Chaos, quantization, and the correspondence principle

R O B E R T W. B A T T E R M A N

C H A O S , Q U A N T I Z A T I O N , AND T H E

C O R R E S P O N D E N C E P R I N C I P L E *

I N T R O D U C T I O N

There have always been interpretive problems with Quantum Mechan- ics (QM). There are problems concerning the completeness of the theory, the possibility of hidden variables, and the interpretation of the act of measurement. Each of these also relates more or less directly to questions about the relationship between QM and its predecessor Classical Mechanics (CM). Even before the problems of completeness and measurement arose, it was recognized that some kind of connection between the two theories must be established. After all, the classical theory had been, and remains, remarkably successful in certain do- mains. This paper is about the relationship between the two theories - a question which has not received much philosophical attention for many years. Here, I discuss this old problem from a new perspective.

Thus, the question to be addressed is how the two theories can fit together. One's initial attitude is most likely to be one of boredom: "It is obvious how they are connected. The answer involves a principle which was first formulated by N. Bohr in the early development of the quantum theory; namely, the Correspondence Principle (CP). Classical physics works for large objects or systems, QM works for small; and in the limit as things 'get big' the classical laws can be recovered from the quantum theory".

Things, as we shall see, are not by any means that simple. Recent work, primarily in CM, has led to the now generally accepted conclusion that classical deterministic systems can exhibit evolutionary behavior that is in a well-defined sense extremely irregular or chaotic. How, if at all, might this fact relate to the quantum theory?

A natural initial response to this fact about CM is that now, using the CP, we might hope to make sense of some of the strange features of indeterminism which are present in QM. But, I want to just point out here how strange a response this really is. If QM is the fundamental theory, of which CM is an imperfect approximation in a limited domain

Synthese 89: 189-227, 1991. © 1991 Kluwer Academic Publishers. Printed in the Netherlands.

190 R O B E R T W . B A T T E R M A N

of validity, why should we think that facts about classical systems will be at all helpful in understanding the behavior of quantum systems? (Of course, this is a question that already occupied Bohr from the beginning of the quantum theory. In fact, such considerations led him to treat the CP as a fundamental law of QM in an attempt to avoid outright inconsistency in the theory. See Jammer, p. 116.) Yet, there is, perhaps, a psychological reason for our hope; namely, the classical point of view is the only one which to us seems to make any sense. When it comes to the standard interpretive problems of QM, we frequently find ourselves grasping at straws.

On the other hand, we may recall that CM does seem to play some kind of role in QM as it is practiced even today. After all, it is only by analogy with certain classical quantities that we are able to suggest the appropriate operators to represent particular observables in the quantum theory. Perhaps, this is another kind of correspondence between the two theories - an analogical correspondence. But, it is important to see right away that this analogical connection or correspondence between CM and QM cannot be taken very far. Such analogies may play some role for quantum systems which do not have any purely quantum mechanical properties such as spin. But, once we are dealing with these, classical quantities cannot be of much help. (It is true that similarities between quantum mechanical spin and classical angular momentum played roles in early attempts to understand spin, but it must be recognized that electrons are not classical spinning tops.)

But the fact is that classical chaos is, prima facie, no help whatsoever when it comes to illuminating the sorts of interpretive problems philoso- phers have worded about for the last sixty years or so. 1 In fact, the existence of classical chaos seems to pose a further problem of its own for QM. This is because quantum systems do not appear to exhibit any kind of chaotic behavior at all. If the quantum systems exhibit nothing like chaos or randomness in evolution, then exactly how does this affect our presumed relationship between QM and CM? What must we say about the CP?

In effect, this paper is really an extended investigation of the Corre- spondence Principle in light of the fact that chaos is generic in CM and virtually absent in QM. (Of course, it is this 'virtual absence' which will be the main focus of attention.) It is also an attempt to resolve, or at least add some direction to, an ongoing debate about how to define chaos in QM. There is a general feeling among physicists working in this area, that there must be some kind chaos in QM. I think the reason

T H E C O R R E S P O N D E N C E P R I N C I P L E 191

for this feeling is some sort of tacit, if not explicit, belief that the CP must be true. Unless the two theories can both be accommodated something is seriously wrong.

In what follows I shall argue for two main points. First, I shall try to show that the current typical understanding of the CP is wrong- headed. In fact, I shall show that Bohr's initial interpretation of the CP is quite different from the usual view. Second, once the CP is properly interpreted, it is not threatened by the fact that QM does not appear to exhibit chaotic behavior of the kind present in CM. In fact, I shall suggest that the CP actually does help settle the definitional debate. This is a claim that has been disputed recently by several authors (see, for example, van Kampen (1985) and Ford (1988)).

Section 1 discusses briefly what I take to be the key defining feature of classical chaotic systems; namely, dynamical instability. In Section 2 I show why QM does not seem at all hospitable to chaos. In particular, there seems to be no quantum analog of the chaotic time evolution ubiquitous in CM. Section 3 gives a detailed discussion of a fundamental, yet relatively unknown, paper by Bohr (1924). This paper, written in 1923, at the apogee of the old quantum theory, sets out what Bohr took to be the logical foundations of the quantum theory at that time. I shall concentrate particularly on what he has to say about the CP and how this differs from the interpretation adopted by most subsequent quantum theorists. In Section 4 1 present the geometrical theory behind the usual quantization method and show how the method depends critically on the existence of invariant regions or tori in the classical phase space. I also explain why it is that, as a general theory of quantization, this method is a failure. The fact is that more often than not, the tori simply do not exist. Finally, in Section 5 I consider recent work on the problem of determining the energy spectrum for classically chaotic systems; that is, in the domain in which the usual quantization method fails. I shall argue that this work can be seen as a relatively straightforward extension of the CP as Bohr understood it in the 1923 paper. It provides the basis for a tentative resolution of the~debate about how to define quantum chaos.

1 . C L A S S I C A L C H A O S

In this section I shall present what I take to be the key defining charac- teristic of classical chaos. The view of classical chaos that I am advocat- ing is one which makes essential reference to dynamical aspects or


properties of the systems involved. This is to be contrasted with certain other approaches which focus on such features as the randomness found in the systems' behavior or upon the failure of predictability associated with that behavior. I have argued elsewhere (Batterman forthcoming) that these latter approaches to defining or characterizing chaos are inadequate. In particular, I argue that these features are necessary but not sufficient conditions for classical chaos.

On the view I adopt, a system is chaotic, roughly, if it exhibits what is sometimes called exponential trajectory instability, or an extreme sensitive dependence on initial conditions. The idea is really quite simple. Consider two classical dynamical systems. 2 Suppose both systems are composed of N particles, where the state of each particle is specified by giving its values for positions and momentum. Such a system has N degrees of freedom. Its state is uniquely represented by a point in a 2N-dimensional Euclidean space, F, called a phase space. Since we are assuming that the system is conservative, we know that its energy is a constant of motion and so the representative point is actually restricted to a 2 N - 1-dimensional energy 'surface' in F. The path or trajectory of this representative point in F represents the dynamical evolution of the system as a whole.

Now, suppose our two systems are in almost identical states. Then their representative points will be found to be very close to each other in F; that is, they will be found to be within some small region of the phase space. Systems exhibiting a sensitive dependence on initial conditions will have representative points which diverge rapidly despite the closeness of their initial points.

A system is unstable with respect to a trajectory through a point 7 ~ F if 38 > 0 such that for any E-neighborhood N~(7) of 3 , (e>0) , 3 7 ' E N ~ ( 7 ) and 3 t > 0 such that d(4~t(3/), ~b,(7))> 8. (Here d(., .) is a Riemannian metric on F.)

Fully chaotic classical systems will characteristically have exponentially- diverging trajectories. The upshot is that two systems which are initially indistinguishable relative to some measurement procedure and governed by the same Hamiltonian (subject to the same forces) may, as a result of this instability or sensitive dependence, yield subsequent macroscopically-distinct behavior. Our ability to predict the subsequent behavior of such a system over a long period of time is severely restric-


ted because of the sensitive dependence on the initial state. Such a system is said to be chaotic. It is clear, intuitively at least, that chaotic systems cannot be periodic. 3 Otherwise, there would be no failure of long time predictability.

Modern ergodic theory has provided us with a complete hierarchy of statistical properties. Certain deterministic classical systems can be shown to possess the most random or 'indeterministic' of these statistical properties, the so-called K and Bernoulli properties. Possession of the strongest of these properties guarantees that the system's behavior is observationally indistinguishable from that of a roulette wheel. The various details and distinctions of the ergodic hierarchy are not crucial for the present investigation. What is crucial is that the proofs or demonstrations that certain classical systems possess these most random properties depend essentially on the character of the systems' phase space trajectories; that is, they depend on the existence of extreme instability of motion.

For nonchaotic conservative systems, there will exist constants of motion other than the energy. These will further confine the path of a system's trajectory in F. 4 The regions or surfaces to which the trajectories are restricted are called invariant tori. This restriction guarantees that the motion of such systems can only be periodic or conditionally (quasi) periodic. For example, a completely integrable (nonchaotic) system of two degrees of freedom will have its motion confined to a two-dimensional torus or doughnut - a two-dimensional surface in the four-dimensional F-space. (See Figure 1 in Section 4.) In general, for an integrable system of N degrees of freedom, the motion is confined to an N-dimensional torus in the 2N-dimensional phase space; whereas, a fully chaotic system of N degrees of freedom has a trajectory which wanders freely over the entire 2N - 1-dimensional energy surface. Mo- tion on an invariant torus does not allow for the extreme divergence of nearby trajectories required for the motion to be chaotic. I shall return to this difference below. The existence or nonexistence of invariant toil in the classical phase space will be discussed in detail in Section 4 where they will be seen to play a crucial role in quantizing classical systems.

It will be useful for our later discussion to very briefly outline one of the popular alternative accounts of chaos mentioned at the beginning of this section. This view might be called the algorithmic or computational complexity account. Here the idea is to use an algorithmic

194 R O B E R T W. B A T T E R M A N

complexity measure of the randomness of a mathematical sequence to determine and define the randomness or degree of chaos in the system generating the sequence. 5 Roughly speaking, a deterministic system is random just in case its orbit sequence is a random sequence. The system's orbit sequence is specified by observing which cell of the coarse-grained phase space the system's representative point is to b e found at fixed temporal intervals. We obtain a sequence of cell occupa- tion numbers. If this is a random sequence, no matter what the 'size' of the coarse-grained cells (as long as it is finite, i.e., nonzero), then the system (or its orbit) is said to be random or chaotic. 6

According to J. Ford the algorithmic complexity definition of chaos (he calls this "deterministic randomness"), "subsumes" accounts based on instability (Ford 1988, p. 130). Since I have discussed this elsewhere (Batterman forthcoming), I shall not rehearse the arguments here. Suffice it to say that I provide an example of a system which is chaotic according to the algorithmic account, but which is integrable and, hence, only linearly unstable. The algorithmic definition focuses on a necessarybut not a sufficient condition for classical chaos.

Now Ford himself repeatedly rejects proposed definitions as providing merely necessary but not sufficient conditions for chaos, particularly in the context of quantum chaos. For example, concerning the results which we consider in Section 5 regarding differences between the statistics of eigenvalue distributions for quantum systems with integrable and chaotic classical counterparts, he argues that we should be careful that their importance is not "overblown" - that they might be symptomatic of quantum chaos but not definitive of it (Ford 1988, p. 130). His arguments against various definitions (as an examination of his articles will show) all presuppose that the correct definition is the algorithmic complexity definition. Consider the following quotation concerning the "definitional confusion regarding quantum chaos" (Ford 1989, p. 369). In referring to work such as that concerning the statistics of quantum spectra mentioned above, Ford says that the definitional confusion forced the theorists

in effect, to attack the problem "sideways". Specifically they have focused their attention on properties a presumed "erratic", "irregular", "disordered", "seemingly unpredict- able" quantum system might possess. Then, if any quantum systems could be shown to exhibit such properties, these properties were immediately nominated as the defining properties of quantum chaos. Unfortunately, this methodology is equivalent to asserting that, since a duck has a broad yellow bill, web feet and oily white feathers, then any


fowl having these properties is a duck• Quite obviously, however, a fowl is a duck if and only if it exhibits all the properties of a duck, just as a quantum system is random (chaotic) if and only if it exhibits all the properties of [complexity] randomness. (Ford 1989, p. 369)

Since none of the algorithms for computing, e.g., eigenvalue distributions have been demonstrated to be sufficiently complex to imply randomness, "no evidence supporting quantum chaos has appeared thus far" (Ford 1989, p. 369). While it is true that we should be careful to avoid identifying consequences of chaos with chaos itself, it is fairly obvious that this general critique of proposed definitions of quantum chaos simply begs the question.

There is another reason Ford offers in favor of the complexity account. This is that the account is "theory neutral". It does not depend on whether the system is classical or quantal. For instance, he says that

• . . of greatest significance to us here, deterministic randomness [i.e. complexity defined chaos] is the sole definition of classical chaos which translates into quantum language without change of syllable: in quantum mechanics as well as in classical, chaos means deterministic randomness. (Ford 1988, p. 131)

But the reason this definition of chaos is theory neutral, is that it is essentially divorced from the dynamics as its failure to provide a sufficient condition for dynamical instability shows. So, this argument does not carry much weight. As we shall see in the next section, it is primarily the nature of quantum dynamics which gets blamed for the lack of chaos in QM.

2. Q U A N T U M M E C H A N I C S IS N O T H O S P I T A B L E TO C H A O S

Since the aim of this paper is in part to investigate what 'chaos' in QM might mean, the title of this section may appear premature. Neverthe- less, it is generally agreed that it is difficult for chaos to get a foothold in the quantum theory. Claims such as this must, therefore, be interpreted as saying that nothing like chaos in classical physics can be found in QM. It is instructive to see why this is so.

In the last section we were concerned with conservative, bounded Hamiltonian systems. Many, in fact most, of these systems exhibit at least some irregular or chaotic behavior. We can easily see why a bounded and isolated time independent quantum system cannot exhibit

196 ROBERT W. BATTERMAN

such behavior. The time evolution of such a system is governed by the familiar Schr6dinger equation:

0 (2.1) - - ~ ( x , t) = - L B , ( x , t)

Ot h

where H is the Hamiltonian operator for the system. For systems of the type considered, (2.1) admits solutions all of the form:

(2.2) ~ ( x , t) = E Cnd)ne -i/~tE" n

where the ~bn are the energy eigenfunctions, and the En are the eigenvalues of H. The c~ are the expansion coefficients. Since the E~ are discrete values, ~(x, t) is an almost periodic function of time. This would be analogous to the classically-integrable motion mentioned briefly in the last section. Here, unlike in the classical case, the dynamical evolution according to the Schr6dinger Equation (2.1) is itself responsible for the recurrence or periodicity of the quantum state function for bound, time independent systems. There is, therefore, no irregular Or chaotic behavior to be found in examining the solutions of (2.1). Unlike the Poincar6 recurrence result for classical systems, the periodicity here in the quantum case is not ignorable.

Another way of seeing that QM is inhospitable to chaos is by recognizing that in QM there is no analog of the classical phase space trajectory. Classical dynamical chaos is understood, as we have seen, in terms of the emergence of complexity at all levels of description in the classical phase space; that is, for chaotic systems, regions in phase space no matter how 'small' will contain microstates leading to completely distinct behavior - diverging trajectories. In QM, the uncertainty relation zXQA~P -~ h 'smooths over' regions in F-space. The very concept of complexity on infinitely fine scales has no meaning in QM.

One might, therefore, look for chaos in time dependent or externally- driven quantum systems. Such systems may have continuous spectra, and one might hope that they may exhibit some kind of chaotic evolution. Unfortunately, here, too, we encounter the "quantum suppression of classical chaos" (Berry 1987, p. 184).7 Indeed, the outlook does appear bleak for finding quantum analogs of the chaotic time evolution which is a generic feature of classical Hamiltonian systems.

But there is a feature of quantum physics which has no classical analog. This aspect concerns the existence of stationary states in QM.


Bound quantum systems are quantized: in general they cannot be found in a continuum of energy states. Instead, they are to be found in one of any number of discrete states or eigenstates. It is this aspect of QM that I want to investigate in light of the fact that there exist classically- chaotic systems. Are there quantum manifestations of classical chaos to be found in the study of the spectra - the distribution of energy eigenvalues - of quantum systems whose classical counterparts exhibit chaotic behavior? If so, the question arises as to whether we can use these properties to formulate a definition of quantum chaos.

I mentioned above that Ford, for one, answers this question in the negative. For Ford the properties of spectra we shall examine below in Section 5 might be necessary conditions for quantum chaos, but they are definitely not sufficient conditions as well. Since on the algorithmic complexity definition of chaos, the algorithms which compute the QM eigenvalues are not sufficiently complex, by 'definition' the properties of the spectra cannot themselves be chaotic properties or properties constitutive of chaos. As I noted above, however, there are good reasons not to give this argument much weight.

More importantly, the entire program of attempting to define quantum chaos rests upon a particular understanding of the relationship between classical and quantum mechanics. As I said in the introduction, the idea here is that, given the CP, classical behavior should be recover- able from quantum mechanical behavior 'in the limit'. If we take the CP to be an a priori or methodological constraint on the quantum theory, as many do, then what do we say about those nonchaotic quantum systems whose classical counterparts are chaotic? How can the quantum theory be correct if it cannot demonstrate the emergence of the chaotic behavior in the (classical) limit as the phenomena 'get bigger'? Ford states the consequence in a particularly strong form: "[e]ither quantum mechanics is a true theory which can eventually be made to expose its deterministic randomness [i.e. chaos], or more likely, contemporary quantum mechanics is a flawed theory which can be modified to include chaos" (Ford 1988, p. 132).

I claim that much of the confusion about finding and defining chaos in QM rests on this interpretation of the CP. In particular, I shall argue, contrary to Ford, that the CP supplies us with an adequate definition of quantum chaos. If the CP is properly understood as Bohr intended, at least in some of his writings, then in a sense the problem of defining quantum chaos does not arise. In fact, once we correctly understand


the CP, we should not be surprised that QM suppresses classical chaos in the ways I have indicated above.

In order to establish this claim, we must examine in detail the CP and its place in QM in some detail. In the next section, I take up this investigation.

3. THE C O R R E S P O N D E N C E PRINCIPLE

What is the role and status of the CP? In contemporary discussions of the relations between CM and QM, the CP is often employed in a manner different from that originally intended by Bohr, or so I shall argue, s What did Bohr say about the CP in the development of the 'old' quantum theory?

It appears that the earliest clear formulation of what later was called the correspondence principle is to be found in Bohr's 1918 paper 'The Quantum Theory of Line Spectra'. Within this single paper, the idea that the quantum theory being developed should be a "natural general- ization of the ordinary [classical] theory of radiation" (Bohr 1967, Section 1, p. 99) changes status from being merely an "expectation" to being a "necessary connection" (Bohr 1967, Section 2, p. 110) between the theories. Before considering what is meant by the statement that the CP expresses a necessary relation between the classical and quantum theories, it will be instructive to examine the context in which the principle is supposed to operate.

The old quantum theory was essentially a patchwork quilt - the result of attempts to reconcile the puzzling experimental results showing discrete spectra with classical electrodynamics. It was not until Heisen- berg's 1925 paper on matrix mechanics that any real attempt was made to formulate a completely distinct quantum theory. Jammer aptly de- scribes the pre-1925 'theory' as follows.

Every single quantum-theoretic problem had to be solved first in terms of classical physics; its classical solution bad then to pass through the mysterious sieve of quantum conditions or, as it happened in the majority of cases, the classical solution had to be translated into the language of quanta in conformance with the correspondence principle. Usually, the process of finding "the correct translation" was a matter of skillful guessing and intuition rather than of deductive and systematic reasoning. (Jammer, p. 196)

What was involved in this "systematic guessing, guided by the Principle of Correspondence?" (van der Waerden, p. 8).


In 1923 (English translation 1924, Bohr 1924) Bohr attempts to give an exact formulation of the postulates and principles of the quantum theory. I propose to examine in detail some of the points of this paper. As Bohr presents it, the foundation of the theory consists of two fundamental postulates and two basic principles. His aim in this paper is to investigate the question of providing a logically-consistent foundation for a theory "which accounts for numerous phenomena for which the classical theory obviously fails, and at the same time closely follows in a natural manner the applications in which the classical theory has given such good service" (Bohr 1924, p. 1). He begins by considering an isolated atomic system - a case for which the classical theory has been found empirically to be completely inadequate.

The "First Fundamental Postulate" as applied to this case states that out of all the conceivable motions of the constituents of an atom (the motions of its electrons), "there exist certain states, the so-called stationary states, which are distinguished by a peculiar stability" (Bohr 1924, p. 2). This "stability" manifests itself in the fact that changes in the system's state of motion involve 'complete' transitions from one distinct stationary state to another. Furthermore, it is only upon transition from one stationary state to another, that electromagnetic radiation can be emitted (or absorbed). This is in direct conflict with classical electrodynamics in which there are no 'preferred' motions - no peculiar stability - and where electromagnetic radiation is continuously emitted by moving charges.

In talking of the motions of electrons in the atom one is, in the old quantum theory, describing the system in classical terms. In other words, the atom is supposed to consist of particles under the influence of electromagnetic forces of attraction and repulsion, whose motions are governed by classical dynamical equations - Hamilton's equations of motion. In essence these are the same equations which govern the motions of the planets. Bohr points out that the attractive and repulsive forces between the constituents of the atom are far greater at any instant than those forces affecting the motions which result from the emission of radiation. This amounts to a purely classical claim and does not yet depend on the first postulate of the quantum theory.

Furthermore, since the reaction due to radiation forces is negligible in this context, the hope is that one will be able to describe the motions within a stationary state by classical, Hamiltonian methods. So, despite the fundamental conflict between the classical theory and the first funda-


mental postulate, according to Bohr, it will be possible "in the description of the motion in stationary states to use to a great extent concep- tions obtained from the classical theory" (Bohr 1924, p. 2).

This is not yet an expression of the CP. As we shall see, the CP plays a role in characterizing and describing transitions between stationary states - transitions which result in radiative processes. Here, no transitions are being considered, and the energy of motion within stationary states remains constant. This fact suggests that these motions can be described by the Hamiltonian equations of motion from which the constancy of energy in conservative systems follows automatically. But the requirement that energy be constant is by no means sufficient to characterize the motion completely. In other words, and this is extremely important for subsequent discussion, motions within stationary

states can take on qualitatively distinct characters. In particular, the motions may be of a simple, regular, periodic or conditionally-periodic nature; or, they might be 'irregular' or chaotic, in the dynamical instability sense considered above. At this stage in the development of the quantum theory, the latter possibility was almost completely ignored. 9

Bohr notes that it is only by considering periodic or conditionally- periodic motions within stationary states that it is possible "to reach a rational method of fixing the stationary states on the basis of the equations of motion" (Bohr 1924, p. 3). A system is periodic or multiply periodic according to Bohr if it is possible to resolve its motion into a series of harmonic vibrations in k distinct fundamental frequencies o~. The number k is called the degree of periodicity of the system. In the limiting case when k = 1, the motion is periodic; for k > 1, the motion is multiply periodic. A system with degree of periodicity k will have, according to the old quantum theory, k stationary fixed states characterized by the following relations:

(3.1) J1 = n l h , J2 = n a h , . . • , Jk = nkh

The nl are the quantum numbers for a given state and the Ji are the so-called actions which are defined roughly as follows: J i = f~0 N ~ = 1 Py dqj, where the pj and qj are, respectively, the generalized momenta and position variables describing the system's state, and the limits Zo, ~" characterize the "mechanical motions of the system". 1° In the next section I shall discuss this in the phase space terminology introduced briefly in Section 1. For now, however, I continue to present

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Bohr's characterization of the fundamental postulates and principles of the old quantum theory.

Bohr next introduces one of the two basic principles of the old quantum theory - Ehrenfest's Adiabatic Principle. This principle plays a role not only in fixing or determining the allowed stationary states, but also in justifying the use of the classical electrodynamical laws to characterize motion in those states. The Adiabatic Principle "demands that the conditions for the stationary states must be of such a kind that they define certain properties of the motion of the system, which will not change during an adiabatic transformation, if the motion is de- scr ibed. . , by help of the usual electrodynamic laws" (Bohr 1924, p. 13). Properties which meet this demand are the actions J,- in the above quantum conditions (3.1).

The Adiabatic Principle states that if a system such as an atom is subjected to slowly varying forces, it will evolve, in the limit of slow variation, from one allowed (stationary) state to another. Bohr notes that the

application of the principle is naturally limited by the demand, that the motion of the system, if it is to be described by use of classical laws, shall exhibit at each moment during the transformation the periodic properties which are necessary for the fixation of the stationary states . . . . (Bohr 1924, p. 12)

In fact, there is some truth to this statement; but it is extremely misleading. la The important point to note now is simply the essential role played by the Adiabatic Principle in the fixation of the stationary states.

We now turn to the second fundamental postulate and the second basic principle of Bohr's foundational picture for the old quantum theory. These both concern the transition from one atomic state to another. The second of Bohr's two fundamental postulates is the following frequency condition:

(3.2) h v = E ' - E "

where E' and E" are the energies of the atom in two different stationary states. It says that the frequency of the radiation emitted when an atomic system undergoes a transition from one stationary state to another is directly proportional to the difference in energies of the two states. In fact, this holds for cases of radiation absorption as well as emission. As Bohr notes, this postulate serves to sharpen the break with the classical theory of radiation.


For example, since a given atom in a particular state may 'jump' to any one of a number of final states, it looks like the process of transition depends as much on the final state as it does on the initial state. In the paper under discussion, Bohr notes that,

in the present state of the theory [1923], it is not possible to bring the occurrence of radiative processes, nor the choice between various possible transitions, into direct relation with any action which finds a place in our description of phenomena, as developed to the present time. (Bohr 1924, pp. 20-21)

The fact is that up until that time, and even until the present, no 'hidden' parameters or variables have been discovered that would re- store a 'causal' connection between the system's initial state and the character of the emitted or absorbed radiation. He notes further that Einstein's use of "the laws of probability", in his derivation of Planck's temperature radiation law, allows for a remarkably accurate subsump- tion of the experimental facts concerning transitions between stationary states.

Having demonstrated the sharp break between the two fundamental postulates and classical electrodynamics, Bohr turned to the correspondence principle, to show how some of the classical results may be applicable in characterizing atomic phenomena after all. After pre- senting Bohr's discussion of the CP in this paper, I want to briefly consider some of its modern formulations. What I shall argue ultimately is that typical contemporary applications of the principle are misleading. In fact, its interpretation, these days, is quite different from that offered by Bohr, at least as set out in the 1923 paper we are discussing. Following this, in Section 4, I shall turn to a more detailed discussion of the dynamical and geometrical considerations involved in the theory of quantization, and then, finally, to an analysis of the problem of finding a place for chaos in the quantum theory.

Bohr introduces the CP as follows. Despite the fundamental break between the classical radiation theory and the quantum theory expressed in (3.1) and (3.2), one may still establish some kind of connection between the two theories.

(I)t has yet been possible, on the basis of the quantitative conditions [(3.1) and (3.2)], to connect the occurrence of radiative processes with the motion in the atom in a way which offers an explanation for the fact that the laws of the classical theory are suitable for the description of the phenomena in a limiting region. This is attained, if the various possible radiative processes are correlated with the harmonic vibrational components, appearing in the motion of the atom. (Bohr 1924, p. 22; my emphasis)


As I mentioned earlier, periodic and multiply periodic systems are characterized by the fact that their motion can be resolved into a series of harmonic vibrations in the fundamental frequencies wl, • • •, ~ok. If there are no integers m l , . . . , m~ such that m l w l + • • • + m k w k = O,

other than ml = m2 . . . . . m~ = 0, then the motion decomposes u n i q u e l y into the harmonic vibrations of the o)i. If n; . . . . , n£ and n~ . . . . . are the quantum numbers for (3.1), respectively for an initial and final stationary state, then the frequencies of the harmonic components of the motion are given by 71w~ + • • • + rkco~ where:

(3.3) r~ = n; - n~ . . . . . r~ = n~ - n~

The frequency of the emitted (or absorbed) radiation coincides (asymp- totically) with the harmonics of the motion - ~'~w~ + • • • + r~wk.

Thus, the relations in (3.3) are called " the 'corresponding' harmonic components in the motion, and the substance of the above statement we designate as the 'Correspondence Principle' for multiply periodic systems" (Bohr 1924, p. 22). Thus, the CP is a statement to the effect that the radiative processes to which the second postulate (3.2) applies are "corre la ted" with, or "correspond" to, mechanical vibrations or periodic motions of the charged particles, the electrons, in the atom. The interesting question, of course, is what this correlation or correspondence amounts to. It cannot be that of an identification of the radiative processes with the classical harmonic motions since the postulates are incompatible with the classical theory. But, whatever the relation is, it is supposed to offer "an e x p l a n a t i o n for the fact that the laws of the classical theory are suitable for the description of the phenomena in a limiting region" (Bohr, 1924, p. 22; my emphasis). This is the limit where the quantum numbers n; and n~' are large compared to their differences n ; - ni'.

Bohr notes that in the limit where the quantum numbers are large there is an asymptotic agreement in the statistical results regarding radiation according to the quantum theory and classical electrodynamics despite the fact that there remains a fundamental incompatibility between the two theories. This asymptotic relation can be regarded "as the i n t i m a t i o n of a general law of the quantum theory for the occurrence of radiation, as it is assumed to be in the correspondence principle" (Bohr 1924, p. 23; my emphasis). Bohr presents this as an argument for the j u s t i f i c a t i o n of the CP - of the connection or correlation "between the frequencies of the components of the motion and those of


the wave-trains which are emitted on the transition" (Bohr 1924, p. 23).

I have quoted Bohr so extensively here because I think we need to be absolutely clear about how he understood the CP, and about what sort of justification he offered for it. In particular, note that as he formulates it, in this paper, the CP does not express a claim (a priori or otherwise) to the effect that the quantum theory must contain the classical theory as a limiting case. Instead, it is a claim that we can 'connect' classical motions with quantum radiation processes. The CP does not say that in the limit of large quantum numbers there must be an asymptotic agreement between the statistical results of the two theories. Instead, it is because o f the asymptotic agreement between results based on the quantum postulates (3.1) and (3.2) and those based on classical electrodynamics that we are justified in correlating radiative transitions with harmonic motions for all - not only large - quantum numbers. Bohr does not understand the CP as simply expressing that the two theories must agree in a limiting domain; rather, it is a substantive claim which connects the character of the classical motion with the quantum transitions.

Consider one final quotation in support of this reading of Bohr. Concerning a particular application of the CP, Bohr claims that

[t]his application of the Correspondence Principle. . . on the whole expresses clearly the close connection between radiation and motion in the quantum theory, which persists in spite of the fundamental difference between the character of the postulates and the continuous description of the classical theory. (Bohr 1924, p. 26)

Now, it does appear that Bohr himself did not always stick to this characterization of the CP. For example, consider again my brief discussion at the very beginning of this section, of what he says in the 1918 paper 'The Quantum Theory of Line Spectra'. There he speaks of a "necessary connection" between the frequency of the radiation as understood in the quantum theory and the limit of slow vibrations in the classical theory. Furthermore, he asserts that a necessary connection must also obtain in the limit of large quantum numbers for the intensities of the line spectra (Bohr 1967, p. 110). This does look like a demand for convergence of the quantum phenomena to the classical in the appropriate limit.

In a later 1928 paper 'The Quantum Postulate and the Recent Devel- opment of Atomic Theory' (Bohr 1983), this point of view is even more

THE C O R R E S P O N D E N C E PRINCIPLE 205

explicit. Discussing his notion of complementarity as it is involved in the characterization of stationary states, Bohr makes the following claim concerning the CP.

In view of the asymptotic connection of atomic properties with classical electrodynamics, demanded by the correspondence principle, the reciprocal exclusion of the conception of stationary states and the description of the behavior of individual particles in the atom might be regarded as a difficulty. (Bohr 1983, p. 120; my emphasis)

This is the attitude toward (or interpretation of) the CP that has become prevalent in the later literature. The CP is typically not presented as a statement of a correspondence between a type of classical motion and a particular quantum phenomenon. Instead, it is a sweeping methodological claim; one which, according to some at least, is in part constitutive of the quantum theory.

For instance, consider the following quote from Gibbins' recent book.

The correspondence principle, Bohr's most important methodological guideline, tells us that the numbers given out by a classical and a quantal explanation of some phenomenon will converge as the phenomenon 'gets bigger,' as the quantum numbers get larger, as in other words the phenomenon gets less specifically quantal. (Gibbins 1987, p. 23)

David Bohm expresses this position in a particularly strong form: "[the Correspondence Principle] states that the laws of quantum physics mus t be so chosen that in the classical limit, where many quanta are involved, the quantum laws lead to the classical equations as an average" (Bohm, p. 31; my emphasis).

Bohm toes the Copenhagen line in which the quantum theory actually "presupposes the correctness of classical concepts" (Bohm, p. 624). In other words, classical concepts cannot be entirely eliminated in favor of quantum concepts. In particular, classical concepts play an essential role in characterizing the measurement process - in describing the outcomes of measurements. On this view, the CP is "simply a consis- tency condition which requires that when the quantum theory plus its classical interpretation is carried to the limit of high quantum numbers, the simple classical theory will be obtained" (Bohm, p. 626).

By way of concluding this section, let me give a summary of the above discussion. I have considered in detail Bohr's foundations for the old quantum theory; in particular, the problem of quantization or the fixation of the so-called stationary states. At this stage the quantum theory is seen to be an amalgamation of nonclassical empirical results and classical etectrodynamics. This connection is in an important sense


mediated by the correspondence principle. I have tried to show that in Bohr's paper on the fundamental postulates - on the conceptual foundations of the theory - the CP plays an essential role. This much is not controversial. However, this role is not adequately expressed by the usual formulation of the principle that one finds in the literature. There the CP functions as a broad methodological requirement necessary for a 'correct' formulation of the quantum theory.

Instead, I have been arguing that the CP actually plays a more substantive role. It expresses a connection or correspondence between classical motions and quantum radiation phenomena. The full title of Bohr's paper is 'On the Application of the Quantum Theory to Atomic Structure: Part I. The Fundamental Postulates of the Quantum Theory'; so, it is not surprising that the CP is expressed there in terms of radiative phenomena. But, it can be generalized so that it applies to other cases of quantizing classical systems - that is, to general cases of determining the energy spectrum of bound states.

To gain a better understanding of both the Adiabatic Principle and the CP, and to bring us closer to a discussion of how chaos functions in all of this, we need to examine more carefully the kind of classical motions referred to in these two principles. Insight may be gained by adopting the geometrical or phase space point of view that I mentioned earlier.

4. Q U A N T I Z A T I O N A N D C L A S S I C A L M O T I O N I N P H A S E S P A C E

I noted in Section 1 that completely nonchaotic Hamiltonian systems have phase space trajectories confined to subregions or surfaces in the phase space called tori. Here I want to show why this is true and illustrate its importance for fixing the stationary states of quantum systems.

We consider again a time independent conservative system of N degrees of freedom. Let H = H(q, p) be the Hamiltonian function characterizing the total energy of the system, or equivalently, the forces acting upon it. q and p are respectively the generalized position coordinates (ql . . . . , qu) and the generalized momenta ( p l , . . . ,pN). Thus, a complete specification of the N qi's and pi's serves to uniquely specify the system's state - it picks out a unique point in the 2N-dimensional phase space F for the system. Hamilton's equations of motion are given as follows:


dqi _ OH(q, p) dpi _ al l (q, p) (4.1) - - , dt 0pi dt 0qi

A time independent Hamiltonian system of N degrees of freedom is integrable if and only if there are N constants of motion (including the energy, i.e., H itself) Fg(q, p) which are analytic functions of q and p, such that they are single-valued, functionally independent, and in involution. This latter condition means that the Poisson brackets defined by

(4.2) {F~(q, p), Fl (q, p)} = ~ OFk OF1 OFI OFk i Oqi Opi Oqi Opi

are all zero; that is, {F~,/71} = 0, Vk, 1 = 1 . . . . . N. An important theorem, sometimes called the Liouville-Arnold theo-

rem, is the following. If a system is integrable, then:

(i) there is a canonical transformation taking the q, p to new 'angle- action' variables O, I (ql, • • •, qN; Pa . . . . . PN) ~ ( 0 1 , • • • , O N ;

11 . . . . . IN) such that the Hamiltonian H(q, p), when expressed interms of these new variables is a function of the actions alone. In other words,

(4.3) H(q, p) ---> H ( I 1 , . . . , IN) ~--- H(I)

The action variables I~ are constants of motion - dlfldt = 0. Further- more, the angle variables evolve (using (4.1)) according to the equations

dO OH (4.4) - = w(I);

dt oI

w~ is the frequency associated with the ith action Ii. (In other words, it is the frequency with which the trajectory winds around the torus in the direction labelled by ®i.)

(ii) For each set of initial conditions, the accessible phase space is a compact manifold with the topology of an N-dimensional torus. A point on the N-torus is fully specified by giving the angle coordinates ( 0 , . . . , ON) (Arnold, p. 272).

This means, as I mentioned in Section 1, that the motion of an integrable system is confined to the N-dimensional torus, unlike that for a chaotic system which can wander over the 2 N - 1-dimensional energy surface.

208 R O B E R T W. B A T T E R M A l q

Fig. 1. 2-Torus.

As an example consider an integrable system of 2 degrees of f reedom. The canonical t ransformation takes us f rom (ql, q2; p l ,p2 )

(02, 02; I1, I2) such that (4.3) and (4.4) hold. The values of I1 and I2 define the torus - they are the radii shown in Figure 1. The angle variables 01, 02 locate points on the torus.

If all the o)g are rationally dependent - that is, if there exist integers m i , m = ( r n l , . . . , m N ) such that mltoa + • • • + m N t o N "= O, o t h e r than ms . . . . . m N = 0, then the trajectory on the torus will be closed. The motion will be periodic, and the trajectory explores only a one- dimensional region in the phase space. In such a case the orbit will close following ms circuits of 02, m2 circuits of 0 2 . . . . . mN circuits of 0N. On the other hand, if a l l of the o)~ are incommensurable so that no relationship r n l w l + • . • + m N t O N = 0 holds except for ml . . . . . mN = 0, then the trajectory will wind its way around the torus in such a way as to densely cover the entire N-dimensional torus. In this case the motion is sometimes called ergodic o n t h e t o r u s . In the intermediate cases where there are k(0 <-k-< N - 1) relations like m . o)= 0, the motion will be confined to N - k-dimensional submanifolds of the full N-torus.

The typical situation for i n t e g r a b l e motion is one where w(I) varies smoothly with the actions I. In this case tori consisting of closed trajectories have measure zero in the phase space, even though they are densely distributed. 12 So, the typical case is that where motion is ergodic on the N-torus in the 2N-dimensional phase space F.


2a. 2b. Fig. 2. C o n t o u r s on the Torus .

It is now time to say something about the actions/~. How are these constants of motion determined? They are defined as follows.

(4.5) L- = pj dq~ 1 = p . d q ( 1 - - i ~ N ) 3'i 3q

The paths of integration yi are N irreducible circuits of the toms; that is, they are topologically-independent closed loops on the toms which cannot be contracted to a point. For any closed loop which can be contracted to a point, the line integral vanishes and so gives zero contribution to the action. For an N-toms there are N distinct irreducible time independent contours Yi. For instance, in the 2-torus of Figure 1, there are 2 irreducible contours shown in Figure 2a. A loop on the toms which can be contracted to a point is shown in 2b. In general, the Yi are not, or need not be, actual time evolving trajectories. Although in the one-dimensional case they obviously are.

As I mentioned earlier, the existence of invariant tori in the classical phase space is essential for determining the energy levels of a bound system using the standard semiclassical quantization scheme developed by Bohr, Sommerfeld, Einstein, Wentzel, Kramers, Brillouin, and Mas- lov. In the textbooks this method is usually named for various combi- nations of these investigators, e.g., WKB, EBK, etc. I shall follow Berry and simply call this method "Toms Quantization" (Berry 1983, p. 214).

One would like exact formulae which yield the energies for each eigenstate. Except in the simplest cases this ideal is not attainable. Instead, what toms quantization gives us is a semiclassical formula that yields the energies with increasing accuracy as h ~ 0. The quantum conditions, due primarily to Einstein who first formulated the


I 2 / ~

19/2

17/2 15/2! 13/2

11/2

9/2

7/2

512 3/2

1/2

, ° ° • ° • ° • ° • ° ,

1 3 5 7 9 11 13 15 1719 2 2 2 2 2 2 2 2 2 2

Fig. 3. Lattice of Quantized Actions.

E(I) = E~

E(I) = Ej

I 1 / ~

Bohr/Sommerfeld rules using the coordinate independent line integrals (4.5), are given by the following formula:

N

(4.6) I n = 2 ~ ~ i~= p jdq j=(n+l l4a)h , Y i

where n = (nl . . . . . rtN) , a = (al . . . . , aN) and the ni and ai are positive integers or zero. The a,- are not quantum numbers. They are called the Maslov indices and characterize certain topological features of how the torus is embedded in the phase space. Thus, a quantum energy state labelled by quantum numbers n = (n, . . . . . nu ) is associated with an N-torus with actions In given by (4.6). The energy eigenvalues En are given by the Hamiltonian (4.3) written as a function of the actions alone.

(4.7) E , = H(In) or, explicitly, En = Enl . . . . . . N = H ( ( n + l l4a)h)

One can see what is going on here by looking at the action space. Consider the special case of a two-dimensional nonlinear oscillator. The Maslov indices for this system are a = (2, 2). From (4.6) we see that the quantum numbers n = (nl, r t 2 ) with the ni positive integers or zero form a regular lattice in /-space as shown in Figure 3 (Ozorio

THE C O R R E S P O N D E N C E P R I N C I P L E 211

deAlmeida, p. 168). The quantum energy levels are those energy contours in/-space which intersect a lattice point. Thus, in Figure 3 we have Ei = ET/e, 13/2 and Ej = E17/2,5/2 using (4.7). This is essentially the quantization method which is taught in the textbooks. The quantum conditions (4.6) are refinements of Bohr's quantum conditions (3.1).

It is now time to show that as a general method for determining energy eigenvalues, the torus quantization method is completely inadequate. This was pointed out by Einstein in 1917 in the very paper where he introduces the quantum conditions (4.6) based on the coordinate independent line integrals ( 4 . 5 ) . 13 E i n s t e i n noted that since the quantum conditions depend essentially on the existence of invariant tori, there are no quantum conditions for those systems whose trajectories are ergodic. (Ergodicity is the weakest property in the ergodic hierarchy, mentioned in Section l, whose strongest properties are manifest in fully chaotic systems.) Ergodic trajectories are those that wander throughout the entire 2 N - 1-dimensional energy surface; however, a system can be ergodic without exponential separation of neighboring trajectories. So, torus quantization can fail even for systems which are not (even close to being) chaotic.

At the time of Einstein's paper, it was generally believed that most Hamiltonian systems were ergodic for reasons having to do with the justification and validity of the methods used in equilibrium statistical mechanics. 14 However, in spite of the fundamental importance of Ein- stein's paper, his warning that no quantization rule exists for the majority of Hamiltonian systems, was in Gutzwiller's words, "totally ignored for forty year" (Gutzwiller 1982, p. 184). So far in our discussion we have ignored this fact as well; that is, the relationship between classical and quantum mechanics in the semiclassical domain where the CP is supposed to apply has only been established insofar as classical trajectories can be shown to lie on invariant tori.

Perhaps it is not quite fair to claim, as Gutzwiller does, that Einstein's remarks were completely ignored. In fact, I believe Bohr addresses the issue in the paper we considered so extensively in the last section. 15 It is instructive to see what he says. This will provide further evidence for my reading of Bohr's attitude toward the CP.

Bohr notes that:

[t]he general solution of [Hamilton's equations (4.1)] frequently yields motions of a far more complicated character [than simply or multiply periodic motion]. In such a case, the considerations [of the periodic/tori based method] are not consistent with the exis-


tence and stability of stationary states, whose energy is fixed with the same exactness as in multiply periodic systems. But now, in order to give an account of the observed properties of the elements, we are f o rced to assume that the atoms, in the absence of external forces at any rate, always possess "sharp" stationary states, although the general solution of the equations of motion [(4.1)] for atoms with several e l e c t r o n s . . , exhibits no simple periodic p r o p e r t i e s . . . (Bohr 1924, p. 15; my emphasis)

For such cases, Bohr says that "we must be prepared to admit that the motion of the particles in the stationary states cannot be described by the use of the classical dynamical laws with greater exactness than the exactness with which the motion exhibits simple periodic properties according to those laws". Finally, he concludes that

this general failure of classical laws shows that, even for the case of a harmonic interplay, we must expect that, neither the fixation of the energy, nor the testing of the stability can be strictly carried out by the use of the principles of ordinary mechanics in cases in which the interactions of the electrons cannot be produced adiabatically [i.e., when the integral invariants Ii cannot be found. But, see note 11.] (Bohr 1924, p. 16)

How are we supposed to interpret these remarks? Bohr claims that to account for the observed phenomena we are always forced to assume the existence of stationary states, despite the fact that in most cases the classically-based method fails to ensure their existence. This as- sumption is a fundamental fact about the quantum theory. For Bohr the fact that most Hamiltonian systems are nonintegrable is just another symptom of the fundamental incompatibility of the quantum theory with classical physics. It is another instance of the 'failure of classical laws'.

Now, if in this paper Bohr had understood the CP to be a general methodological principle of, or constraint upon, the quantum theory of the sort discussed at the end of Section 3, then he may very well have reached a conclusion about the quantum theory similar to Ford's. That is, since there appears to be no way to quantize classically-nonintegrable systems using the only method (torus quantization) known, he might have concluded that the quantum theory does not contain the classical theory as a limiting case. (Of course, Ford takes this a step further arguing that this constitutes a flaw of the quantum theory.) Instead, even in the face of Einstein's remark, Bohr continues to maintain that

it still seems possible, even for atoms with many electrons [the classical (and nonintegrable) many body problem], to characterize their motion in a rational manner by the introduction of quantum numbers. In the fixation of these quantum numbers,

THE CORRESPONDENCE PRINCIPLE 213

considerations which rest on the Adiabatic Principle, as well as on the Correspondence p r i n c i p l e . . , play an important role. (Bohr 1924, p. 16; my emphasis)

As I mentioned in Section 2, one of the main points of this paper is to show that Bohr's CP, properly interpreted, continues to play an important role even when the classical motion is chaotic. Furthermore, the CP settles the question about a proper definition of quantum chaos in a natural way. But before we get to these issues I must for the sake of completeness discuss one more extremely important aspect of the classical theory of nonintegrable systems.

The discussion above might easily give one the impression that classical motion is either integrable or fully ergodic; and that the latter is the rule and the former is the exception . It is true that most conservative Hamiltonian systems of the kind we are considering are not integrable, but there is an important theorem which guarantees in a large number of cases that this nonintegrable motion is not ergodic either. This theorem, is called the KAM theorem, after Kolmogorov, Arnold, and Moser.

We can call systems which are neither integrable nor ergodic, quasi- integrable systems. They are characterized by smooth Hamiltonians of the form:

(4.8) H(®, I) = Ho (I) + ~H1 (O, I)

Ho is an integrable Hamiltonian, and a small parameter ~ turns on a perturbation represented by H1 which we can take to be a multiply periodic function of the angles O. Clearly, when ~ = 0 we have integrable motion and the phase space is filled with invariant tori. The KAM theorem guarantees that for sufficiently small a there remain regions in the phase space of f ini te measure on which trajectories are regular; that is, there remain invariant tori associated with the first integrals of the motion. The remaining phase space contains trajectories which exhibit stochastic or chaotic behavior and appear to explore regions of dimension 2 N - 1.16 These chaotic and regular trajectories are closely intertwined so that there will be chaotic trajectories lying arbitrarily closely to 'regular' points on the entire 2 N - 1-dimensional energy surface.

The characterization of this complex classical motion is a difficult problem in classical mechanics, although it is now fairly well understood through various techniques (see Lichtenberg and Liebermann). To the


extent that these classical techniques can provide the actions and energies for the remaining undestroyed tori, we can expect the method of torus quantization to yield a corresponding finite proportion of the energy levels we are after. But for the remaining stochastic regions, torus quantization is doomed to failure. Even worse is the ergodic case where, as we have seen, no tori exist.

So, the question remains: Does there exist a quantization method for classically-ergodic, afortiori, chaotic systems? It would be overly optimistic to answer this in the affirmitive. Nevertheless, in recent years some remarkable results have been forthcoming. These results do not succeed in providing the desired energy levels except in certain very special cases, but they show that the distribution of energy levels exhibit different statistics, depending on whether the classical motion is regular or chaotic. In the next section I shall present briefly some of these results and discuss the theory through which they may be explained.

As in the case of torus quantization, we shall see that classical periodic motion plays the key role. As I shall try to argue, on a proper understanding of the CP, this is to be expected. Furthermore, the CP suggests (contrary to Ford and others 17) that we take the existence of the different energy level statistics as definitive of quantum chaos.

5 . Q U A N T U M C H A O S

Section 2 discussed arguments which purport to show that QM is inhospitable to chaos. The most persuasive of these arguments is the one which shows that the quantum analogs of bound conservative Hamilton- ian systems all exhibit periodic motion. The solutions to the Schr6dinger equation, unlike the solutions to the classical Hamiltonian equations of motion, do not allow for chaotic time evolution. The mathematical structures of the equations of motion in the two theories are of an entirely different type. Therefore, it is not very surprising that we can find no chaos in quantum evolutions. If the CP is interpreted as being some general methodological principle (as we have seen, many do), then this fact would naturally suggest a failure of correspondence between the two theories.

In Section 3 I examined in detail Bohr's views about the CP and the relations between CM and the quantum theory at a crucial point in its evolution - just before Heisenberg and Schrrdinger equipped the theory with its own dynamics, its own equations of motion. The so-called


'old' quantum theory has now been reinterpreted as semi-classical mechanics; that is, it is now a discipline which examines the relationships between CM and QM as h --~ 0. It seems entirely reasonable to question whether and how the CP, as used in the old quantum theory, carries over into the new quantum theory. I claim that much of the confusion surrounding the problem of defining chaos in QM is due to the fact that the CP does not carry over into every aspect of the new quantum theory. It is a mistake to interpret the CP in such a way that it does. It has restricted validity and cannot be used to formulate genuine correspondences between all features of CM and QM - particularly, those involving quantum time evolution.

Once we notice that the CP was introduced by Bohr in the context of the old quantum theory, where it was understood to express a far-reaching 'connection' or correspondence between classical periodic motions and discrete quantum properties or processes, we should be led to examine the consequences of the existence of classical chaos in the semi-classical domain. There seem to be two possible outcomes of such an investigation. Either (1) classical chaos demonstrates that the CP fails (i.e., that it can be shown to lack physical content) andso the sorts of ties between CM and QM which are cursorily presented in most textbooks are generally false, or at best misleading (since the ties would then hold only in the exceptional case of integrable motion).18 Or, (2) the CP can be maintained and, in fact, used to further investigate the connections between classical motions and quantum properties and processes. If (2) holds, then it may be quite reasonable to think that the CP can serve as a guide or signpost to finding a proper definition of quantum chaos.

To decide between these alternatives we must examine some recent theoretical results. First, we return to the integrable case and consider again the method of torus quantization.

One can rewrite the torus quantization rule given in (4.7) in terms of the level density or density of states d(E).

(5.1) d(E) = E 6 [ E - H(n + 1/4a)h] = ~ 6 [ E - En] n i1

Note that each state gives a unit &function contribution. This completely characterizes the spectrum of the bound system. Through rather detailed manipulation of (5.1) Berry and Taylor (1976) show that it can be written as a "topological" sum, the terms of which depend only on


properties of those tori formed by trajectories which close. In other words, the quantum conditions (5.1) or (4.7) can be expressed solely in terms of the rational tori whose frequencies oJi are commensurable (see Section 4). Recall that this means that each trajectory on such a torus will be periodic, with period determined by the number of circuits mi around each irreducible contour labelled by ®i, (the mi are integers, positive or negative). 19 As I noted in Section 4, these rational tori are the exception rather than the rule. They have (Lebesgue) measure zero in the phase space. Nevertheless, they are densely distributed, and it is this fact that is crucial. Basically, the fact that the closed trajectories - the rational tori, can be used to generate the whole spectrum is analogous to the fact that any irrational number can be approximated arbitrarily well by a series of rationals.

So, we have a highly-nontrivial reformulation of the method of torus quantization. The key feature of which to take note is that closed classical trajectories can be used to yield the quantum energy spectrum for bounded integrable motion. The big question remains: Can we say anything about the nonintegrable case?

As I mentioned at the end of Section 4, the generic classical motion is quasi-integrable. The KAM theorem applies, and there will be a highly-nontrivial torus structure throughout the phase space. In what follows, however, I shall be concerned only with the limiting case of ergodicity and, afortiori, chaos. This is the regime in which, as Einstein noted, torus quantization must fail completely since the tori do not exist. Is there a method which will yield, if not the complete spectrum, at least some information about the density of states d(E) for systems whose classical counterparts exhibit chaos? The answer is 'yes'. And, perhaps surprisingly, it is the closed classical orbits that once again play the crucial role.

To begin, note that the density of states d(E) given in (5.1) can be decomposed into a sum containing a term representing the average density of states (d(E)), and an infinite set of oscillatory corrections, d(E) to that average. For reference let me give the definitions of these terms and then explain them.

d f

(5.2)a. d(E) = ~ 6[E - En] = (d(E)) + d(E) where n


d(E) >

Fig. 4. d(E) = (d(E)) + d(E).

i

E

b.

C.

(d(e)) = - H(q, p)] dq do)

(](E) = 1 ~ Aj (E) el=' (=/2) e (i/h) I](E)

h j

(5.2b) is the usual average density of states, sometimes called the Weyl density. It expresses the fact that on the average there should be one quantum state in a phase space region of volume h N. This is to be expected on account of the uncertainty relation. (Aqi)(Api)~ h, so ( A q ) ( A p ) ~ h N. There is strong computational evidence suggesting that (5.2b) applies to all systems, integrable, quasi-integrable, and ergodic. Thus, if differences do exist, we must look elswhere. In fact, we must look for differences in fluctuations. These are provided by (5.2c) - the periodic orbit sum. It is the topological sum mentioned earlier. As we shall see this sum is completely general; it is the result of some spectacu- lar theoretical work initiated by Gutzwiller. (For a review see Gutz- willer 1979.)

An idea of what is going on here can be gotten by examining Figure 4 (Hannay and Ozorio de Almeida, p. 3430). The curvy line represents one of an infinite number of oscillatory corrections to (d(E)), provided by (5.2c). As more and more corrections are superposed, the result is the true &function distribution of states d(E) - the spikes in Figure 4.


The quantities appearing within the argument of the sum (5.2c) are all classical, with the exception of h, of course.

Aj is the amplitude of the jth periodic orbit, the Ij are the classical actions: mj~ p • dq, where m/is the repetition number of the jth orbit. (Multiple traversals of a given orbit are counted as separate orbits.) aj is an integer related to the Maslov index discussed earlier; v = 1 for isolated periodic orbits (the chaotic case); and 1/2(N + 1) for rational tori (the integrable case) (Hannay and Ozorio de Almeida, p. 3431).

The remarkable thing is that in the semiclassical limit as h--~ 0 this sum over periodic orbits suffices to give the density of states d(E) for both integrable and ergodic systems. Prima facie it is hard to see how this is possible in the ergodic case. After all, trajectories in such cases wander densely over the entire energy surface spending equal time in regions of equal measure. In the case where the motion is fully chaotic, we have the further fact that nearby trajectories separate exponentially, so how can there be a sum over closed orbits? The dynamical instability definition of chaos depends essentially on the fact that the trajectories are everywhere dense and that they separate exponentially. But, charac- teristic of chaotic systems is also the fact that there exist dense sets of periodic trajectories. As should by now be almost expected, this set of trajectories has Lebesgue measure zero.

As I mentioned, (5.2a) is the result of some highly-nontrivial theoretical work. Unfortunately, however, in practice it is almost always impossible to use the periodic orbit sum to determine d(E), that is, to quantize the classical motion. The reason for this is that it is virtually impossible to systematically identify and enumerate the periodic orbits of a typical Hamiltonian system. These orbits are "deeply buried" in the structure of the Hamiltonian (Hannay and Ozorio de Almeida, p. 3430).

Let us for the moment take stock of the situation. We have been led, by treating the CP as a substantive principle which connects or correlates classical periodic motions with discrete quantum properties, to consider in some detail the semiclassical formula for the density of states. Further, having made the periodic orbit sum explicit, we are beginning to get a better understanding of what that correlation or connection amounts to. Now, we also see that the decomposition of d(E) as in (5.2) holds for both regular as well as ergodic classical motion. This takes us a long way toward justifying the ties between CM and QM that, as I mentioned, are briefly presented in most


textbooks: the fact that classical motion can be ergodic does not vitiate the semiclassical theory of bound states. It is true that torus quantization fails for ergodic systems, and because of this the textbook presentations are misleading. But, despite this failure, an expression for the density of states is still possible, and it is one which depends on classical periodic motion.

I think that this is sufficient to reject alternative (1) from the beginning of this section. The existence of classical chaos does not 'nail the coffin shut' on the CP. There remains a highly-nontrivial correspondence between CM and QM. However, the question of both the existence and definition of quantum chaos remains. A proper understanding of the CP has led us to examine the properties of bound, time-independent quantum systems. But, as we saw in Section 2, these systems cannot exhibit chaotic time evolution. Despite this, has our journey, guided by the CP, ted us to our 'elusive object of desire', namely, quantum chaos? 2°

Due to the practical impossibility of evaluating the periodic orbit sum one might feel that recognizing signatures of classical chaos in the density of states is hopeless. Hannay and Ozorio de Almeida, however, have shown that the behavior of a certain correlation function involving the periodic orbit sum does not depend on the details of the periodic orbits. It depends only on whether the orbits come in families, on tori (the integrable case), or whether they are isolated (the ergodic case) (Hannay and Ozorio de Almeida, p. 3430 and Ozorio de Almeida, p. 228ff.) The two cases give markedly different results.

This result can be related to various different statistics concerning the spectra of the systems. 21 Classical integrable systems have energy levels which are "locally uncorrelated and well described by a Poisson distribution; in contrast, classical chaotic systems have levels with strong local repulsion, well described by the eigenvalues of matrices drawn randomly from appropriate ensembles of Hamiltonians" (Berry 1985, p. 229). 22

These two quantum spectral universality classes are directly related to differences in the distribution of periodic orbits in the distinct classes (integrable and chaotic) of classical systems. These are the differences to which Hannay and Ozorio de Almeida appeal. They do not depend on a detailed knowledge of the periodic orbits. Their result depends on the fact that periodic orbits are uniformly dense in the phase space.


(In the appendix I shall give an illustrative example of the uniformity principle for the simple case of a chaotic two-dimensional billiards system.)

It turns out that the number of periodic trajectories increases with the period of the orbit; that is, for high periods, periodic trajectories will proliferate throughout the phase space. However, the rate of proliferation with increasing period is different for integrable and chaotic systems. Consequently, the weights assigned to the orbits in the orbit sum will be different for the two cases. 23 For integrable systems, the number of periodic orbits with period T increases as T N, a power law, where N is the number of degrees of freedom of the system. On the other hand, for chaotic systems which have positive Kolmogorov-Sinai entropy, 24 the number of periodic orbits increases exponentially with period a s ehT/hT, where h is the metric or K-S entropy. It is this difference which ultimately explains the distinct classes of quantum spectra. This, finally, is responsible for a universal quantum signature of classical chaos. I think that this fact along with the preceding discussion of the CP makes a strong case for defining chaos in QM in terms of the spectral statistics of the quantum systems.

6. CONCLUSION

We see that Bohr had an important insight which was formulated as the CP. This principle expresses an intimate connection between periodic classical motion and the stationary states of quantum systems. Of course, Bohr had no idea how this was eventually going to work out. For him, as we saw, the fact that typical classical motion of the constituents of an atom is not even multiply periodic was just another indication of the fundamental conflict between CM and QM. Despite this he still felt, as I pointed out in Section 4, that the CP would continue to "play an important role, even in the cases where the classical motion is not multiply periodic" (Bohr 1924, p. 16). With the derivation of the periodic orbit sum we now have a much better idea of what that role is.

Most of the confusion, or as Ford says "definitional chaos", regarding the problem of defining chaos in QM rests upon a mistaken interpretation of the CP. Ford claims that "an unbiased contemporary observer carefully reviewing the evidence would likely conclude that the notion of chaos, whether classical or quantal, has 3)et to be adequately reconciled with the correspondence principle" (Ford 1988, p. 129). My goal


in this paper was to review this evidence carefully and, hopefully, without bias.

The result of this review can be summarized in the following state- ments: (i) the fact that there exists chaos in classical systems, but no close analog of this chaos in QM, does not imply that the CP is false; (ii) classical chaos together with the fact that there is no close analog of this chaos in QM does not itself imply that QM is false, flawed, or that it requires modification. (Recall Ford's statement that perhaps the quantum theory is a "flawed theory which can be modified to include chaos" (Ford 1988, p. 132)); (iii) the CP is a substantive claim which finds its modern expression in the periodic orbit sum as it functions in (5.2a); and, finally, (iv) the CP, because of (iii), makes plausible a definition of quantum chaos in terms of the statistics of spectra.

Regarding (iv) we can say the following. Bohr maintained that the CP must be considered a fundamental principle or law of QM both in the old and new quantum theory. In this he was followed by many other quantum theorists, including Bohm as the quotes in Section 3 indicate. If we adopt this point of view as well, only with the new understanding of the CP, then it would be difficult to dismiss these results on the statistics of spectra as merely necessary conditions for quantum chaos°

The arguments for (i) and (ii) depend crucially on not interpreting the CP as a sweeping methodological constraint on an 'appropriate' quantum theory. As I tried to argue in Section 3, there is quite good textual evidence that Bohr, at least in 1923, did not take it to be such a general constraint. Once this is recognized, we can conclude that the CP has been 'adequately reconciled' with the chaos in CM and QM.

A P P E N D I X

The Principle of Uniformity

We consider the phase space mapping induced by the trajectory of a particle on a two-dimensional billiards table. For example, the Bunimo- vich stadium shown in Figure A1 which consists of two semicircles of radius R joined by parallel lines of length L.

The particle bounces around at constant speed with mirror reflections at the boundary. The trajectory of the particle is completely specified by providing its position and direction immediately following each of


Fig. A1. Bunimovich Stadium.

its collisions with the boundary; that is, we can parameterize its position q in terms of arc-length d from some point 0 and its 'momentum' p = cos ~ in terms of the angle from the forward tangent as shown in Figure A1. The phase space will be a rectangle of area A and the dynamical evolution is completely characterized by a measure preserving mapping of the rectangle into itself. An ergodic system such as the one being considered will have (with probability one) its iterates from some initial point Xo -- (qo, P0) spread throughout the entire phase space.

The principle of uniformity asserts that the set of periodic orbits (which are represented by points which after n iterations - the period - return to their starting point) are uniformly dense throughout the phase space. To see what this means consider the following function (Ozorio de Almeida, p. 70):

(i) ~(x. - xo)

This is a function of x0 = (qo, Po) which is peaked in those regions of the phase space in which orbits return very close to their initial starting points after exactly n iterations. In the limit as e--+ 0, 6~(x,, -Xo) be- comes a Dirac 6-function as shown in Figure A2.

Now, we can also consider the following function of x0: N

(2) A,N(Xo) = 1 E 6~(x,, - Xo) Nn=L

This function peaks in all those regions in which orbits return to within


X0 / d A)/I I /

0 q

Fig. A2. 6(xn - xo).

some e-neighborhood of their starting point for some iteration n -> N. By the (discrete analog of the) Poincar6 recurrence theorem we know that almost all points Xo eventually have iterates which return to within some small neighborhood of Xo; that is, the image of any neighborhood of Xo eventually intersects that neighborhood under the dynamical evolution.

We now take the limit of A~,N(Xo) a s N---> o~. This is the infinite 'time average' of 8~(xn- Xo). Since the mapping is ergodic, we know that this time average equals the 'phase average' of 3~(x , - Xo); namely, A -1. That is,

N

(3) lim 1 ~ 8~(xn - Xo) = 1 N ~ = N n = I A

The principle of uniformity is now expressed by the fact that we can take the limit as e--~ 0 before taking the limit as N ~ co in the left hand side of (3). In other words, the ergodic identity (3) remains valid even when we neglect the contribution to A,,N by the nonperiodic orbits in the e-neighborhood of xo. This means that

N

(4) lim 1 ~ 6(xn --Xo) = _1 N---'.~Nn=I A '

where the sum is a sum over periodic orbits only. (Strictly speaking, (4) makes sense only for arbitrarily small smoothings of the 8 functions (Ozorio de Almeida, pp. 70-71).) This is possible because the number of periodic orbits proliferate exponentially throughout the phase space. All the orbits of period N contribute the same weight to the sum in (4). The greater the period, the less weight each B(Xn -- XO) contributes.


Nevertheless, because of the exponential proliferation of orbits with increasing period, high period orbits are themselves uniformly distributed. They "just overwhelm the low orders on the average" (Hannay and Ozorio de Almeida, p. 3440).

N O T E S

* Work on this paper was supported by the National Science Foundat ion under grant No. DIR-9012010.

1 However, there have been suggestions to the effect that replacing the Schr6dinger equation with some nonlinear equat ion may help with the measurement problem. But the sort of randomness found in classical systems is distinct from that captured in the statistical algorithm of QM. The ' s tandard ' philosophical problems of QM will be completely ignored in what follows.

2 When talking about such classical systems, I shall always mean conservative Hamilton- Jan systems whose mot ion is confined to bounded regions in a phase space. As an example, one can always take a hard sphere 'gas' in a box. 3 Paradoxically perhaps, all of the systems we are considering are subject to the Poincar6 recurrence theorem. According to this theorem, over an infinite amount of t ime the system's trajectory will visit a neighborhood of its initial state an infinite number of times. However, there are two reasons that this sort of almost periodicity does not rule out chaotic behavior. First, the recurrence times are typically extremely long - greater than the age of the universe. Second, and more importantly, nearby initial states will in general have widely different recurrence times, so given a sensitive dependence on the initial state long time prediction will obviously fail in spite of recurrence. 4 Recall that I consider here only conservative systems. Dissipative systems can have their trajectories confined to subregions of the phases space as well as regular nonchaotic conservative systems. The interesting feature of dissipative systems is that, in some instances, mot ion within these regions or 'at tractors ' can be extremely irregular. These regions are called strange attractors. 5 See Fine (1973) for a good discussion of the computat ional complexity definition of randomness . 6 See Bat te rman (forthcoming) for a detailed critique of this approach. 7 A brief discussion of this suppression in the case of t ime dependent quan tum systems is given by Ford (1989, pp. 366-67). 8 Bohr, of course, is notorious for changing his views and interpretations over time. However, as I shall try to show, there is evidence which suggests that at an important point in the development of the quan tum theory he held what might today be considered a nons tandard interpretation of the CP. 9 Einstein was an important exception. See note 13. lo See any text on classical mechanics. The roughness of this definition is due to two factors. First, more must be said about the path of integration. Second, certain corrections in the general case need to be taken into account. These are the so-called Maslov corrections. 11 It follows from a special case of the Poincar~-Cartan Integral theorem that the integral


invariants required for the quantum conditions will exist at all points during the 'adiabatic' transformation. Given ~v P ' dq for some contour 3' on the torus (see Section 4), then if under the transformation, 7 evolves to 7, in time t, it is always the case that 56~ p • dq = 5¢~,p. dq. On the other hand, it is not the case that for all t during the 'adiabatic' transformation, the integral invariant fv, p . dq is an invariant o f the Hamiltonian. Only if it is such an invariant will the transformation be adiabatic. In general, the evolution of the angle variables will pass through values associated with resonances where the required tori won't exist. See Reinhardt and Dana (1987). 12 This is no more surprising than it is that the rational numbers are dense in the reals; yet they, nevertheless, have measure zero. 13 See Einstein (1917). A loose English translation of the parts relevant for this discussion can be found in Percival (1977). i4 In fact, the original definition of an ergodic system due to Boltzmann turned out to be contradictory. It was later replaced by a weaker condition sometimes called the quasi- ergodic hypothesis. Today, ergodicity is simply meant to express the equivalence of infinite time averages with phase averages - which is what the earlier 'definitions' were trying to show. 15 Bohr does not cite Einstein's paper in Bohr (1924). But, he does present the actions li (Jr in his notation) in the coordinate independent form first proposed by Einstein in that paper. 16 An excellent discussion of the classical mechanics of integrahle and quasi-integrable systems can be found in Lichtenberg and Lieberman (1983). 17 For example, van Kampen (1985). 18 Bohm's Quantum Theory is fairly unique in that the relationships between CM and QM receive substantial attention. 19 The sum is 'topological' because the contribution of a given closed orbit to the sum depends on the ratio of the mi's. Different ratios yield different orbit topologies (Berry and Tabor 1976, pp. 106-07). 20 I borrow this phrase from the title of a paper by Sklar (1986): 'The Elusive Object of Desire: In Pursuit of the Kinetic Equation and the Second Law'. 21 See Bohigas and Giannoni (1984) for a good introduction to the different statistical measures. See Berry (1985) for a detailed discussion of one such statistic - the spectral rigidity. 22 See Porter (1965) and Bohigas and Giannoni (1984) for the notion of ensemble used here. 23 The weights or "orbit intensities" for integrable and ergodic motions are calculated in Hannay and Ozorio de Almeida (1984, pp. 3436-38). In turn they are related to the orbit amplitudes Aj in (5.2c). 24 This is a quanti~:y which is related to the average rate of exponential divergence of the system's trajectories. Integrable systems have zero K-S entropy.

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Chaos, quantization, and the correspondence principle