REVIEWS OF MODERN PHYSICS VOLUME 42, NUMBER 4 O C T 0 B E R 19 7 0 The Statistical Interpretation of Quantum M echanics L. E. BALLENTINE Department of Physics, Simon Fraser University, Burnaby, B.C., Canada The Statistical Interpretation of quantum theory is formulated for the purpose of providing a sound interpretation using a minimum of assumptions. Several arguments are advanced in favor of considering the quantum state description to apply only to an ensemble of similarity prepared systems, rather than supposing, as is often done, that it exhaustively represents an individual physical system. Most of the problems associated with the quantum theory of measurement are artifacts of the attempt to maintain the latter interpretation. The introduction of hidden variables to determine the outcome of individual events is fully compatible with the statistical predictions of quantum theory. However, a theorem due to Bell seems to require that any such hidden-variable theory which reproduces all of quantum mechanics exactly (i.e., not merely in some limiting case) must possess a rather pathological character with respect to correlated, but spacially separated, systems. CONTENTS pretive attempts is promised for a later volume. 1. Introduction.................................................................................. 358 • • • pBut] even the cautious word ‘attempts’ may be 1.0 Preface and Outline............................................................. 358 too positive a description for what are only pro­ 1.1 Mathematical Formalism of Quantum Theory............ 359 grams • • • that are not yet clear even in outline.” 1.2 Correspondence Rules......................................................... 360 1.3 Statistical Interpretation................................ 360 If, as appears to be the case, the latter remarks by 2. The Theorem of Einstein, Podolsky, and Rosen...................362 2.1 A Thought Experiment and the Theorem..................... 362 Peierls refer to the models known as hidden-variable 2.2 Discussion of the EPR Theorem...................................... 364 theories (see Sec. 6), we agree that these should be 3. The Uncertainty Principle......................................................... 364 3.1 Derivation.................................................................................364 treated as new theories, and that they are not new 3.2 Relation to Experiments.................................................... 365 interpretations of quantum mechanics “any more than 3.3 Angular and Energy-Time Relations.............................. 367 quantum mechanics is a new interpretation of classical 4. The Theory of Measurement.................................................... 367 4.1 Analysis of the Measurement Process............................ 368 physics.” However we shall show, contrary to the view 4.2 The Difficulties of the “Orthodox” Interpretation.... 368 expressed by Peierls, that the Copenhagen interpreta­ 4.3 Other Approaches to the Problem of Measurement. 370 tion contains assumptions which are not “essential 4.4 Conclusion—Theory of Measurement............................ 371 5. Joint Probability Distributions................................................ 372 parts of the structure of quantum mechanics,” and that 6. Hidden Variables......................................................................... 374 one such assumption is at the root of most of the 6.1 Von Neumann’s Theorem.................................................. 374 6.2 Bell’s Rebuttal...................................................................... 376 controversy surrounding “the interpretation of quantum 6.3 Bell’s Theorem.......................................... 377 mechanics.” It is the assumption that the quantum 6.4 Suggested Experiments....................................................... 378 state description is the most complete possible descrip­ 7. Concluding Remarks................................................................... 378 References..................................................................................... 380 tion of an individual physical system. An interpretation which is more nearly minimal in the INTRODUCTION sense of including all verifiable predictions of quantum theory, but without the contestable features of the 1.0 Preface and Outline Copenhagen interpretation, we shall call the Statistical Interpretation. The distinction between these inter­ This article is not a historical review of how the pretations (which share many features in common) quantum theory and its statistical interpretation came will be made in the following sections. Suffice it to say, to be. That task has been admirably carried out by for now, that if we identify the Copenhagen Inter­ Max Jammer (1966) in his book The Conceptual pretation with the opinions of Bohr, then the Statistical Development of Quantum Mechanics, and also by van Interpretation is rather like those of Einstein. Contrary der Waerden (1967). Our point of departure can to what seems to be a widespread misunderstanding, conveniently be introduced by considering the follow­ Einstein’s interpretation corresponds very closely with ing statement made by Peierls (1967) in a review of the one which is almost universally used by physicists Jammer’s book: in practice; the additional assumptions of the Copen­ “Chapter 7 • • • is headed ‘The Copenhagen Inter­ hagen interpretation playing no real role in the applica­ pretation.’ • • • the phrase suggests that this is only one tions of quantum theory. of several conceivable interpretations of the same The outline of this paper is as follows. First we give a theory, whereas most physicists are today convinced brief summary of the mathematical formalism of that the uncertainty relations and the ideas of comple­ quantum theory in order to distinguish the formalism, mentarity are essential parts of the structure of quan­ which we accept, from the physical interpretation, tum mechanics • • •. A discussion of alternative inter­ which we shall examine critically. We then consider L. E. B allentine Statistical Interpretation of Quantum Mechanics 359 different interpretations, and expound the Statistical Here the numbers rn are the eigenvalues of R, and the Interpretation in detail. parameter a labels the degenerate eigenvectors which The Secs. 2, 3, and 4 could be prefaced by Appendix belong to the same eigenvalue of R. The sums become *xi of Popper (1959) on the proper use of imaginary integrals in the case of continuous spectra. Equation experiments, in which he points out that Gedanken (1.1) is equivalent to the statement that an observable experiments can be used to criticize a theory but not to must possess a complete orthogonal set of eigenvectors. justify or prove a theory. Our discussions of the Gedanken experiment of Einstein, Podolsky, and Rosen, of the F2 A state is represented by a state operator (also uncertainty principle, and of the measurement process called a statistical operator or density matrix) which are undertaken to criticize the assumption that a state must be self-adjoint, nonnegative definite, and of unit vector provides a complete description of an individual trace. This implies that any state operator may be system. Although arguments of this type can refute the diagonalized in terms of its eigenvalues and eigenvectors, hypothesis being criticized, they cannot, of course, p — ^L,Pn | 4*11) (<t>n I y (1.3a) “prove” the Statistical Interpretation but can only n illustrate its advantages. with Sections 5 and 6 deal with two concepts (joint 0<P„<1 (1.3b) probability distributions for position and momentum, and II and hidden variables) which have often been thought H-* 3s 3s (1.3c) to be incompatible with quantum theory. That belief, however, was based in an essential way upon the above This state operator formalism ’ Ms is reviewed by Fano hypothesis which we criticize and reject. It turns out (1957). that, within certain limits, the formalism of quantum theory can be extended (not modified) to include these F3 A pure state can be defined by the condition p2=p. concepts within the Statistical Interpretation. It follows that for a pure state there is exactly one Finally we summarize our conclusions. nonzero eigenvalue of p, say, p»=l, pn' = 0 for n ^n '. (1.4) 1.1 Mathematical Formalism of Quantum Theory In this case we have Quantum theory, and indeed any theory, can be divided [[see Prugovecki (1967), or Tisza (1963)] P^ i I? (1.5) into: and so a pure state may be represented by a vector in (a) A mathematical formalism consisting of a set of the Hilbert space. A general state which is not pure is primitive concepts, relations between these concepts commonly called a mixed state. (either postulated or obtainable by given rules of F4 The average value of an observable R in the state deduction), and a dynamical law. p is given by (b) Correspondence rules which relate the theoretical (R )= Tr(ptf), (1.6) concepts of (a) to the world of experience. where Tr means the trace of the operator in parentheses. This division is not absolute—'dearly one must have For a pure state represented by the normalized vector a formalism in order to make correspondence rules, but \xp), (1.6) reduces to (R)= (p | R | p). By introducing unless one has at least some partial idea of corre­ the characteristic function (e^'R)j we can obtain the spondence rules, one would not know what one was entire statistical distribution of the observable R in the talking about while constructing the formalism—