Theory of Quantum-Mechanical Description by Walter M. Elsasser*

THEORY OF QUANTUM-MECHANICAL DESCRIPTION BY WALTER M. ELSASSER*

INSTITUTE FOR FLUID DYNAMICS AND APPLIED MATHEMATICS, UNIVERSITY OF MARYLAND, COLLEGE PARK Communicated January 4, 1968 In a recent book, L. Brillouin' has gathered a great many illustrative examples to show how in problems of classical physics any initial uncertainty increases with time. His work is clearly related to the fact that since the advent of quantum mechanics there have been the two schools of thought: those who tried to return toward classical determinism and those who found in quantum theory a challenge for investigating all possible ramifications or generalizations of indeterminacy which may be part of physical description and prediction. Bril- louin's work belongs to the second category; so does this note. Consider now a process of prediction which we schematize as follows. Given a system whose future is to be predicted, assume that all the relevant information refers to one time, to, and all the predictions to a later time, t1. That is to say, from a description at time to (which may be more or less incomplete) we obtain information about the system at time t1 on integrating with respect to time the equations of motion of corpuscles or fields. But note the radical difference in this predictive process as between classical and quantum theory. In classical physics one assumes as a matter of course that all variables of the system can be measured as precisely as one wishes. But one assumes, implicitly or explicitly, somewhat more, namely, that there exists for any object a univocal pattern of numbers which is in effect its exhaustive scientific description; if complete, it says everything that can be said by science about the properties of that object at that time. To have a term for this, we shall speak of a univocal Cartesian description (UCD). ANo special reference to Descartes' philosophy is implied, of course; this is just terminology. Classical physicists consistently emphasized the need for postulating a UCD by way of a semiphilosophical foundation; in fact this assumption has become virtually coextensive with the realm of validity of classical physics. Unexpected and profound difficulties appear, however, in the integration of the generally nonlinear equations of mechanics. Brillouin discusses these at great length, basing himself upon Poincar6's proof that, except for a limited number of known special cases, the equations of motion of classical dynamics do not possess analytical integrals. In quantum mechanics, the situation is exactly the reverse: given a description at time to, the equations of motion (i.e., Schr6dinger equations) are a set of coupled linear, first-order, ordinary differential equations with time as inde- pendent variable. Lengthy as this set of equations may be when the number of dependent variables is large, the difficulties of integration are of a practical order; it is impossible to see how there could be mathematical difficulties even re- motely resembling the profound ones of nonlinear classical dynamics. We conclude that such serious problems as exist in the process of prediction in quantum mechanics will be found almost wholly in the initial description of the 738 Downloaded by guest on September 28, 2021 VOL. 59, 1968 PHYSICS: W. M. ELSASSER 739

system. Thus we are led to the question: How does one define modalities of description which can be taken as the quantum-mechanical equivalent of the UCD? In the ensuing discussion we have in mind conventional nonrelativistic quantum mechanics, without thinking as yet of extension to quantum field theories. The following conceptual distinction will be found convenient. A collection of similar material objects or events, that is, objects at given times, will be designated as a class, while a collection of entirely abstract symbols or abstract structures composed of such symbols will be designated, as is usual in mathematics, as a set. Now mathematicians deal mostly with infinite sets, whereas classes are necessarily finite. This last proposition may be considered a direct consequence of the finiteness of the empirical universe of astronomy. The fundamental importance of this limitation of classes for all scientific methodology has repeatedly been stressed by the writer.2-4 In quantum mechanics proper (i.e. "below" the classical limit) all propositions contain statistical elements, and statistical distributions are expressed by means of wave functions. A wave function in turn is, mathematically speaking, an infinite point set. Thus there exists a clear discrepancy between wave functions or probabilities which are infinite point sets and classes of events or objects which can only contain a finite number of members. (The writer admits to having toyed for years with the idea that description should be in terms of finite abstract structures, but he has wholly abandoned this view for reasons that will appear below.) A further difficulty of description arises if we try to find a class of objects which all have the same Hamiltonian such that this class can be described by a single wave function (pure state). This can be operationally achieved in limiting cases, the most familiar one being that of the spin state in atomic beams. However, as Niels Bohr5' 6 has shown long ago for a system with any consider- able degree of complexity, the physical operations which would transform a system into a pure state with respect to some extensive set of microscopic dynamical variables are prohibitive; they will interfere with the structure of the system so as to alter it radically. Even the minimum of physical interaction required to generate a pure state will involve an enormous energy of interaction which engenders an altogether radical change in the physical condition of the system after this interaction. Although Bohr formulated these ideas primarily with respect to organisms, a little reflection shows that they cannot be assigned to biology proper but apply to any class of physical systems whatever, provided these are of an appreciable degree of complexity. We must therefore look for a realistic description of an object, an event, or a class, meaning one which involves no radical change caused by the measurements preceding the description. Note that such a concept as "realistic" does not even exist in classical physics, except if applied to approximations (which would in- deed be worthless unless to some degree realistic). But in quantum mechanics we do not encounter a UCD, hence nothing has been, even ideally, defined that is to be approximated. In order to deal with the logical problem appearing here, we proceed as follows: We take the point of view that in general (with the exception, that is, of certain limiting cases dealt with in books on atomic physics) Downloaded by guest on September 28, 2021 740 PHYSICS: W. M. ELSASSER PROc. N. A. S.

the question of what exact state the system is in has no operational meaning. This question resembles formally the well-known classical one: which of a set of inertial systems is at rest, the others then being, by definition, in motion. We shall make the following assumptions regarding the prerequisites of description. In the first place we shall deal with a class of objects whose members all have one given Hamiltonian. In the second place, we shall require certain data about the class, meaning by this a set of numbers, assumed to be the results of measurements. Quantum-mechanically the data correspond to a set of expectation values of quantities (operators) pertaining to the system. We shall now limit our discussion to a class of objects which not only have the same Hamiltonian but also have the same data (where included in the data might be limits of accuracy, etc). Limited in this way, the problem of description is not the most general we could formulate but it is quite general enough to be illustrative. Since there is nothing here resembling a UCD, the data can be said to be the invariants of all sets of descriptions; this is asserted prior to having found or defined any such set. In making description a function of the invariants, we circumvent questions of a metaphysical nature regarding the system's "true state." If the objects of the class to be described are not too simple, for instance if these objects approach macroscopic dimensions, the number of pure-state descriptive wave functions compatible with the data is found to be extraordinarily large. As we have indicated, this large number of pure states cannot in general be substantially reduced without making the description unrealistic. One understands readily that the problem of description so formulated is nothing but a generalization of quantum statistical mechanics. The separation of mechanics into two parts, mechanics proper on the one hand and statistical mechanics on the other, is quite meaningless from the viewpoint of quantum-mechanical description. This separation is based upon mental remnants of a UCD. From our viewpoint the unity of quantum mechanics and quantum statistical mechanics is complete; it is indispensable except for the most minute systems, i.e., those of strictly atomic size. A complete descriptor is a wave function which assigns a probability distribution to each of the 2N dynamical variables appearing in the Hamiltonian. By the rules of the operator calculus one can assign a probability distribution to any quantity that is a function of these 2N variables. The total description of a class of objects having the same invariant data is the set of all complete descriptors compatible with the invariants. Furthermore, any complete descriptor can be assigned a probability of being the right one, by the use of Bayes' theorem of inverse probabilities. The total description, therefore, consists of the set of all complete descriptors, each appearing with a suitable a priori probability. It will in general not be possible to narrow down the description any further without introducing additional assumptions which, from the viewpoint of quantum mechanics, are wholly arbitrary. We note that the famous Heisenberg "reduction" of a wave packet which results from measurements is a special case of this theory of description. We are now, however, dealing with an extensive generalization of this idea, since the description is in each case dependent on the invariants of the class. This does Downloaded by guest on September 28, 2021 VOL. 59, 1968 PHYSICS: W. M. ELSASSER 741

not imply any philosophical extravagance: We do not have to dispense with that convenient auxiliary commonly used to connect classical physics with philosophical generalization-the Laplacian Spirit. We must merely be careful to prevent him from making measurements that would modify the structure of an object he measures, i.e., we must limit him to "classical" measurements; but inasmuch as such measurements represent a limiting concept, we must also limit our predictions so that they are not radically unstable relative to those perturba- tions which result from quantum effects. In these considerations, the writer has drawn extensively upon the fundamental work of E. T. Jaynes7-9 on the principles of statistical mechanics. As Jaynes clearly states, statistical mechanics must be thought of as a tool of inductive inference, and in this respect he follows precisely the point of view of Gibbs. We fully agree with this view of Jaynes: If Gibbs at no place mentions the concept of ergodicity (which was well known at the time he wrote), he did so not out of ignorance, but because he considered it irrelevant to statistical mechanics. We should add that the numerous discussions to be found in the literature on ergodic theory, reversals, etc. are obviously the result of a (perhaps not even conscious) desire to introduce a UCD into statistical mechanics, usually beyond limits of operational verifiability. In classical statistical mechanics, the phase space is defined as the domain of variability of the 2N variables of a system. The ensemble is a probability distribution; in contradistinction, the image of an event, a system at a given time (considered as a UCD), is one point in phase space. This implies that any phase space of a class is sparsely populated by image points which represent actual members of the class (since the class is necessarily finite); rigorously speaking, the image points form a set of measure zero compared to the probability distribution which is the ensemble. We see here clearly that the attempt at using either a UCD or else an ensemble for descriptive purposes is far from introducing just a vague philosophical ambiguity; instead it leads one into quite specific mathematical problems. The simplest and most convenient quantum-mechanical analogue of the classical ensemble is the statistical matrix. But whereas a probability distribution in classical phase space is a quite general form of statistical description, the corresponding situation for the statistical matrix appears more complex. Jaynes (ref. 7, part II) mentions more general statistical structures, and the question of how far the usual statistical matrix is a comprehensive tool requires future in- vestigations. Undoubtedly, for most concrete problems the statistical matrix represents a convenient and also an adequate tool. Given a statistical matrix p, an entropy S can be defined by the formula S = -k Tr (p In p), where Tr designates the trace or spur of the matrix in parentheses. This writer" long ago pointed out that a statistically appropriate description of a class of objects is obtained by maximizing S with subsidiary conditions. The conditions say that the expectation values of the operators corresponding to the data must have the numerical values given as invariants of the description. This procedure reduces the set of complete descriptors to a single statistical matrix which may be considered the "best" description of the class of objects. (Note, however, that "best" has no rigorous meaning since we are not able to Downloaded by guest on September 28, 2021 742 PHYSICS: W. M. ELSASSER PROc. N. A. S.

define a UCD or other "meta" structure to which to approximate.) Since I was not able to master the mathematical intricacies of this procedure, my own early efforts were not continued. Jaynes has not only given the general, formal solu- tion of the extremum problem, but he and his pupils have applied the method to a number of concrete problems of statistical mechanics and have exhibited its versatility and analytical power. More recently, Jaynes9 has tackled with con- siderable success the approach, by this method, to the most basic and still un- solved problem of nonequilibrium statistical mechanics, namely, the representation of an irreversible process in terms of a time-dependent statistical matrix or ensemble. As is well known, the traditional methods of Boltzmann for dealing with irreversibility are not only very ingenious but also very much ad hoc; they are rather resistant to generalization. The methods of Jaynes, on the other hand, which are extensions of Gibbs' inductive scheme, appear outstand- ingly suited for very general descriptions of irreversible processes. On generalizing a little beyond the level that seems now accessible by these techniques, we arrive at a very significant conclusion. Let us assume that all systems of the type found in physical and chemical laboratories can be adequately described in space and time by using the methods of irreversible quantum statistical mechanics. This has an immediate bearing on the subject known as the theory of quantum measurement. As Bohr has so often emphasized, one can speak of a true measurement only when its ultimate outcome is some modification of the macroscopic world. Now all the manipulations that precede or ac- company a measurement are macroscopic; we have no means for conducting manipulations directly on the atomic scale. Thus if we are able to describe all the usual processes carried out by the physical scientist in terms of irreversible quantum statistical mechanics, it must also be possible to express the data, which we also called the invariants, as macroscopic parameters of the system. Thus if one possesses a sufficiently comprehensive statistical mechanics of irreversible systems (including sometimes downright unstable ones, as in counters, for instance), the quantum theory of measurement reduces to a special case of the theory of description of laboratory equipment in terms of irreversible statistical mechanics. As Jaynes has made clear, this does not imply a diminution of the assurance with which the basic principles of quantum mechanics can be experi- mentally verified: it is always possible to design experiments such that their outcome would be different depending on whether the fundamental laws were the ones now accepted or others. Another, purely verbal interpretation of our point of view may help to visualize it: The formal laws of quantum mechanics play, within a statistical description, the role of constraints, and this is true with respect to both space and time. Ex- perimentally cases can be realized where these constraints are equivalent to rigorous determinism (classical limits) but this is not always so. When it comes to systems as utterly complicated as, say, organisms, we do not propose to change the constraints but, as the authors4 has endeavored to show, we are then con- fronted with novel and more varied descriptions which are possible in the case of these more intricate structures, and it is this new variety which requires more quantitative investigation. Downloaded by guest on September 28, 2021 VOL. 59, 1968 PHYSICS: W. M. ELSASSER 743

We finally mention a point of interpretation on which we disagree with Jaynes. This is the use of the concept of "subjective" probabilities meant as a generalization of inductive probabilities. It has been given its most telling and elaborate form in the treatise on probability by Harold Jeffreys,"' whom Jaynes has followed in matters of the basic definition of probabilities. Fortunately, as Jeffreys remarks, experienced statisticians seem to arrive at the same quantitative results in special cases, whether they base themselves on the traditional frequency concept of probabilities or on a "subjective" interpretation. There must, however, be some basic problems of interpretation if such conceptual discrepancies can arise at all. Philosophers of science tend to agree that any scientific reasoning of some length and complexity contains in general both inductive and deductive elements, frequently intercalated among each other in a complicated manner. Here, language plays us a trick because the superficial symmetry between the terms "deduction" and "induction" turns out to be purely verbal; there is nothing in these two methods that would be indicative of any symmetry. Hence their joint use might be better accompanied by operational specifications for each of them. Deduction should, if possible, be completely rigorous, and this is made possible for us by using formal, mathematical structures composed of infinite point sets as descriptors. Probabilistic deduction proceeds then in the conventional manner based on the frequency interpretation of probability in the usual set-theoretical sense. If one tried to replace this by a definition of probability which relates it to subjective judgment, one would succeed in breaking down the usefulness of deductive processes which, while not becoming formally false, would appear to lose just about any reference to experience. Induction consists in the choice of a given set of abstract descriptors for a class. Used in this sense the term is all but synonymous with generalization. (Whether generalization is what mathematicians would call an undefinable concept, I do not know, but I rather suspect that it is.) From such a point of view, the problem of inductive representation might be alternately described as an embedding of the given class which, like all classes, is necessarily finite into the descriptors which by their definition are representative of an infinite collection of samples. Once a description is given, the embedding procedure as such (corresponding to the placing of image points into a phase space) is not likely to lead to quandaries of a fundamental nature. On the other hand, there may be a great deal of freedom (especially a certain "fuzziness at the edges") in the choice of the set of descriptors. This indeterminacy of the formal description is intrinsic; it is just another aspect of the same general indeterminacy that makes it impossible to find a general rule which would determine unambiguously the mapping of finite classes onto descriptors which, from the set-theoretical viewpoint, are infinite point sets. The more homogeneous a class and the more structurally simple its objects, the less difficulty of description will arise as a consequence of the indeterminacies just outlined. More serious difficulties intervene when the membership of the class becomes very small. We must then be clear that we have in fact arrived at the limits of the conventional methodology of physical science (and the "sub- Downloaded by guest on September 28, 2021 744 PHYSICS: W. M. ELSASSER PROc. N. A. S.

jective" interpretation of probability simply claims that there are no such limits). Science, by the common consensus of the scientist and the ordinary man, deals with regularities, which in more abstract language means with classes. When the number of elements in a class becomes so small that ob- servational frequencies in the class begin to lose their meaning, we are moving out of the domain of analytical science and we retain only primitive description in the everyday sense of the word. Nevertheless, the domain of abstraction in which the regularity of classes is interspersed with events which are each structurally unique is of great interest since, as I have endeavored to show in the work quoted, abstractions of this sort must play the basic role in the formulation of theoretical biology. The writer is indebted to the Office of Naval Research for support. * On leave of absence from Princeton University, Princeton, New Jersey. 'Brillouin, L., Scientific Uncertainty and Information (New York: Academic Press, 1964). 2Elsasser, W. M., The Physical Foundation of Biology (New York: Pergamon Press, 1958). 3Elsasser, W. M., Atom and Organism (Princeton: Princeton University Press, 1966). 4Elsasser, W. M., J. Theoret. Biol., 7, 53 (1964). 5 Bohr, N., Nature, 131, 421, 457 (1933). 6Bohr, N., Atomic Physics and Human Knowledge (New York: John Wiley, 1958). 7Jaynes, E. T., Phys. Rev., 106, 620; 108, 171 (1957). 8Jaynes, E. T., "Information theory and statistical mechanics," in Brandeis Summer Insti- tute, 1962 (New York: W. A. Benjamin, Inc., 1963), pp. 181-218. 9 Jaynes, E. T., "Foundations of probability and statistical mechanics," in Delaware Seminar on the Foundations of Physics, ed. M. Bunge (Berlin and New York: Springer-Verlag, 1967). 10 Elsasser, W. M., Phys. Rev., 52, 987 (1937); see also Z. f. Physik, 171, 66 (1963). 11 Jeffreys, H., Theory of Probability (Oxford: Clarendon Press, 1961), 3rd ed. Downloaded by guest on September 28, 2021