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 quan- tum 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 descrip- tion (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 descrip- tion 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 mathe- matics, 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 dy- namical 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 mem- bers 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 illustra- tive. 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 de- scriptive 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 mechan- ics 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 de- scriptors 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.