us to identify the conditions that a quantum system must satisfy to lead to non-integrability in the classical limit.

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2. The relative scarcity of quantum chaos There are some reasons to doubt that classical chaos actually exists in non-dissipative Hamiltonian systems. In fact, chaos has been demonstrated only in some highly idealized models; but the existence of chaotic behavior in realistic models of classical phenomena is far from being clear. Nevertheless, a definite position about the effective existence of classical chaos does not need to be taken for addressing the problem of quantum chaos. In fact, the relevant point in the discussion is that quantum mechanics should not represent an obstacle to the emergence of classical chaos. Here our aim is not to discuss the problem of the existence of chaos in classical mechanics; we shall only address the question about quantum chaos in relative, conditional terms: if classical chaotic behavior exists, quantum systems should manifest such a chaotic behavior in the adequate limit. A different, non-theoretical but empirical matter is the ‘‘widely shared sense that chaos presents a challenge either to quantum mechanics itself or to our understanding of the correspondence principle’’ (Belot & Earman, 1997, p. 149). The discussions about quantum chaos show that many authors find in this field a sort of incompatibility between classical mechanics and quantum mechanics; ‘‘some have used the quantum chaos problem to cast doubt on the correspondence principle and, as a result, on the empirical adequacy of quantum mechanics’’ (Kronz, 1998, p. 51). Even if one does not need to share these extreme positions for addressing the problem of quantum chaos, it is interesting to examine the different arguments used to ground them. At least two general strategies have been used to stress the supposed tension between chaos and quantum mechanics. The first one is a ‘‘top-down’’ strategy, which consists in obtaining the quantization of simple classical chaotic models by replacing classical functions with the corresponding quantum operators: the conflict arises because the resulting quantum models are usually non-chaotic according to some feature considered as an indicator of chaos. This is the argumentative path followed by Ford and his collaborators (Ford, Mantica, & Ristow, 1991; Ford & Mantica, 1992): by taking the notion of complexity as the key concept for defining chaos, they argue that the quantization of a classical chaotic system has null complexity and, therefore, is intrinsically non-chaotic.1 On the basis of this argument, the authors conclude the incompleteness of the quantum formalism: in the light of the supposed ubiquity of classical chaos, quantum mechanics should be replaced with a theory capable of accounting for chaotic behavior. The second general strategy consists in seeking the usual indicators of chaos directly in quantum systems and verifying that those indicators are not present in quantum evolutions. The particular arguments differ from each other with respect to the specific feature to be regarded as the relevant indicator of chaotic behavior. For instance, when the exponential divergence of trajectories is the focus, the usual claim is that quantum mechanics suppresses chaos because it is not possible to define precise trajectories in quantum evolutions as a consequence of the uncertainty principle (see Schuster, 1984; Batterman, 1991). One condition also considered as necessary for chaos in classical systems is non-linearity: there must be non-linear components coupling at least two variables together in the equation of motion. Since the quantum equations of motion are solutions of the linear 1

Ford claims to have proved a conclusive refutation of the correspondence principle. For a detailed criticism of Ford’s argument, see Belot & Earman (1997).

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Schro¨dinger equation, some researchers have concluded that quantum systems are necessarily non-chaotic (Berry, 1989).2 One way out to this conclusion is the attempt to recover quantum chaos by introducing non-linear terms in the Schro¨dinger equation; this could be achieved, for instance, by means of the general framework for introducing nonlinear operators into quantum mechanics developed by Weinberg (1989). Another feature of quantum mechanics that has been used to explain why quantum models are in general non-chaotic is the unitary character of the evolution described by the Schro¨dinger equation: since the time evolution of a quantum system changes neither the angle nor the distance between vectors corresponding to different states, quantum systems are not sensitive to initial conditions and, therefore, they are non-chaotic. This conclusion has led some authors to seek the way to quantum chaos in non-unitary approaches to quantum mechanics like the GRW theory (Ghirardi, Rimini, & Weber, 1986), where the collapse of the wave function is a physical process whose frequency of occurrence increases with the size of the quantum system. A different path is followed by Zurek and his collaborators (Paz & Zurek, 2000; Zurek, 2003; Zurek & Paz, 1994), who search for chaotic behavior in open quantum systems which can undergo non-unitary time evolutions; for these authors, only the coupling between the quantum system and its environment can explain the emergence of classical chaos. It is interesting to note that the pessimistic conclusions about the compatibility between chaos and quantum mechanics are usually based on a too narrow conception of the problem. The real conflict, which would pose a threat to the correspondence principle, would arise only if the classical limit of quantum systems did not display chaotic behavior. But the results obtained from the quantization of classical chaotic models are, at least, inconclusive, since at present it is well known that the explanation of the classical limit of quantum mechanics involves certain elements that are much more subtle than the inverse of the traditional quantization (we shall return to this point in Section 5). On the other hand, the fact that there is no chaos at the quantum level—under the assumption that somebody knows what ‘chaos at the quantum level’ means—does not prove yet the absence of chaos in the classical description emerging as the result of the classical limit: only if the transference of chaotic behavior from the quantum level to the classical level is assumed in advance, the scarcity of chaos in quantum systems could be considered a real problem in the light of the supposed ubiquity of classical chaos. For these reasons, in order to address the problem of quantum chaos we shall follow a strategy where the detailed explanation of the classical limit of quantum mechanics plays a central role. 3. Belot–Earman’s program In their paper on quantum mechanics, chaos and the correspondence principle, Belot and Earman (1997) present a general framework for the discussion of the problem of quantum chaos. On the basis of the fact that there is no consensus as to what ‘quantum chaos’ means, they propose four requirements that any definition of quantum chaos should fulfill (Belot & Earman, 1997, p. 159): (i) that it possess generality and mathematical rigor, 2 Berry (1989, 1991) invented the term ‘quantum chaology’ for the study of quantum phenomena arising in the small _, large t regime. But this terminology has not been taken up because it was censored by Physical Review Letters.

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(ii) that it agree with common intuitions, (iii) that it be clearly related to criteria of classical chaos, and (iv) that it be physically relevant. We shall take Belot–Earman’s program as the starting point of our argumentation because, among the different strategies for addressing the problem of quantum chaos, this program is the approach with more points of contact with our perspective. In the first place we consider, like Belot and Earman, that the problem of quantum chaos amounts to the problem of the emergence of classical chaos from quantum descriptions of physical systems. And we also share their view that, in turn, this is a particular aspect of the more general issue of the classical limit of quantum mechanics, that is, how classical behavior can emerge from the quantum realm. On the other hand, we also agree with Belot and Earman about the kind of quantum systems considered in the problem. Whereas many authors look for quantum chaos in open systems in order to obtain non-unitary time evolutions (see Kronz, 1998; Paz & Zurek, 2000; Zurek, 2003), Belot and Earman restrict their attention to the standard quantum-mechanical treatments of closed quantum systems: they focus exclusively on Schro¨dinger evolutions and ignore the measurement problem. According to them, this is a reasonable strategy given the absence of a consensus about quantum measurement. However, our main reason for adopting this perspective is the very nature of the problem: if the task is to explain the emergence of classical chaos from quantum descriptions, the explanation should permit quantum systems to lead to classical chaotic behavior in the classical limit even in situations where there is no measurement involved. Our viewpoint about the matter is grounded in our detailed explanation of the classical limit of quantum mechanics (Castagnino, 2004; Castagnino & Lombardi, 2003, 2005a; Castagnino & Gadella, 2006) according to which, under certain spectral conditions, a sufficiently macroscopic closed quantum system behaves as a classical statistical system, independently of any measurement process (we shall develop this point in Section 5). The third point of agreement is related to the formalism used to address the problem. Belot and Earman point out that physicists are able to derive testable predictions from quantum mechanics with no reference to the measurement problem, and this fact can be justified by a reliance on the notion of expectation values of observables and their evolutions. On this basis, the authors develop their argumentation in the language provided by the algebraic formalism of quantum mechanics. Our account of the classical limit also agrees with Belot–Earman’s approach regarding this point since, as we shall see, our analysis is completely expressed in the context of the algebraic formalism and relies on the time behavior of the expectation values of the relevant observables of the quantum system. On the basis of these points of contact between Belot–Earman’s program and our approach to the classical limit, it might be expected that the conclusions drawn from the two perspectives should also be close to each other. However, we shall see that this is not the case, and that the divergence between both approaches begins precisely when the particular view about the classical limit of quantum mechanics is considered. 4. The definition of quantum chaos As we have already noted, perhaps the main difficulty to be found in the problem of quantum chaos is the absence of a precise definition shared by all the participants in the discussion. Belot and Earman do not try to identify the ‘‘essence’’ of quantum chaos, but they attempt to find an acceptable characterization (according to the four requirements previously proposed) by looking for some of the symptoms of classical chaos at the quantum level. For

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this purpose they appeal to the Koopman formalism (Koopman, 1931), that is, the Hilbert space formulation of classical statistical mechanics, whose similarity with the standard formalism of quantum mechanics makes it easier to find the quantum counterparts to classical chaos. The task is, then, to translate the classical definitions of the properties of the ergodic hierarchy into the algebraic language, and to carry them over to the quantum case. Belot and Earman focus their attention on the definition of quantum mixing, since mixing provides a strong necessary condition for chaos. Their starting point is the characterization of an abstract dynamical system as (X, T, m), where X is a phase space that comes with a s-algebra A of subsets, m is a normalized measure on ðX ; AÞ (m(X) ¼ 1), and T: X-X is a measure preserving invertible map (m(T(A)) ¼ m(A)). In the Koopman formalism, T does not act on a single point of X, but rather on a smooth r: X-[0, N] such R that X r dm ¼ 1. If r is conceived as modeling an ensemble of systems, T acts on r by acting on each member of the ensemble; then, we can define a linear map U such that U(r(x)) ¼ r(T(x)). When rAL2(X, m), then U is a unitary map. Therefore, an abstract dynamical system is also represented by (L2(X, m), U) in the Koopman formalism. On the other hand, the functions OAL2(X, m) can be considered as linear operators on L2(X, m) R representing the observables of the system. In this case, hOir ¼ X Or dm is the expectation value of the observable represented by O in the state given by the density r. With these elements, mixing can be defined as (Belot & Earman, 1997, p. 153, Definition 2.4): Definition 1. An abstract dynamical system (X, T, m) is mixing if limn!1 mðT n A \ BÞ ¼ mðAÞmðBÞ for all measurable A; B  X . This definition says that, in a mixing system, an ensemble initially concentrated in a measurable set gets smeared out evenly over the entire phase space, in spite of the fact that T is measuring preserving. This means that, independently of the initial state r, in the infinite time limit the expectation value /OSr(n) of any observable O converges to its equilibrium value hOireq , with req unique (Belot & Earman, 1997, pp. 154–155): Proposition 1. An abstract dynamical system (L2(X, m), U) is mixing iff Z Z limn!1 hOirðnÞ ¼ limn!1 OU n ðrÞ dm ¼ O dm ¼ hOireq X

X

for any O; r 2 L2 ðX ; mÞ. In the algebraic language, an abstract dynamical system is a pair ðA; yÞ, where A is a C*-algebra representing the algebra of observables, and y is an automorphism of A representing the time evolution of the system. States j are normalized positive functionals on A. The expectation value of any observable A 2 A in a state j is given by the action of j on A: j(A). On this basis, Definition 1 can be translated into the algebraic language (Belot & Earman, 1997, p. 160, Definition 3.1): Definition 2. An abstract dynamical system ðA; yÞ is mixing if there exists a y-invariant state jE such that limn!1 jE ðAyn BÞ ¼ jE ðAÞjE ðBÞ for all A; B 2 A: Since in this definition jE plays the role of the equilibrium state, it can be proved that (Belot & Earman, 1997, p. 169):

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Proposition 2. An abstract dynamical system ðA; yÞ is mixing iff there exists a y-invariant state jE such that limn!1 jðyn AÞ ¼ jE ðAÞ

for any j acting on A and any A 2 A:

As in the case of Proposition 1, this proposition expresses the convergence to equilibrium of a mixing system: independently of the initial state j, the expectation value of any observable tends to its equilibrium value jE(A) in the infinite time limit. In other words, in a mixing system, the y-invariant state jE satisfying Proposition 2 exists and is unique. Belot and Earman propose to consider the above definitions as the characterization of the concept of mixing for any dynamical system, either classical or quantum. Therefore, a quantum system will be mixing when its algebraic representation satisfies Definition 2 —or Proposition 2—. This strategy sounds reasonable since Definition 2 seems to capture a strong sense of unpredictability usually considered as a necessary condition for chaos. Nevertheless, Belot–Earman’s proposal should also be assessed in the light of the authors’ original idea (and one of the points of agreement with our position): the problem of quantum chaos amounts to the problem of the emergence of classical chaos from quantum descriptions. Therefore, a quantum system should be defined as chaotic when its classical limit is chaotic. This requirement also applies to the case of mixing: Definition A. A quantum system is mixing if its classical limit leads to a classical mixing system. When we bear in mind this original idea, the question about the compatibility between the two definitions immediately arises: Question A. Given a quantum system that is mixing according to Definition 2 (or to Proposition 2), is it also mixing according to Definition A? Of course, the answer to this question strongly depends on how the classical limit of quantum mechanics is conceived. Belot and Earman introduce an index N to measure the ‘‘size’’ of the quantum system, in such a way that letting N-N corresponds to taking the _-0 limit. On this basis, they focus their attention on how ‘‘mixing emerges in the N-N limit’’ (Belot & Earman, 1997, p. 170). This means that, for the authors, the classical limit of quantum mechanics is given by the macroscopic limit. However, as we shall see, there are good reasons for thinking that macroscopicity is not sufficient to explain the emergence of classical behavior from the quantum realm. 5. The classical limit of quantum mechanics Often it is supposed that the classical limit obtains when the quantum system is macroscopic enough, that is, when its dimensions and masses are large, of the order of centimeters and grams: in these cases, the characteristic action S of the system is much greater than _, and the limit _-0—strictly speaking, the limit _/S-0—3 acquires a physical meaning. However, at present it is known that classicality and macroscopicity are 3 In the rest of the paper we shall use the counterfactual limit _-0 in order to simplify expressions, but in any case such a limit represents the factual limit _/S-0. For an explanation of the difference between factual and counterfactual limits, see Rohrlich (1989) and, with a different terminology, Bruer (1982).

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different concepts: there are macroscopic quantum phenomena where macroscopic interference and coherence do exist.4 This means that macroscopicity is a necessary but not a sufficient condition for the emergence of classicality. For this reason, in previous works we have developed a general explanation of the classical limit of quantum mechanics, which involves two elements: self-induced decoherence and macroscopicity (Castagnino, 2004; Castagnino & Gadella, 2006; Castagnino & Lombardi, 2005a). Decoherence, in its self-induced version, transforms quantum mechanics into a Boolean quantum mechanics where the interference terms that preclude classicality have vanished. Macroscopicity turns the resulting Boolean quantum mechanics into classical statistical mechanics, which can be adequately expressed in phase space. In the following subsections we shall outline the main elements of this approach to the classical limit. 5.1. Self-induced decoherence In the last decades, the study of the phenomenon of decoherence has acquired a great relevance in many areas of physics. The ‘‘orthodox’’ view in the field is given by the environment-induced approach, proposed by Zurek and his collaborators (Paz & Zurek, 2000; Zurek, 2003). According to these authors, decoherence is a process resulting from the interaction between a quantum system and its environment; by eliminating the ‘‘nonclassical’’ states in the Hilbert space, such a process leads to the emergence of the classical world. However, since the ‘‘openness’’ of the system is essential for environment-induced decoherence, this approach cannot explain the emergence of classicality in closed systems. In order to face this problem, we have adopted the self-induced approach to decoherence (Castagnino & Laura, 2000a, b; Castagnino & Lombardi, 2003, 2004, 2005b; Castagnino & Ordon˜ez, 2004), according to which the phenomenon of decoherence can be explained as the result of the dynamics of a closed quantum system governed by a Hamiltonian with continuous spectrum. From this perspective, classical behavior may arise in closed quantum systems, in particular, in the Universe as a whole. In the original formulation of the algebraic formalism of quantum mechanics, the algebra of observables was a C*-algebra which does not admit unbounded operators. For this reason, the self-induced approach to decoherence appeals to a nuclear algebra (see Treves, 1967; Castagnino & Ordon˜ez, 2004) whose elements are nuclei or kernels.5 This nuclear algebra is used to generate two additional topologies by means of the Nelson operator: one of them corresponds to a nuclear space V0 of generalized observables; the other topology corresponds to the space VS of states, dual of V0. Let us consider the simple case of a quantum system whose Hamiltonian has a continuous spectrum jHÞjoi ¼ ojoi

o 2 ½0; 1Þ,

(5.1)

where o and |oS are the generalized eigenvalues and eigenvectors of |H), respectively. The task is now to express a generic observable |O) in the eigenbasis of the Hamiltonian. Following the formalism of van Hove (1955), a generic observable |O) belonging to the 4

A well-known example is the case of superconducting SQUIDs (see, for instance, Leggett, 1986). Quantum phenomena at the macroscopic level have also been observed in certain ferromagnetic systems (see Stamp, 1992). 5 By means of a generalized version of the GNS theorem (Gel’fand-Naimark-Segal), it can be proved that this nuclear formalism has a representation in a rigged Hilbert space (Iguri & Castagnino, 1999).

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space VO reads Z Z Z jOÞ ¼ OðoÞjoÞ do þ Oðo; o0 Þjo; o0 Þ do do0 ,

(5.2)

where O(o) and O(o, o0 ) are generic distributions, and |o) ¼ |oS/o| and |o; o0 ) ¼ |oS/o0 | are the generalized eigenvectors of the observable |O); fjoÞ; j o; o0 Þg is the basis of VO (for the formalism of round bras and kets used in the case of continuous spectrum, see Antoniou, Suchanecki, Laura, & Tasaki, 1997; Laura & Castagnino, 1998a, b). On the other hand, states are represented by linear functionals belonging to the space VS, which is the dual of VO; therefore, a generic state (r| belonging to VS can be expressed as Z ðrj ¼

Z Z rðoÞðoj do þ

rðo; o0 Þðo; o0 j do do0 ,

(5.3)

where the first term of the r.h.s. is the singular part of (r|, and {(o|, (o; o0 |} is the basis of VS, that is, the cobasis of {|o), |o; o0 )}.6 The only condition that the distributions r(o) and r(o, o0 ) must satisfy is that of leading to a well defined expectation value of the observable |O) in the state (r|. With respect to the formalism of van Hove, the new approach introduces two restrictions (for details, see Castagnino & Laura, 2000a; Castagnino & Lombardi, 2004). The first one consists in considering only the observables |O) whose O(o) and O(o, o0 ) are regular functions; these observables define what we have called ‘van Hove space’, VVH O CVO. Since the states are represented by linear functionals over the space of observables, in this case they belong to the dual of the space VVH O . The second restriction consists in considering only the states (r| whose r(o) and r(o, o0 ) are regular functions; these states belong to a convex set S included in the dual of VVH O , since they satisfy the usual conditions of quantum states: hermiticity, non-negativity and normalization of the diagonal terms.7 Under these restrictions, decoherence follows in a straightforward way. According to the unitary von Neumann equation, (r(t)| ¼ eiHt (r0|eiHt. Therefore, the 6 The cobasis of {|o), |o; o0 )} is defined by the relations (o|o0 ) ¼ d(oo0 ), (o; o00 |o0 ; o000 ) ¼ d(oo0 ) d(o00 o000 ) and (o|o0 ; o00 ) ¼ 0. These relations are just the generalization of the relationship between the basis {|iS} and the cobasis {/j|} in the discrete case: /j|iS ¼ dij. 7 These conditions do not diminish the physical generality of the self-induced approach, since the observables 0 not belonging to VVH O (that is, whose O(o) and/or O(o, o ) are singular functions) and the states not belonging to S (that is, whose r(o) and/or r(o, o0 ) are singular functions) are not experimentally accessible. In fact, the functions (of o in our case) representing the coordinates of observables and states (in the eigenbasis of |H) in our case) are always experimentally determined on the basis of discrete measurements by interpolation. So, in our case, no matter how small the discrete energy step Do (the minimal energy difference that our instruments can discriminate) is, singular functions are observationally indistinguishable from regular functions. Therefore, in practice the functions O(o), O(o, o0 ) and r(o), r(o, o0 ) are approximated, with the desired precision, by regular functions corresponding to observables and states for which the self-induced approach works satisfactorily (for a full argument, see Castagnino & Lombardi, 2004). Of course, experimental inaccessibility cannot be used to draw ontological conclusions, but represents an empirical limitation to the possibility of disconfirming the theory. In fact, due to their experimental inaccessibility, observables not belonging to VVH O and states not belonging to S cannot be used to refute the self-induced approach: this approach should be judged in the light of its ability to account for an extensively studied phenomenon as decoherence. This is what we mean when we say that those observables and states do not diminish the physical generality of the self-induced approach.

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expectation value of the observable |O)AVVH O in the state (r(t)|AS reads Z Z Z 0 rðo; o0 Þeiðoo Þt Oðo; o0 Þdo do0 . hOirðtÞ ¼ ðrðtÞjOÞ ¼ rðoÞOðoÞdo þ

491

(5.4)

Since O(o, o0 ) and r(o, o0 ) satisfy the condition of being regular functions, when we take the limit for t-N, we can apply the Riemann-Lebesgue theorem8 according to which the second term of the right-hand side of Eq. (5.4) vanishes. Therefore, Z (5.5) limt!1 hOirðtÞ ¼ limt!1 ðrðtÞjOÞ ¼ rðoÞOðoÞ do. But this integral is equivalent to the expectation value of the observable |O) in a new state ðr j where the off-diagonal terms have vanished Z Z (5.6) ðr j ¼ rðoÞðoj do ) hOir ¼ ðr jOÞ ¼ rðoÞOðoÞ do. Therefore, for all |O)AVVH O and for all (r|AS, we obtain the limit limt!1 hOirðtÞ ¼ hOir .

(5.7)

This equation shows that the definition of self-induced decoherence involves the convergence of the expectation value of any observable belonging to VVH to a value O that can be computed as if the system were in a state represented by a diagonal density operator ðr j.9 Up to this point we have considered a simplified case where the Hamiltonian was the only dynamical variable. But in a general case we must consider a complete set of commuting observables (CSCO) {|H), |O1), y, |ON)}, whose eigenvectors are |o, o1, y, oNS. In this case, ðr j will be diagonal in the variables o, o0 , but not in general in the remaining variables. Therefore, a further diagonalization of ðr j is necessary: as the result, a new set of eigenvectors {|o, r1, y, rNS}, corresponding to a new CSCO {|H), |R1), y, |RN)}, emerges. This set defines the eigenbasis {|o, r1, y, rN), |o, r1, y, rN; o0 , r01 ; . . . ; r0N )} of the van Hove space of observables VVH O , where |o, r1, y, rN) ¼ |o, r1, y, rNS/o, r1, y, rN| and |o, r1, y, rN; o0 , r01 , y, r0N ) ¼ |o, r1, y, rNS/o0 , r01 , y, r0N |. ðr j will be completely diagonal in the cobasis of states, {(o, r1, y, rN|, (o, r1, y, rN; o0 , r01 , y, r0N |} corresponding to the new eigenbasis of VVH O (for details, see Castagnino & Laura, 2000a). This new CSCO can be called ‘preferred CSCO’ since its eigenvectors define the basis that diagonalizes ðr j.10 In this case, decoherence occurs when the expectation value /OSr(t) of R R The Riemann–Lebesgue theorem states that limx-N eixyf(y) dy ¼ 0 iff f(y)AL1 (that is, iff jf(y)j dyoN). It can be proved that decoherence also occurs when the spectrum of the Hamiltonian has a single discrete value non-overlapping with the continuous part; but if the Hamiltonian’s spectrum has more than one non-overlapping value in its discrete part, the system does not decohere (for details, see Castagnino & Lombardi, 2004). Although the self-induced approach strictly applies in the continuous case, it also leads to approximate decoherence in quasi-continuous models, that is, discrete models where (i) the energy spectrum is quasi-continuous, i.e., has a small discrete energy spacing, and (ii) the functions of energy used in the formalism are such that the sums in which they are involved can be approximated by Riemann integrals. 10 Decoherence times are computed with this method for microscopic and macroscopic systems (see Castagnino & Lombardi, 2005b); they essentially coincide with those obtained with the environment-induced approach. 8 9

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any observable |O)AVVH O converges to a value hOir , where Z Z Z ðr j ¼ rðo; r1 ; . . . ; rN Þðo; r1 ; . . . ; rN j do dr1 . . . drN . r1 ...

rN

(5.8)

o

Summing up, self-induced decoherence does not require the openness of the system of interest and its interaction with the environment: a single closed system can decohere since the diagonalization of the density operator does not depend on the openness of the system but on the continuous spectrum of the system’s Hamiltonian. 5.2. Classical limit In order to obtain the classical limit, the second step is to represent the diagonal state ðr j resulting from decoherence in the corresponding phase space and to apply the macroscopic limit _-0.11 As it is well known, the Wigner transformation W is a map from the product Hilbert space H  H onto the phase space G, which transforms quantum operators A belonging to H  H onto functions A(f) on G (see Hillery, O’Connell, Scully, & Wigner, 1984), WA ¼ AðfÞ,

(5.9) 2ðNþ1Þ

where f ¼ ðq; pÞ ¼ ðq1 ; . . . ; qNþ1 ; p1 ; . . . ; pNþ1 Þ 2 G ¼ R . With this transformation, the product of quantum operators becomes the star product in phase space, that is, the classical operation between functions on G corresponding to the multiplication of operators on H  H (see Bayen, Flato, Fronsdal, Lichnerowicz, & Sternheimer, 1978) W ðABÞ ¼ WA n WB ¼ AðfÞ n BðfÞ

(5.10)

and the quantum commutator becomes the Moyal bracket W ½A; B ¼ i_ fAðfÞ; BðfÞgmb ¼ AðfÞ n BðfÞ  BðfÞ n AðfÞ.

(5.11)

Therefore, since the quantum evolution law, dr/dt ¼ (i_)1[r, H], is given by the commutator [r, H] where H is the Hamiltonian operator, its Wigner transformation results drðfÞ=dt ¼ frðfÞ; HðfÞgmb ,

(5.12)

where H(f) ¼ WH and r(f) ¼ Wr. From a physical point of view, the most interesting feature of the Wigner transformation is that, when applied to quantum observables and quantum states, WO ¼ O(f) and Wr ¼ r(f), yields the correct expectation value of any observable O in any state r:12 Z Z hOir ¼ TrðrOÞ ¼ rðfÞOðfÞ df ¼ hOðfÞirðfÞ . (5.13) 11

Let us note that, with respect to this point, our explanation of the classical limit follows a strategy opposed to that of Belot and Earman, who exploit the similarity between the Koopman formalism of classical statistical mechanics and the standard formalism of quantum mechanics. Our approach, on the contrary, appeals to the Wigner transformation in order to compare classical statistical mechanics with quantum mechanics in its phase space representation. For a detailed discussion on this point, see Appendix A. 12 Since r(f), although always real, is not non-negatively defined, it cannot be interpreted as a probability distribution. This fact discouraged from the beginning the belief that quantum theory could be formulated as a classical probabilistic theory by means of just the Wigner transformation.

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In the traditional presentations, the Wigner transformation is mathematically defined for operators having regular functions as their coordinates; however, it has not been defined when singular functions are involved. But the peculiarity of our case is that ðr j is precisely the singular part of the initial state (r| (compare Eqs. (5.3) and (5.6)). Therefore, the Wigner transformation of singular states must be defined in this situation. In fact, the task is to find the classical distribution rc ðfÞ resulting from applying the limit _-0 to the Wigner transformation of ðr j, Z (5.14) rc ðfÞ ¼ lim_!0 W ðr j ¼ rðoÞ½lim_!0 W ðoj do. Using only the reasonable requirement that the Wigner transformation leads to the correct expectation value of any observable in a given state also in the singular case, it can be proved that, when |H) is the only dynamical variable and, therefore, G ¼ R2 (for a detailed proof, see Castagnino, 2004; Castagnino & Gadella, 2006) then 13 lim_!1 W ðoj ¼ dðHðfÞ  oÞ ¼ dðHðq; pÞ  oÞ,

(5.15)

where H(f) ¼ H(q, p) is the Wigner transformation of the quantum Hamiltonian |H), H(q, p) ¼ W|H). As a consequence, the classical distribution rc (q, p) becomes Z (5.16) rc ðq; pÞ ¼ rðoÞ dðHðq; pÞ  oÞ do. This result has a clear physical interpretation. We know that, in this simple case, the classical distribution rc (q, p) is defined in a 2-dimensional phase space, and H(q, p) ¼ o is the only global constant of motion. As a consequence, rc (q, p) is an infinite sum of classical densities d(H(q, p)o), represented by the hypersurfaces H(q, p) ¼ o in phase space and averaged by the corresponding value of the function r(o). On the other hand, r(o) is normalized and non-negatively defined due to its origin, since it represents the diagonal components of the initial quantum state (r| (see Eq. (5.3)); this fact is what permits it to be interpreted as a probability function. Therefore, rc (q, p) can be conceived as the infinite sum of the classical densities defined by the global constants of motion H(q, p) ¼ o and weighted by their corresponding probabilities given by the quantum initial condition (r(t ¼ 0)|. Once rc (q, p) has been obtained, we can explain the classical limit in phase space language. In fact, decoherence involves the convergence of the expectation value of any |O)AVVH O to a final value hOir (see Eq. (5.7)), limt!1 hOirðtÞ ¼ hOir .

(5.17)

In turn, we know that the Wigner transformation has the property of preserving the expectation values hOirðtÞ ¼ hOðq; pÞirðq;p;tÞ ,

(5.18)

hOir ¼ hOðq; pÞirðq;pÞ ,

(5.19)

where O(q, p) ¼ W|O), r(q, p, t) ¼ W(r(t)| and r(q, p) ¼ W ðr j. We also know that, for _-0, W ðr j turns out to be rc (q, p). Therefore, in the macroscopic limit, the expectation As in Castagnino & Gadella (2006), for simplicity here we have omitted a constant factor 1/(2p)N+1 that multiplies the d(H(f)o), where 2(N+1) is the dimension of G. 13

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value of any observable belonging to the van Hove space converges to a final value that can be computed in classical terms as limt!1 hOirðtÞ ¼ limt!1 hOðq; pÞirðq;p;tÞ ¼ hOðq; pÞircðq;pÞ .

(5.20)

In the general case, |H) is not the only dynamical variable, but the system has a CSCO consisting of N+1 observables {|H), |O1), y, |ON)}. As we have seen, in this situation a preferred CSCO {|H), |R1), y, |RN)} emerges, in such a way that ðr j becomes diagonal in the new basis of states {(o, r1, y, rN|, (o, r1, y, rN; o0 , r01 , y, r0N |}. In this case, ðr j is given by Eq. (5.8), and the classical distribution rc (f) becomes (for a detailed proof, see Castagnino & Gadella, 2006): Z Z Z rc ðfÞ ¼ rðo; r1 ; . . . ; rN ÞdðHðfÞ  oÞ r1 ...

rN

o

dðR1 ðfÞ  r1 Þ . . . dðRN ðfÞ  rN Þ do dr1 . . . drN ,

ð5:21Þ

where now f ¼ (q, p) ¼ (q1, y, qN+1, p1, y, pN+1) and RI(f) ¼ W|RI), with I ¼ 1 to N. This means that rc (f) is defined on a 2(N+1)-dimensional phase space, and there are N+1 global constants of motion H(f) ¼ o, RI(f) ¼ rI. Therefore, in this general case, the classical distribution rc (f) defined on R2ðNþ1Þ can be conceived as an infinite sum of classical densities d(H(f)o) d(R1(f)r1) y d(RN(f)rN), defined by the corresponding global constants of motion H(f) ¼ o, R1(f) ¼ r1, y, RN(f) ¼ rN and weighted by the corresponding probability r(o, r1, y, rN).14,15 Analogously to the 2-dimensional case, here the classical limit (decoherence and macroscopicity) can be expressed as (see Eq. (5.20)) limt!1 hOirðtÞ ¼ limt!1 hOðfÞirðf;tÞ ¼ hOðfÞircðfÞ .

(5.22)

In the next sections we shall see the points of contact between this characterization of the classical limit and Belot–Earman’s definition of quantum mixing. 6. Mixing in physical systems In order to compare the two approaches, it is necessary to begin with applying the mathematical definitions given by Belot and Earman to physical situations, since here we are not dealing with abstract dynamical systems but with classical and quantum systems. Let us consider a classical system S characterized by its canonical variables f ¼ (q, p). The state of the system, defined by the value of those variables, is represented by a point in a 2(N+1)-dimensional phase space G ¼ R2ðNþ1Þ . If S is closed, its energy is constant, and the relevant region of the phase space the region on which the trajectory representing the system’s evolution is confined—is the hypersurface SHCG defined by HðfÞ ¼ o ¼ const. But, in general, the system S has A+1 global constants of motion, 14

It is worth stressing the conceptual meaning of this account of the classical limit. The classical distribution rc(f) tells us that the system behaves as a classical statistical systems from the perspective given by any observable |O)AVVH O , but this fact does not mean that there exist real trajectories in the quantum level. The classical distribution is a coarse-grained magnitude whose features derive from the coarse-grained nature of the process of decoherence (for a detailed discussion of this conceptual point, see Castagnino & Lombardi, 2004, 2005a). 15 This account of the classical limit has been applied to the so called ‘Mott problem’ (Castagnino & Laura, 2000b) and to the description of a closed quantum universe (Castagnino & Lombardi, 2003).

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H(f) ¼ o, R1(f) ¼ r1, y, RA(f) ¼ rA with 0pApN, which define an hypersurface SCCG where the system’s evolution is confined. The measure m is defined on this hypersurface SC, and it is usually identified with the Liouville measure—or any measure absolutely continuous with respect to it. R Then, the probability function r(f) is defined on SC (r: SC[0, 1]) in such a way that Sc r(f) dm ¼ 1. Finally, the time evolution governed by the Hamilton equations is represented by a measure preserving transformation T: SC-SC, or by a unitary transformation such that U(r(f)) ¼ r(T(f)).16 With these elements we can now apply Belot–Earman’s Definition 1 to the classical system S in order to obtain the definition of classical mixing: Definition 3. Let S be a closed classical system whose states are represented on a phase space G ¼ R2ðNþ1Þ and which possesses A+1 global constants of motion, with 0pApN. S is a classical mixing system if limt!1 mðT t A \ BÞ ¼ mðAÞmðBÞ for all measurable A; B 2 SC , where SCCG is the hypersurface defined by the A+1 global constants of motion, m is the Liouville measure defined on SC, and T: SC-SC is a measure preserving transformation. In turn, if the observables of the system are represented by functions O: S C ! R, then the approach to equilibrium of the expectation value of any observable O can be expressed in similar terms to those of Proposition 1. Proposition 3. The closed classical system S is a classical mixing system iff Z Z limt!1 hOirðtÞ ¼ limt!1 OU t ðrÞ dm ¼ O dm ¼ hOireq Sc

Sc

2

for any O, rAL (X, m), and where req is the equilibrium value of the state r and it is unique on SC. Following Belot–Earman’s strategy, the next step is to translate this definition of classical mixing into the quantum language; this should amount to apply Belot–Earman’s Definition 2 to a quantum system described in the algebraic language. But the application of Definition 1 to a classical system has required to take into account the relevant region of the phase space defined by the global constants of motion of the system. Thus, the problem here is how to express the quantum counterpart of the hypersurface SC over which the initial ensemble smears out. As we have seen in the previous section, our account of the classical limit of quantum mechanics shows that the observables belonging to the preferred CSCO turn out to be the global constants of motion in the classical description. Therefore, it seems reasonable to require that quantum mixing leads to a unique time-invariant quantum state jE for each set of values of the observables {|H), |R1), y, |RN)} belonging to the preferred CSCO, that is, the CSCO that defines the basis in which the time-invariant state jE becomes diagonal. Under this requirement, the definition of classical mixing can be translated into the quantum language as 16

If the definition of classical mixing is not restricted to the relevant region of the phase space defined by the system’s constants of motion, conceptual problems arise. In particular, systems that are mixing in a relevant physical sense turn out to be non-mixing according to the definition (for a detailed discussion about this point, see Lombardi, 2003).

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Definition 4. Let S be a closed quantum system whose evolution is governed by the unitary transformation Ut defined by the Schro¨dinger equation. S is a quantum mixing system if there exists a Ut-invariant state jE such that limt!1 jE ðAU t BÞ ¼ jE ðAÞjE ðBÞ

for all A; B 2 A;

where A is the algebra of observables, and jE is unique for each set of values of the observables belonging to the preferred CSCO that defines the basis in which jE becomes diagonal. In this case, the approach to equilibrium is expressed by Proposition 4. The closed quantum system S is a quantum mixing system iff there exists a Ut-invariant state jE such that limt!1 jðU t AÞ ¼ jE ðAÞ for any j acting on A and any A 2 A; where jE is an equilibrium state that is unique for each set of values of the observables belonging to the preferred CSCO. Of course, the subtler point of this argument has been the question of deciding which quantum magnitudes are the right counterparts of the global constants of motion of classical mechanics. For this reason, such a question deserves a further attention, and we shall consider it from a different, more general viewpoint. In order to compare classical mechanics with quantum mechanics, Belot and Earman take advantage of the similarity between the Koopman formalism and the ordinary formalism of quantum mechanics. The limitation of this strategy is that the notion of constants of motion gets lost in a Hilbert space of states. But classical mechanics and quantum mechanics can also be compared by means of the inverse strategy: instead of expressing classical mechanics in the mathematical language of the Hilbert spaces, we can formulate quantum mechanics in phase space through the Wigner transformation.17 This strategy allows us to identify the quantum elements that play the same role as the global constants of motion in phase space. As we have seen, the Wigner transformation is a map from H  H onto G, which turns the product of operators into the star product and the quantum commutator into the Moyal bracket. It can be proved that (see Hillery, O’Connell, Scully, & Wigner, 1984) AðfÞ n BðfÞ ¼ AðfÞBðfÞ þ 0ð_Þ,

(6.1)

fAðfÞ; BðfÞgmb ¼ fAðfÞ; BðfÞgpb þ 0ð_2 Þ,

(6.2)

where fAðfÞ; BðfÞgpb is the well-known Poisson bracket. Therefore, in the macroscopic limit _-0, the Wigner transformation maps the algebra A of quantum operators onto the algebra AC of classical operators, the product in A onto the product in AC , and the evolution law of quantum states onto the evolution law of classical states

17

A ! AðfÞ,

(6.3)

AB ! AðfÞBðfÞ,

(6.4)

dr=dt ¼ ði_Þ1 ½r; H ! drðfÞ= dt ¼ frðfÞ; HðfÞgpb .

(6.5)

See footnote 11.

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Let us now consider a quantum system endowed with a CSCO {O0, y, ON} with O0 ¼ H. By definition of CSCO we know that ½Oi ; Oj  ¼ 0

with i; j ¼ 0 to N

(6.6)

and the corresponding Moyal brackets will be W ½Oi ; Oj  ¼ i_fOi ðfÞ; Oj ðfÞgmb ¼ i_½fOi ðfÞ; Oj ðfÞgpb þ 0ð_2 Þ ¼ 0.

(6.7)

Therefore, for _-0, {Oi(f)} is a complete set of constants of motion in involution, globally defined on the corresponding phase space fOi ðfÞ; Oj ðfÞgpb ¼ 0.

(6.8)

This shows, from a general point of view, that the quantum correlates of the global constants of motion of classical mechanics are the observables of a CSCO that contains the Hamiltonian H. It is worth stressing that this result is completely independent of decoherence. On the other hand, among all the possible CSCOs of the quantum system, decoherence selects the preferred one, which defines the basis in which the time-invariant state ðr j becomes diagonal. This means that, when viewed from the phase space perspective, the role of decoherence is to select a preferred set of global constants of motion in the Wigner representation of the quantum system. Summing up, quantum mixing must lead to classical mixing in the classical limit. On the other hand, we have shown that the observables belonging to the preferred CSCO become global constants of motion in the classical phase space description arising from the classical limit. Therefore, in the translation of the definition of classical mixing into the quantum language, we are entitled to replace the restriction to the hypersurface defined by the constants of motion of the classical system with the restriction to each set of values of the observables belonging to the preferred CSCO of the quantum system. Now we have all the elements necessary to answer the Question A posed at the end of Section 4, but now in its new version: Question B. If a quantum system is quantum mixing according to Definition 4 (or Proposition 4), is it also quantum mixing in the sense that its classical limit leads to a classical mixing system? 7. Quantum mixing? When the definition of quantum mixing is applied to physical systems according to Definition 4, it is not difficult to realize the equivalence between Proposition 4 and our characterization of self-induced decoherence. In fact, in the expression limt!1 jðU t AÞ ¼ limt!1 U t jðAÞ ¼ jE ðAÞ

(7.1)

we can replace:



Utj(A) with (r(t)|A), since these two expressions represent the expectation value of the observable A in the time-dependent state represented as Utj in the first case and as (r(t)| in the second case, and

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jE(A) with ðr jA), because both jE and ðr j represent the equilibrium state where the expectation value of A does not undergo further changes.

We thus obtain limt!1 ðrðtÞjAÞ ¼ limt!1 hAirðtÞ ¼ hAir ¼ ðr jAÞ

(7.2)

that is precisely the definition of self-induced decoherence (see Eq. (5.7)). Furthermore, it is easy to prove that ðr j is unique for each set of values of the observables of the preferred CSCO, as Proposition 4 requires. Let us remember that, when we consider the preferred CSCO {|H), |R1), y, |RN)}, ðr j becomes diagonal in the preferred basis of states; as a consequence, it can be expressed as (see Eq. (5.8)) Z Z Z ðr j ¼ ... rðo; r1 ; . . . ; rN Þðo; r1 ; . . . ; rN j do dr1 ; . . . ; drN . (7.3) r1

rN

o

If the observables of the preferred CSCO have the set of values fo ¼ a0 ; r1 ¼ a1 ; . . . ; rN ¼ aN g; then the coordinates of the diagonal part of the initial state (r| at t ¼ 0 are given by rðo; r1 ; . . . ; rN Þ ¼ dðo  a0 Þdðr1  a1 Þ . . . dðrN  aN Þ.

(7.4)

By introducing Eq. (7.4) into Eq. (7.3) Z Z Z ðr j ¼ ... dðo  a0 Þdðr1  a1 Þ . . . dðrN  aN Þðo; r1 ; . . . ; rN j do dr1 ; . . . ; drN r1

rN

o

ðr j ¼ ða0 ; a1 ; . . . ; aN j.

ð7:5Þ

This shows that ðr j ¼ (a0, a1, y, aN| is unique for each set of values fo ¼ a0 ; r1 ¼ a1 ; . . . ; rN ¼ aN g of the observables belonging to the preferred CSCO {|H), |R1), y, |RN)}. At this stage, the argument seems to show that we have arrived to the same point from two different ways. In other words, it seems right to conclude that, if a quantum system satisfies the conditions for decoherence, it is a quantum mixing system according to Definition 4. In fact, as Belot and Earman point out, if a quantum system is mixing according to their definition, then 1 is the only eigenvalue of the unitary transformation Ut and is a simple eigenvalue. It is quite clear that this property is also satisfied in the case of decoherence: ðr j is the only eigenvector of Ut and its corresponding eigenvalue is 1. Can we conclude that self-induced decoherence is a necessary and sufficient condition for quantum mixing? The answer to this question is negative, and not because there exists some hidden element in Definition 4 or in the characterization of self-induced decoherence. The reason turns out to be clear when Definition 4 is considered in the light of the Question B posed at the end of the previous section: if a quantum system is quantum mixing according to Definition 4 (or Proposition 4), is it also quantum mixing in the sense that its classical limit leads to a classical mixing system? Definition 4 does not supply the desired positive answer to this question, since the classical limit of a quantum system that fulfills the conditions imposed by such a definition may be non-classical mixing. In fact, the results of Section 5 have shown that, when the quantum system has a CSCO of N+1 observables that is sufficient to define an eigenbasis for the system’s states, the classical distribution rc (f) resulting from the classical limit is defined on a phase space R2ðNþ1Þ and has N+1 global constants of motion in involution. But, as it is well known, a classical system with n degrees of freedom and n/2 global constants of motion is integrable and, as a consequence,

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non-mixing. Therefore, a quantum system that satisfies the conditions imposed by Definition 4 cannot legitimately be considered as quantum mixing since its classical limit may lead to an integrable classical system. This conclusion makes clear that there is something missing in the general strategy of Belot and Earman, based on the mere translation of the notions associated with classical chaos to the quantum language. In particular, it shows the need of studying how nonintegrability can be expressed in quantum mechanics. 8. Non-integrable quantum systems Up to this point we have considered quantum systems endowed with a CSCO {|H), |O1), y, |ON)} of N+1 observables, that defines an eigenbasis in terms of which the state of the system can be expressed. In this case we shall say that the system has a total CSCO. As we have seen, the observables of a total CSCO turn out to be, for _-0, global constants of motion in involution in the phase space Wigner representation of the system. We have also proved that the classical limit of a quantum system endowed with a total CSCO leads to an integrable classical system and, therefore, the quantum system cannot be considered as quantum mixing. But a quantum system can also have a CSCO {|H), |O1), y, |OA)} of A+1 observables, with 0pAoN; this is the case of the systems, as a single non-ionized Helium atom (the quantum analogue to a classical three-bodies system), whose ‘‘good’’ quantum numbers are not sufficient to define their states. In this situation, we shall say that the quantum system is endowed with a partial CSCO. Since total CSCOs lead to integrability in the classical limit, it seems reasonable to search for non-integrability—and for the higher ergodic properties—in the case of partial CSCOs. Here the problem is how to treat a quantum system endowed with a partial CSCO and, in particular, how to obtain the classical limit in this case. In this section we shall present a formalism that permits this kind of quantum systems to be theoretically described in such a way that their classical limit can be obtained (for details, see Castagnino, 2005; Castagnino & Lombardi, 2006). The general idea consists in taking the Wigner representation of quantum mechanics as the starting point: in the phase space representation of the quantum system, the features leading to non-integrability in the classical limit can be studied. On this basis, we shall define the notion of local CSCO, which in turn, will allow us to obtain the classical limit for quantum systems endowed with partial CSCOs. 8.1. Local CSCO Let us begin with considering a classical system whose only global constant of motion is the Hamiltonian H(f), where f ¼ (q, p) denotes a point of the phase space R2ðNþ1Þ . However, nothing prevents us from seeking the remaining N constants of motion OI(f), with I ¼ 1 to N. These N constants of motion, of course, will be non-global; nevertheless, they will satisfy fHðfÞ; OI ðfÞgpb ¼ 0, that is, X j¼1;N

ðqHðfÞ=qpj qOI ðfÞ=qqj  HðfÞ=qqj qOI ðfÞ=qpj Þ ¼ 0.

(8.1)

(8.2)

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This system of N partial differential equations, one for each OI(f), does not have a unique solution for all points f 2 R2ðNþ1Þ since, in such a case, the OI(f) would be global constants of motion, contrary to our original assumption. Nevertheless, if the functions satisfy certain reasonable mathematical conditions, for any point fi 2 R2ðNþ1Þ this system of equations will have solutions fOIfi ðfi Þg in a maximal domain of integration Dfi around fi, which define N local constants of motion OIfi(f) in involution in such a domain (see Abraham & Marsden, 1967, in particular, the Charathe´odory-Jacobi Theorem 16.29, p. 112). When H(f) is added to the OIfi(f), we obtain a complete set of N+1 constants of motion for the domain Dfi ; fHðfÞ OIfi(f), y, ONfi(f)} (see Castagnino & Lombardi, 2006). As it is quite clear, the same procedure can be followed to obtain the complete set of constants of motion {H(f), OIfj(f), y, ONfj(f)} in any different domain Dfj . Let us now consider the case of a quantum system with a partial CSCO consisting only of the Hamiltonian |H) and whose Wigner transformation HðfÞ ¼ W jH) is defined on the phase space R2ðNþ1Þ . We know that, for _-0, H(f) is a global constant of motion in the Wigner representation on R2ðNþ1Þ . In this case we can be interested in finding the remaining N functions OIfi(f) which, for _-0, will became local constants of motion in the domain Dfi . The procedure for obtaining these OIfi(f) is analogous to the procedure followed in the classical case, but now the Moyal brackets must be used instead of the Poisson brackets fHðfÞ; OI ðfÞgmb ¼ 0 fOI ðfÞ; OJ ðfÞgmb ¼ 0.

(8.3)

In this quantum case it is possible to find N functions OIfi(f) defined in the domain Dfi , which turn out to be local constants of motion in the limit _-0; in turn, when H(f) is added, we obtain a complete set of N+1 constants of motion for the domain Dfi (for a detailed presentation, see Castagnino & Lombardi, 2006). Of course, the same strategy described above can be used to obtain the N functions corresponding to any different domain Dfj . This means that we can cover the entire phase space with regions Dfi where the corresponding functions OIfi(f) are defined.18 Now we can introduce a partition of the phase space by means of a positive partition of the identity (see Benatti, 1993, Section 3.4) X IðfÞ ¼ I fi ðfÞ ¼ 1, (8.4) i

where Ifi(f) is the characteristic or index function ( 1 if f 2 Pfi ; I fi ðfÞ ¼ 0 if fePfi

(8.5)

and Pfi\Pfj ¼ +. If we choose a partition of the identity such that always Pfi 2 Dfi , then we can define the functions AIfi(f) as ( OIfi ðfÞ if f 2 Pfi ; AIfi ðfÞ ¼ . (8.6) 0 if fePfi 18 If we replace the natural global coordinates f ¼ (q, p) with the local coordinates (aIfi, OIfi) in each domain Dfi , where the aIfi are the coordinates canonically conjugated to the OIfi, then (aIfi, OIfi) is a chart of G ¼ R2ðNþ1Þ in the domain Dfi and the set of all the charts is an atlas of G (see Castagnino & Lombardi, 2006).

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As a result, we have obtained a partition Pfi of the entire phase space in such a way that the functions AIfi(f) are non-zero only at the points fAPfi and, by construction, in the limit _-0 they will represent the local constants of motion in the region Pfi.19 Up to this point, we were working in the phase space representation of quantum mechanics, but now we have to come back to the quantum formalism by translating the AIfi(f) into the quantum language. In other words, we have to obtain the quantum operators corresponding to the phase space functions AIfi(f) by means of the Weyl transformation W1, that is, the inverse of the Wigner transformation W. This task is not possible with the OIfi(f), because they are defined in a domain Dfi 2 R2ðNþ1Þ and the Weyl transformation can be applied only on phase space functions defined on the whole phase space. But since the AIfi(f) were defined for all f 2 R2ðNþ1Þ , we can obtain the corresponding quantum operators jAIfi Þ ¼ W 1 AIfi ðfÞ.

(8.7)

Let us remember that the original functions OIfi were obtained by making zero the corresponding Moyal brackets in the domain Dfi (see Eq. (8.3)), and that Pfi  Dfi . Therefore, the AIfi(f) make zero the corresponding Moyal brackets for all f 2 R2ðNþ1Þ : for fAPfi because AIfi(f) ¼ OIfi(f), and trivially for fePfi. Since the Moyal bracket is the phase space counterpart of the quantum commutator, we can guarantee that all the operators of the set {|H), |AIfi)} commute with each other: ½jHÞ; jAIfi Þ ¼ 0

½jAIfi Þ; jAJfi Þ ¼ 0 for I; J ¼ 1 to N and for each Pfi .

(8.8)

Therefore, we shall call the set fjHÞ; jAIfi Þ; . . . ; jANfi Þg the local CSCO corresponding to the domain Dfi  R2ðNþ1Þ . This terminology may sound strange in the quantum context to the extent that the notion of locality as opposed to globality is not used in the common problems treated in quantum mechanics. Nevertheless, the concept of local CSCO acquires its meaning from the way in which it was obtained in the phase space Wigner representation, and it will be an essential element for treating those quantum systems whose classical limits turn out to be non-integrable. 8.2. The classical limit of systems with partial CSCOs As we have said at the beginning of the present section, a partial CSCO is not sufficient to define an eigenbasis in terms of which the state of the system can be expressed. Therefore, if we do not add observables to the CSCO of the system, we cannot apply the formalism presented in Section 5, which was developed for obtaining the classical limit of quantum mechanics. The observables of the local CSCOs 19

For the sake of simplicity, here we have used a characteristic function Ifi(f) to define the AIfi(f). In Castagnino & Lombardi (2006), we have used a ‘‘bump’’ smooth function Bfi(f), which has values belonging to [0,1] in a boundary zone of the corresponding domain, and we have defined AIfi ðfÞ ¼ OIfi ðfÞBfi ðfÞ. This strategy guarantees smooth connections between the functions AIfi(f) defined in adjacent domains; in particular, it can be shown that any possible discontinuity in the boundary zones introduces just a 0(_2), which vanishes when _-0 and, therefore, the Moyal brackets can be replaced with Poisson brackets in such a limit (see Castagnino & Lombardi, 2006). It can be expected that non highly idealized models have such smooth properties, since in these models the potentials are represented, not by step functions, but by continuous potential barriers that smoothly connect adjacent domains (see the example of the Sinai billiard presented in the Appendix of Castagnino & Lombardi (2006) and used to describe the Casati–Prosen model in Castagnino (2006)).

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defined in the previous subsection are precisely the observables that will allow us to apply that formalism in the case of quantum systems with partial CSCOs. Only slight modifications will be needed: the main difference will consist in the introduction of the sums and the subindices corresponding to the local CSCOs previously computed. Let us consider a quantum system endowed with a partial CSCO consisting only of the Hamiltonian |H). In order to complete the basis, let us add the observables belonging to the local CSCOs as defined in the previous subsection; thus, we obtain the set {|H), |AIfi)}, with I ¼ 1 to N and i corresponding to all the domains Pfi obtained from the partition of the phase space. We shall consider, for simplicity, that |H) as well as the |AIfi) have continuous spectra, 0pooN and NomIfioN, respectively jHÞjo; mIfi i ¼ ojo; mIfi i

jAIfi Þjo; mIfi i ¼ mIfi jo; mIfi i.

(8.9)

The set of vectors {|o; mIfi i} with I ¼ 1 to N and i corresponding to all the Pfi, is a basis of the Hilbert space since, by construction, these vectors are orthogonal to each other (see detailed proof in Castagnino & Lombardi, 2006). Then, the elements of any local CSCO can be expressed in terms of such a basis as Z XZ jHÞ ¼ ojo; mIfi Þ do dmIfi , (8.10) mIfi

i

o

XZ

jAIfi Þ ¼

Z mIfi jo; mIfi Þ do dmIfi .

mIfi

i

(8.11)

o

The observables |H) and |AIfi) belong to the van Hove space VVH O since they are particular cases of the generic van Hove observables defined in Eq. (5.2), where the second term is zero. In the basis {|o; mIfi i} a generic observable |O) belonging to the van Hove space VVH O will have the following form: Z Z XZ Ofi ðo; mIfi ; m0Ifi Þjo; mIfi ; m0Ifi Þdo dmIfi dm0Ifi jOÞ ¼ m0 Ifi

mIfi

i

þ

XZ

Z Z

m0 Ifi

mIfi

i

o

Z

o

o0

Ofi ðo; mIfi ; o0 ; m0Ifi Þjo; mIfi ; o0 ; m0Ifi Þ

0

do do dmIfi dm0Ifi ,

ð8:12Þ

where |o, mIfi; m0 Ifi) ¼ |o, mIfiS/o, m0 Ifi|, |o, mIfi; o0 , m0 Ifi) ¼ |o, mIfiS/o0 , m0 Ifi|, and Ofi(o, mIfi, m0 Ifi), Ofi(o, mIfi, o0 , m0 Ifi) are regular functions. On the other hand, the states (r(t)| belonging to the convex space S included in the dual of VVH O will be Z Z XZ rfi ðo; mIfi ; m0Ifi Þðo; mIfi ; m0Ifi jdo dmIfi dm0Ifi ðrðtÞj ¼ mIfi

i

þ

XZ i

m0 Ifi

o

Z

mIfi 0

m0 Ifi

Z Z o

o0

rfi ðo; mIfi ; o0 ; m0Ifi Þ

eiðoo Þt ðo; mIfi ; o0 ; m0Ifi jdo do0 dmIfi dm0Ifi .

ð8:13Þ

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Therefore, when we obtain the limit of the expectation value /OSr(t) for t-N, the equilibrium state ðr j results (see Section 5.1) Z Z XZ ðr j ¼ rfi ðo; mIfi ; m0Ifi Þðo; mIfi ; m0Ifi j do dmIfi dm0Ifi . (8.14) m0 Ifi

mIfi

i

o

At this stage, ðr j is diagonal in the variables (o, o0 ), but not in general in the remaining variables. As we have seen, a further diagonalization of ðr j leads to a new basis of states {(o, rIfi|, (o, rIfI; o0 r0 IfI|} such that (see Eq. (5.8)) Z XZ ðr j ¼ rfi ðo; rIfi Þðo; rIfi j do drIfi . (8.15) rIfi

i

o

The set of the relevant operators |RIfi) can be defined in terms of the corresponding new basis of observables as Z Z jRIfi Þ ¼ rIfi jo; rIfi Þ do drIfi , (8.16) rIfi

o

where rIfi and |o, rIfi) are the eigenvalues and eigenprojectors of |RIfi). In each domain Dfi , {|H), |RIfi)} is the preferred local CSCO, and {|H)} can be called ‘preferred partial CSCO’ (this result will be generalized in Section 9). Up to this point we have arrived to the same result as in Section 5.1, with the only difference of the added subindex fi; then, it is not difficult to see that the classical distribution rc (f) resulting from applying the limit _-0 to the Wigner transformation of ðr j will be (see Eq. (5.21)) Z XZ rc ðfÞ ¼ rfi ðo; rIfi ÞdðHðfÞ  oÞPI dðRIfi ðfÞ  rIfi ÞÞ do drIfi , (8.17) i

rIfi

o

where HðfÞ ¼ W jH), and RIfi ðfÞ ¼ W jRIfi ). In this case, each term of the sum is valid in the corresponding domain Dfi or Pfi, since Pfi  Dfi – and in such a domain it has the same interpretation as in the integrable case: in each local context around fi there are N local constants of motion RIfi ðfÞ ¼ rIfi and one global constant of motion HðfÞ ¼ o; thus, the classical distribution rc (f) can be conceived as an infinite sum of classical densities, each one of them defined by the constants of motion HðfÞ ¼ o, RIfi ðfÞ ¼ rIfi , and weighted by the corresponding probability rfi(o, rIfi). If the phase space functions were originally constructed in such a way that they satisfy a smooth connection at any point belonging to the intersection between different domains,20 we can guarantee that the trajectories in the domain Dfi are continuously connected with the trajectories in the domain Dfj through the points f 2 Dfi \ Dfj : one and only one trajectory passes by each point of the phase space (for details, see Castagnino & Lombardi, 2006). At this point, it is time to bring all the pieces of the argument together. Our starting point was the idea that a quantum system is mixing if its classical limit leads to a classical mixing system, that is, a classical system for which (see Proposition 3) limt!1 hOðfÞirðf;tÞ ¼ hOðfÞireq ðfÞ ,

(8.18)

where req (f) is unique on the hypersurface SC defined by the global constants of motion of the system. As we have seen, the classical limit of a quantum system endowed with a 20

See footnote 19.

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total CSCO is integrable; then, a mixing quantum system must have a partial CSCO. We have also seen that, if a quantum system with partial CSCO satisfies the conditions for having a classical limit (decoherence and macroscopicity), its classical limit leads to a rc (f) given by Eq. (8.17). As a consequence, in the phase space, the expectation value of the Wigner transformation of any observable belonging to the van Hove space VVH O settles down in a final equilibrium value as in the total CSCO case (see Eq. (5.22)) limt!1 hOðfÞirðf;tÞ ¼ hOðfÞircðfÞ .

(8.19)

By comparing Eqs. (8.19) and (8.18), we can conclude that a quantum system with a partial CSCO and a well defined classical limit is mixing when rc (f) is unique on each hypersurface HðfÞ ¼ o (since, in our case, |H) is the only observable of the partial CSCO and, therefore, HðfÞ ¼ o is the only global constant of motion). When this condition holds, rc (f) ¼ rmix c (f) will be a function only of H(f), but not of the local constants of motion RIfi(f) ¼ rIfI : Z mix rc ðfÞ ¼ rðoÞdðHðfÞ  oÞ do. (8.20) o

In the quantum language, this means that the classical distribution emerging from the classical limit is unique for each value o of the Hamiltonian |H), the only observable of the partial CSCO. 9. Quantum mixing In this section, we shall generalize the results obtained for a particular case in the previous section. This generalization will clarify the physical meaning of quantum nonintegrability and will allow us to formulate an alternative definition of quantum mixing. Given a quantum system that would require N+1 quantum numbers for defining its states, we can always distinguish between two kinds of observables:





The A+1 (0pApN) observables |GI), including |H), that we shall call ‘global’ because they lead to the global constants of motion in the classical limit described in R2ðNþ1Þ . These are the observables belonging to the CSCO {|G0) ¼ |H),y,|GA)} of the system. The observables |LKfi), with K ¼ 1 to NA, that we shall call ‘local’ because in the classical limit they lead to the local constants of motion in each domain Pfi of the phase space R2ðNþ1Þ . These are the observables that constitute, with the |GI), the local CSCO {|G0) ¼ |H),y,|GA),|L1fi),y,|L(NA)fi)} corresponding to the domain Pfi. On the basis of this characterization, we can distinguish two cases:





A ¼ N. This means that the quantum system has a total CSCO {|G0) ¼ |H),y,|GN)} whose observables turn out to be the N+1 global constants of motion in the classical description on R2ðNþ1Þ resulting from the classical limit. As a consequence, such a classical description will be integrable and we shall say that we have an integrable quantum system. AoN. This means that the quantum system has a partial CSCO {|G0) ¼ |H),y,|GA)} whose observables turn out to be the A+1 global constants of motion in the classical

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description on R2ðNþ1Þ resulting from the classical limit. But since A+1oN+1, such a classical description will be non-integrable and we shall say that we have a nonintegrable quantum system. Let us consider each case in more detail. In the case of quantum integrability, if {|G0) ¼ |H),y,|GN)} is the preferred CSCO of the system, the classical distribution rc (f) resulting from the classical limit can be expressed as21 Z rc ðfÞ ¼ rðgI ÞPI dðG I ðfÞ  gI Þ dgI (9.1) gI

with I ¼ 0 to N. The N+1 global constants of motion G I ðfÞ ¼ gI foliate the phase space R2ðNþ1Þ into submanifolds Mðg0 ; . . . ; gN Þ of dimension N, labeled by the constants gI ¼ g0 ; . . . ; gN . If the emergent classical system is endowed with actionangle variables, those submanifolds are the tori on which the trajectories are confined. On these tori, the trajectories can only be periodic or quasi-periodic and, as a consequence, they cannot give rise to mixing behavior – and, therefore, even less to chaotic behavior. In the case of quantum non-integrability, if {|G0) ¼ |H),y,|GA)} is the preferred CSCO of the system, the classical distribution rc (f) resulting from the classical limit will have the following form: Z XZ rc ðfÞ ¼ rðgI ; l Kfi Þfi PI dðG I ðfÞ  gI ÞPK dðLKfi ðfÞ  l Kfi ÞÞdgI dl Kfi , l Kfi

i

gI

(9.2) with I ¼ 0 to A and K ¼ 1 to NA. In this case, the emergent classical system has A+1 global constants of motion G I ðfÞ ¼ gI , and NA local constants of motion LKfi ðfÞ ¼ l Kfi in each domain Pfi of the phase space R2ðNþ1Þ . As in the previous case, the A+1 global constants of motion foliate the phase space into submanifolds Mðg0 ; . . . ; gA Þ, but now the submanifolds have dimension 2NA+1 and, therefore, they are not tori. At this stage we can say that we have a non-integrable quantum system because its classical limit leads to a classical non-integrable system. But quantum mixing imposes an additional requirement. We know that, if the classical system emerging from the classical limit is mixing, its time evolution must converge, for t-N, to a time independent state corresponding to the microcanonical distribution in each submanifold M. This means that the corresponding classical distribution rc (f) must be a final equilibrium distribution rc mix(f) independent of the local variables and dependent only on the global variables that define the submanifolds M: Z mix rc ðfÞ ¼ rðgI ÞPI dðG I ðfÞ  gI ÞdgI . (9.3) gI

If we remember the quantum origin of the global constants of motion that foliate the phase space into the manifolds M, we can say that rmix c ðfÞ must have a constant value for each set of values of the observables belonging to the preferred partial CSCO of the 21

For simplicity, in this subsection we shall suppose that all the observables GI have continuous spectra.

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quantum system.22 In fact, if the observables of the preferred partial CSCO {|G0) ¼ |H),y, |GA)} have the set of values {gI ¼ aI}, then (see Eq. (7.4)): rðgI Þ ¼ PI dðgI  aI Þ. By introducing Eq. (9.4) into Eq. (9.3) we obtain Z rmix ðfÞ ¼ PI dðgI  aI ÞPJ dðGJ ðfÞ  gJ Þ dgJ ¼ PI dðG I ðfÞ  aI Þ. c

(9.4)

(9.5)

gI

This expression shows that rmix c ðfÞ is unique and constant in each hypersurface defined by GI ðfÞ ¼ aI . Summing up, our theoretical account of the classical limit of quantum mechanics supplies a general framework where the problem of quantum chaos can be addressed. In particular, it provides a precise characterization of the notions of quantum non-integrability and quantum mixing. A quantum system is quantum non-integrable if its classical limit leads to a classical non-integrable system, that is, if the following two conditions hold: (1) The system has a well defined classical limit, that is: (1.1) the system satisfies the necessary conditions for decoherence, and (1.2) the system is macroscopic enough to consider that _-0–strictly speaking, that _/ S-0. (2) The system has a partial CSCO, that is, a CSCO that is not sufficient to define an eigenbasis in terms of which the system’s states can be expressed. In turn, a quantum system is quantum mixing when its classical limit leads to a classical mixing system, that is, if the following two conditions hold: (1) The system is quantum non-integrable, according to the previous conditions. (2) The value of the classical distribution emerging from the classical limit is constant and unique for each set of values of the observables belonging to the preferred partial CSCO of the system. These characterizations of quantum non-integrability and quantum mixing seem to fulfill the four requirements that, according to Belot and Earman (1997, p. 159), any definition of quantum chaos should satisfy: (i) They possess generality and mathematical rigor since they are based on a general and mathematically precise explanation of the classical limit of quantum mechanics, which does not appeal to particular examples or to approximations. (ii) They agree with common intuitions about classical non-integrability and mixing to the extent that they were designed in terms of the features of the phase space Wigner representation of quantum mechanics. 22

Let us remember that rc(f) is the classical distribution that yields the expectation value of any relevant observable |O) after decoherence and in the macroscopic limit. Therefore, it can be expected that rc(f) be a rmix c (f) (that is, that rc(f) be independent of the local variables) when the domains Dfi are very small and, as a consequence, the local variables have strong fluctuations in phase space in comparison with the variation of the functions O(f): in this situation, the expectation value /O(f)Src(f) averages out the local variables and mixing is obtained.

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(iii) They are clearly related to criteria of classical non-integrability and mixing since they lead to classical non-integrable and mixing systems, respectively, in the classical limit. (iv) They are physically relevant because they supply the theoretical tools for treating quantum systems with partial CSCOs, as the well-known case of multielectronic atoms. From this general framework, the seemingly paradoxical situation about quantum chaos can be assessed from a new perspective. The supposed scarcity of quantum chaos is due to the fact that the conclusions about this topic are drawn from studying quantum models having total CSCOs. But total CSCOs are a particular case when real quantum systems are considered. In fact, only the Hydrogen atom can be studied as a quantum system with a total CSCO; starting by the non-ionized Helium atom, all multielectronic atoms have partial CSCOs. This means that, in spite of the fact that quantum systems with total CSCOs have been the paradigmatic examples treated in quantum mechanics, when considered from a broader viewpoint, they turn out to be only a small fraction of the wide range of possible quantum systems. This situation is analogous to what happened in the classical domain: while the mathematical and computational tools required to treat non-integrable systems were not developed, the typical systems studied in classical mechanics were integrable; but once those tools were available, suddenly high instability begun to be conceived as an ubiquitous feature of the classical world. 10. Conclusions If quantum mechanics is a fundamental theory, it should be capable of describing any kind of systems, even those macroscopic systems that can also be adequately described by means of classical theories; in particular, quantum mechanics should be able to account for macroscopic chaotic behavior. The problem of quantum chaos is, then, to find out which properties a quantum system must possess in order to lead to chaotic behavior in the classical limit. Then, the relevant task is not to search for the usual indicators of classical chaos in the quantum domain, as many authors believe; what really matters is whether the quantum system possesses the right properties necessary to manifest chaotic behavior in the appropriate classical limit. When the problem is viewed from this perspective, the claim that quantum chaos puts pressure on the correspondence principle is not legitimate without a clear and precise account of the classical limit of quantum mechanics, since this is the key element for explaining how classical properties can emerge from quantum systems. In this paper we have addressed the problem of quantum chaos by means of our account of the classical limit of quantum mechanics, which in turn is based on the selfinduced approach to decoherence. We have seen that, if a quantum system satisfies the conditions required for decoherence and is macroscopic enough, it has a definite classical limit. In the usual case, when the quantum system is endowed with a CSCO whose observables define an eigenbasis for the system’s states, the emerging classical description is integrable and, therefore, non-chaotic: in the classical limit the system can be described as an ensemble of classical densities defined by global constants of motion and weighted by their corresponding probabilities. This result has led us to search

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for chaos in quantum systems whose CSCOs are not sufficient to define an eigenbasis of the Hilbert space of the system. We have shown that, in these cases, the classical limit results in a classical non-integrable description characterized by global and local constants of motion; therefore, such a description may possess the higher ergodic properties. This theoretical framework was applied to the example of the Sinai billiard (Castagnino & Lombardi, 2006, Appendix A), and used to describe the Casati–Prosen model (Castagnino, 2006), which combines a typical case of classical chaos–the Sinai billiard– with a paradigmatic quantum phenomenon—the double slit experiment. The model consists in a triangular upper billiard with perfectly reflecting walls and two slices in its base, placed on the top of a box—the radiating region—with a photographic film in its base and absorbent walls (Casati & Prosen, 2004). A quantum state with a Gaussian wave packet as initial condition bounces inside the triangle, and produces two centers of radiation in the two slices, which radiate from the billiard to the radiating zone. Computer simulations show that, when the billiard is perfectly triangular and, therefore, integrable, interference fringes appear in the film; but when the billiard is a Sinai billiard and, therefore, non-integrable, the system decoheres and interference vanishes. This model shows that decoherence may occur in a closed system, that is, without the interaction with an environment or with external noise. The results, explained by Casati and Prosen by means of a kinematical average, can be recovered and the decoherence time can be estimated when the model is treated in the framework of self-induced decoherence and the classical limit of quantum systems with partial CSCOs. Of course, this work is far from exhausting all the aspects involved in a problem as complex as that of quantum chaos. Our proposal is merely programmatic. Nevertheless, we consider that the theoretical framework presented in this paper may supply the conceptual and formal elements required to face the many still open questions in the attempt to account for the classical ergodic properties of macroscopic systems by means of quantum mechanics. Acknowledgements We are extremely grateful to Fred Kronz for his encouragement and stimulating comments. We are also grateful to the referees, whose comments have greatly improved the final version of this work. This paper was supported by grants of CONICET and of FONCYT. Appendix A We shall devote this appendix to compare Belot–Earman’s strategy with our approach to the problem of defining mixing in quantum mechanics. This task will allow us to show how the weak limit found in our account of decoherence is related with the traditional definition of mixing. Belot and Earman (1997, p. 60) summarize their strategy in the following words: ‘‘The point of this exercise is that we can now translate Definitions 2.2–2.7 into the algebraic language, and carry them over without modification to the quantum case’’. If we want to

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refer to the definitions as given in the present article, we can describe this strategy as consisting in two steps: (a) To translate the traditional definition of mixing in terms of measurable sets belonging to a s-algebra (Definition 1) or in terms of the convergence of expectation values to their equilibrium values (Proposition 1) into the algebraic language (Definition 2 and Proposition 2). This step is justified on the basis of the Koopman formalism for classical statistical mechanics. (b) To transfer the algebraic definition of classical mixing (Definition 2 and Proposition 2) without modification to the quantum level. It seems quite clear that the second step, essential for the purposes of the authors, is based on an assumption adopted by analogy to the classical case. But nothing guarantees that the definition of classical mixing, even if expressed in the algebraic language, has to be still valid in the quantum case when we take into account an independent criterion for mixing in quantum mechanics, namely, that the classical limit of a quantum mixing system be a classical mixing system. In fact, we have shown that such an assumption is not valid: a quantum system that satisfies the definition of mixing just transferred from the classical level may lead to an integrable classical system in its classical limit. Our strategy, on the contrary, does not need to translate the traditional definition of mixing into the algebraic language,23 since it translates the quantum results into the phase space language via the Wigner transformation. Once the quantum results are formulated in phase space, the convergence of the expectation values to their equilibrium values can be expressed in the same language as the traditional definition of mixing. Summing up, our strategy consists in three steps: (a0 ) (b0 ) (c0 )

To characterize decoherence and the classical limit in the algebraic formalism of quantum mechanics. To impose certain additional conditions to the quantum systems having a definite classical limit (see the last paragraphs of Section 9). To verify, via the Wigner transformation, that the classical limit of the systems that satisfy those additional conditions leads to classical mixing systems according to the traditional definition of mixing (Definition 1 and Proposition 1).

In this procedure there is no assumption adopted by analogy: the Wigner transformation is a precise formal way of expressing quantum mechanics in the language of phase space. It is worthwhile to reexamine this strategy in precise formal terms, in order to verify that each step of the argument is mathematically correct. Let us recall the mathematical definitions relevant to the discussion: Definition A1. A classical dynamical system S is represented by a triple (G, Tt, m), where:

 

G is a phase space coming with a s-algebra A of measurable subsets. Tt: G-G ðt 2 RÞ is a one-parameter family of invertible transformations with group properties.

23 We use the algebraic formalism only in quantum mechanics. Our appeal to Definition 2 and Proposition 2 (the algebraic definitions of mixing for classical systems, assumed as valid also for quantum systems by Belot and Earman) has the only aim of comparing them with our characterization of decoherence.

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m : A ! Rþ is a measure invariant under the transformation T t : 8A 2 A, m(Tt(A)) ¼ m(A).

Definition A2. (Lasota & Mackey, 1985, p. 59, Definition 4.3.1). A classical dynamical system S represented by (G, Tt, m) is mixing iff: limt!1 mðA \ T t ðBÞÞ ¼ mðAÞmðBÞ 8A; B 2 A: Definition A3. (Lasota & Mackey, 1985, p. 42). Given a classical dynamical system S represented by (G, Tt, m), the Frobenius-Perron operator Ut corresponding to Tt is a unitary evolution operator such that, for any integrable function f: G ! R, U t ðf ðxÞÞ ¼ f ðT t ðxÞÞ. Proposition A1. (Lasota & Mackey, 1985, p. 65, Theorem 4.4.1). Given a classical dynamical system S represented by (G, Tt, m) and the Frobenius-Perron operator Ut corresponding to Tt, S is mixing iff, for any integrable functions f: G ! R and g: G ! R, there is a unique function f  : G ! R such that Z Z limt!1 U t ðf Þg dm ¼ f  g dm. G

G

Definition A4. Given a classical dynamical system S represented by (G, Tt, m) and two integrable functions f : G ! R and g : G ! R, the expectation value /gSf is computed as the scalar product /f, gS: Z hgif ¼ hf ; gi ¼ fg dm. G

On the basis of Definition A4, and calling U t ðf Þ ¼ f ðtÞ, Proposition A1 can be restated by saying that S is mixing iff limt!1 hgif ðtÞ ¼ hgif  .

(A.1)

In quantum mechanics, we have used the algebraic formalism under the van Hove conditions, according to which: ^ Definition A5. A quantum dynamical system S is represented by a triple (VVH O , U t , (r0|), where:

  

VH VVH O is the nuclear space of observables: |O)AVO . ^ U t is the unitary evolution operator corresponding to the Schro¨dinger equation. VH (r0| is the initial state: (r0|AVVH is the nuclear space of states, the dual S , where VS VH space of VO .

We have proved that, under definite conditions (see Subsection 5.1), the system decoheres, VH ^ that is, 8|O)AVVH O and 8ðrj ¼ U t ðr0 j 2 V S , limt!1 hOirðtÞ ¼ hOir .

(A.2)

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Then we have imposed additional conditions (see the last paragraphs of Section 9) for the system be considered quantum mixing. The central step of the argument was to show that, under those conditions, in the classical limit Eq. (A.2) leads to the traditional characterization of mixing, that is, Eq. (A.1). For a moment we shall leave aside the additional conditions (partial CSCO, ðr j unique for each set of values of the observables belonging to the preferred partial CSCO), to examine in detail the mathematical basis of this central step. Although Eq. (A.2) is a quantum statement (|O), (r| and ðr j are elements of nuclear algebras in the algebraic formalism of quantum mechanics) and Eq. (A.1) is a classical statement (f, g and f  are functions on the phase space G), they can be related via the Wigner transformation W, which transforms the elements of the nuclear algebras into functions on G OðfÞ ¼ W jOÞ;

rðf; tÞ ¼ W ðrðtÞj;

r ðfÞ ¼ W ðr j.

Since W preserves the expectation values (as we have shown in Subsection 5.2, W can be extended to satisfy this requirement also in the singular case), we can guarantee that, VH 8|O)AVVH O and 8(r|AVS , hOirðtÞ ¼ hOðfÞirðf;tÞ . Therefore, if we apply the Wigner transformation to Eq. (A.2), we obtain limt!1 hOðfÞirðf;tÞ ¼ hOðfÞirðfÞ . Under the additional conditions mentioned above, this equation is no more than another way of expressing Eq. (A.1). As we can see, the elements of the nuclear spaces VVH O and VVH are transformed into functions on the phase space. S However, Proposition A1 (Lasota & Mackey’s Theorem 4.4.1) requires that f and g (O(f) and r(f) in our case) be integrable functions on G, that is, be ‘‘regular’’ functions. The question is if O(f) and r(f) satisfy this requirement. It can be proved that this is the case in the macroscopic limit _-0. Let us consider the simple case where |H) is the only observable of the CSCO. Following the van Hove formalism, a generic observable |O)AVVH O reads Z Z ~ jOÞ ¼ Oðo; o0 Þjo; o0 Þ do do0 ~ and the coordinates Oðo; o0 Þ have the following form: ~ Oðo; o0 Þ ¼ OðoÞdðo  o0 Þ þ Oðo; o0 Þ, where O(o) and O(o, o0 ) are regular functions. So, the observable |O) can be expressed as Z Z Z Z 0 0 0 jOÞ ¼ OðoÞ dðo  o Þjo; o Þ do do þ Oðo; o0 Þjo; o0 Þ do do0 ¼ jOS Þ þ jOR Þ, where |OS) is the ‘‘singular part’’ and |OR) is the ‘‘regular part’’ of |O). The function OR ðfÞ ¼ W jOR ) is regular since it is obtained by means of the traditional Wigner transformation (given that O(o, o0 ) is a regular function). The question is, then, if the function OS ðfÞ ¼ W jOS ) is also a regular function when obtained by means of the Wigner transformation adequately extended to deal with operators having components that contain a d(oo0 ) (being this the only extension needed for our purposes).

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Only on the basis of the reasonable requirement that the Wigner transformation leads to the correct expectation value of any observable in a given state also in the singular case, it can be proved that (see Castagnino, 2004; Castagnino & Gadella, 2006) lim_!0 W joÞ ¼ dðHðfÞ  oÞ,

(A.3)

lim_!0 W ðoj ¼ dðHðfÞ  oÞ.

(A.4)

Therefore, by using Eq. (A.3), in the macroscopic limit _-0 the Wigner transformation of |OS) results Z Z Z OðoÞdðo  o0 Þjo; o0 Þ do do0  ¼ W ½ OðoÞjoÞ do W jOS Þ ¼ W ½ Z Z ¼ OðoÞW joÞ do ¼ OðoÞdðHðfÞ  oÞ do ¼ OðHðfÞÞ ¼ OS ðfÞ. Since O(o) is a regular function, OðHðfÞÞ ¼ OS ðfÞ is also a regular function. In turn, since OS(f) and OR(f) are both regular functions, we can guarantee that OðfÞ ¼ OS ðfÞ þ OR ðfÞ will also be a regular function. By appealing to Eq. (A.4) instead of to eq. (A.3), an analogous argument can be used to prove that r(o) is also a regular function. On the other hand, it is well known that, for _-0, the quantum von Neumann equation becomes the classical Liouville equation under the Wigner transformation. This means that lim_!0 W U^ t ¼ U t , where Ut is a Frobenius-Perron operator. Therefore, all the conditions necessary for the application of Proposition A1 (Lasota & Mackey’s Theorem 4.4.1) to the case of O(f) and r(f, t) are satisfied. VH Summing up, the elements of the nuclear spaces VVH are Wigner transformed O and VS into functions on the phase space G that turn out to be regular functions in the macroscopic limit. Therefore, under certain definite conditions, the convergence of the expectation value resulting from decoherence leads to the standard definition of mixing in the classical limit. References Abraham, R., & Marsden, J. E. (1967). Foundations of Mechanics. New York: Benjamin. Antoniou, I., Suchanecki, Z., Laura, R., & Tasaki, S. (1997). Intrinsic irreversibility of quantum systems with diagonal singularity. Physica A, 241, 737–772. Batterman, R. W. (1991). Chaos, quantization and the correspondence principle. Synthese, 89, 189–227. Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., & Sternheimer, D. (1978). Deformation theory and quantization II. Physical applications. Annals of Physics, 110, 111–151. Belot, G., & Earman, J. (1997). Chaos out of order: Quantum mechanics, the correspondence principle and chaos. Studies in History and Philosophy of Modern Physics, 28, 147–182. Benatti, F. (1993). Deterministic Chaos in Infinite Quantum Systems. New York: Springer. Berry, M. V. (1989). Quantum chaology, not quantum chaos. Physica Scripta, 40, 335–336. Berry, M. V. (1991). Some quantum-to-classical asymptotics. In M. J. Giannoni, A. Voros, & J. Zinn-Justin (Eds.), Chaos and Quantum Physics. Amsterdam: North Holland. Bruer, J. T. (1982). The classical limit of quantum theory. Synthese, 50, 167–212. Casati, G., & Prosen, T. (2004). Quantum chaos and the double-slit experiment. Los Alamos National Laboratory, CD/0403038. Castagnino, M. (2004). The classical-statistical limit of quantum mechanics. Physica A, 335, 511–518. Castagnino, M. (2005). Classical limit of non-integrable sytems. Brazilian Journal of Physics, 35, 375–379.

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