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) (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

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 ’ M s 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— state p. It follows that: nevertheless it is convenient for the present task. The mathematical formalism of quantum theory is F5 The only values which an observable may take on well known and can be abstracted from any of several are its eigenvalues, and the probabilities of each of the textbooks (Dirac, 1958; Messiah, 1964). The primitive eigenvalues can be calculated. In the case of a pure concepts are those of state and of observable. state represented by the normalized vector \ p), the probability of eigenvalue rn of R is X) | (p | a, rn) |2. FI An observable is represented by a self-adjoint This is a generalization of Born’s (1926) famous operator on a Hilbert space. It has a spectral repre­ postulate that the square of a wave function represents sentation, a probability density. R=ZrnPn, (1.1) n So far we have only given necessary, but not sufficient, conditions for the mathematical representations of where the Pn are orthogonal projection operators observables and of states. To complete the specification, related to the orthonormal eigenvectors of R by the following is usually postulated: Pn = ’£\ a , r n)(a,rn \. (1.2) a F6 The Hilbert space is a direct sum of coherent subspaces, within each of which (almost) every vector to arise in practice. We can, for example, unambiguously may represent a pure state. This is a formal statement determine the Hamiltonian operator for any finite of the superposition principle with allowance being number of particles interacting through velocity- made for superselection rules.1 The set of all mixed independent potentials in the presence of an arbitrary states can be constructed from the set of all pure states external electromagnetic field. using (1.3). The different interpretations of quantum theory are most sharply distinguished by their interpretations of F7 Any self-adjoint operator which commutes with the concept of state. Although there are many shades of the generators of superselection rules, or equivalently, interpretation (Bunge, 1956), we wish to distinguish all of whose eigenvectors lie within coherent subspaces only two: of F6, represents an observable. This postulate may be criticized on the grounds that it is difficult to imagine a (I) The Statistical Interpretation, according to which procedure for observing a quantity like (x2p^-\-pxzx2), a pure state (and hence also a general state) provides which should be an observable according to this a description of certain statistical properties of an postulate. The general form of F7 is unnecessary ensemble of similarily prepared systems, but need not in most, if not all practical applications, but it is used provide a complete description of an individual system. in Von Neumann’s theorem (Sec. 6.1). This interpretation is upheld, for example, by Einstein F8 The dynamical law or equation of motion depends (1949), by Popper (1967), and by Blokhintsev (1968). in detail upon the physical system under consideration Throughout this paper the capitalized name “Statistical (i.e., number of degrees of freedom, whether relativistic Interpretation” refers to this specific interpretation or nonrelativistic), but in every case it can be written (described in detail in Sec. 1.3). in the form, pit) = Up(to) U~l (1.7a) (II) Interpretations which assert that a pure state in general, or provides a complete and exhaustive description of an I *(*>)> (1.7b) individual system (e.g., an electron). for a pure state, where U — U(t, t0) is a unitary operator. This class contains a great variety of members, from The above is not intended to be an axiomatization of Schrodinger’s original attempt to identify the electron quantum theory, but merely a compact summary of with a wave packet solution of his equation to the the mathematical formalism of the theory as it exists at several versions of the Copenhagen Interpretation.2 present and in practice. Except for the reservation Indeed many physicists implicitly make assumption II noted in F7 it should be noncontroversial. Such is not without apparently being aware that it is an additional the case with the correspondence rules. assumption with peculiar consequences. It is a major aim of this paper to point out that the hypothesis II is 1.2 Correspondence Rules unnecessary for quantum theory, and moreover that it leads to serious difficulties. The correspondence rules must relate the primitive concepts of state and observable to empirical reality. 1.3 The Statistical Interpretation In so doing they will provide a more specific inter­ pretation for the averages and probabilities introduced The term, Statistical Interpretation, will be used in F4 and F5. throughout this paper in the specific sense here ex­ The natural requirement placed upon an observable pounded, and should not be confused with the less is that we should be able to observe it. More precisely, specific usage of this term by other writers (e.g., an observable is a dynamical variable whose value can, Messiah, 1964, Chap. IV) who do not distinguish it in principle, be measured. For canonically conjugate from the Copenhagen interpretation. variables the corresponding operators are obtained Of primary importance is the assertion that a quantum through Dirac’s canonical commutation relation, 2 Bohr (1935, 1949) argued against Einstein, who rejected II. qp—pq=fii. (1.8) Heisenberg’s position is somewhat unclear. One reference (1930, p. 33) contains a statement fully in accord with the Statistical There is no general rule for constructing a unique Interpretation; however, in a later writing specifically in defense of the Copenhagen interpretation (1955, p. 26) he states “an operator to represent an arbitrary function f(q,p) individual atomic system can be represented by a wave func­ because of the noncommutability of q and p (Shewell, tion .. . ”. His interpretation of probability (1958, Chap. 3) was 1959). Fortunately the most general case does not seem based on the Aristotelian notion of “potentia”, which is quite different from the Statistical Interpretation. Messiah (1964) is quite explicit (p. 152, 158) in favoring II over I. Although both 1A superselection rule (Wick et at., 1952; Galindo et at., 1962) claim orthodoxy, there now seems to be a difference of opinion is a restriction on the superposition principle. For example, a between what may be called the Copenhagen school represented vector which is a linear combination of integer and half-integer by Rosenfeld, and the Princeton school represented by Wigner angular momentum eigenvectors cannot represent a physical (see Rosenfeld, 1968 and references therein). But since both fac­ state. A coherent space is one in which the superposition principle tions appear to accept hypothesis II, our criticism will apply to has unrestricted validity. both. L. E. B allentine Statistical Interpretation of Quantum Mechanics 361 state (pure or otherwise) represents an ensemble of of the results of an ensemble of similar experiments. similarly prepared systems. For example, the system Because this ensemble is not merely a representative or may be a single electron. Then the ensemble will be the calculational device, but rather it can and must be conceptual (infinite) set of all single electrons which realized experimentally, it does not inject into quantum have been subjected to some state preparation tech­ theory the same conceptual problems posed in statistical nique (to be specified for each state), generally by thermodynamics. interaction with a suitable apparatus. Thus a momen­ In general quantum theory will not predict the result tum eigenstate (plane wave in configuration space) of a measurement of some observable R. But the represents the ensemble whose members are single probability of each possible result rn, calculated ac­ electrons each having the same momentum, but dis­ cording to F5, may be verified by repeating the state tributed uniformly over all positions. A more realistic preparation and the measurement many times, and example which occurs in scattering problems is a finite then constructing the statistical distribution of the wave train with an approximately well-defined wave­ results. As pointed out by Popper (see Korner, 1957, length. It represents the ensemble of single electrons p. 65, p. 88), one should distinguish between the which result from the following schematically described probability, which is the relative frequency (or measure) procedure—acceleration in a machine, the output from of the various eigenvalues of the observable in the which can take place in only some finite time interval conceptual infinite ensemble of all possible outcomes of (due to a “chopper”), and collimation which rejects identical experiments (the sample space), and the any particle whose momentum is outside certain limits. statistical frequency of results in an actual sequence of We see that a quantum state is a mathematical repre­ experiments. The probabilities are properties of the sentation of the result of a certain state preparation state preparation method and are logically independent procedure. Physical systems which have been subjected of the subsequent measurement, although the statistical to the same state preparation will be similar in some of frequencies of a long sequence of similar measurements their properties, but not in all of them (similar in (each preceded by state preparation) may be expected momentum but not position in the first example). to approximate the probability distribution. If hy­ Indeed the physical implication of the uncertainty pothesis I is adopted, we may say simply that the principle (discussed in detail in Sec. 3) is that no state probabilities are properties of the state. preparation procedure is possible which would yield an The various interpretations of a quantum state are ensemble of systems identical in all of their observable related to differences in the interpretation of proba­ properties. Thus it is most natural to assert that a bility (see Popper, 1967 for a good exposition of this quantum state represents an ensemble of similarily point). In contrast to the Statistical Interpretation, prepared systems, but does not provide a complete some mathematicians and physicists regard probability description of an individual system. as a measure of knowledge, and assert that the use of When the physical system is a single particle, as in probability is necessitated only by the incompleteness the above examples, one must not confuse the ensemble, of one’s knowledge. This interpretation can legitimately which is a conceptual set of replicas of one particle in be applied to games like bridge or poker, where one’s its experimental surroundings, with a beam of particles, best strategy will indeed be influenced by any knowledge which is another kind of (many-particle) system. A (accidentally or illegally obtained) about an opponent’s beam may simulate an ensemble of single-particle cards. But physics is not such a game, and as Popper systems if the intensity of the beam is so low that only has emphasized, one cannot logically deduce new and one particle is present at a time. verifiable knowledge—statistical knowledge—literally The ensembles contemplated here are different in from a lack of knowledge. principle from those used in statistical thermodynamics, Heisenberg (1958), Chap. 3, combined this “sub­ where we employ a representative ensemble for cal­ jective” interpretation with the Aristotelian notion of culations, but the result of a calculation may be com­ “potentia.” He considers a particle to be “potentially pared with a measurement on a single system. Also present” over all regions for which the wave function there is some arbitrariness in the choice of a repre­ p(r) is nonzero, in some “intermediate kind of reality,” sentative ensemble (microcanonical, canonical, grand until an act of observation induces a “transition from canonical). But, in general, quantum theory predicts the possible to the actual.” In contrast, the Statistical nothing which is relevant to a single measurement Interpretation considers a particle to always be at some (excluding strict conservation laws like those of charge, position in space, each position being realized with energy, or momentum), and the result of a calculation relative frequency | \p(r) |2 in an ensemble of similarily pertains directly to an ensemble of similar measure­ prepared experiments. The “subjective” and Aristotelian ments. For example, a single scattering experiment ideas are primarily responsible for the suggestion that consists in shooting a single particle at a target and the observer plays a peculiar and essential role in measuring its angle of scatter. Quantum theory does quantum theory. Whether or not they can be con­ not deal with such an experiment, but rather with the sistently developed, the existence of the Statistical statistical distribution (the differential cross section) Interpretation demonstrates that they are not necessary, and in my opinion they bring with them no advantages 2. THE THEOREM OF EINSTEIN, PODOLSKY, to compensate for the additional metaphysical com­ AND ROSEN plication. The tenability of hypothesis II, Sec. 1.2, was chal­ If the expression “wave-particle duality” is to be lenged in a paper by Einstein, Podolsky, and Rosen used at all, it must not be interpreted literally. In the (1935) (abbreviated EPR) entitled “Can Quantum- above-mentioned scattering experiment, the scattered Mechanical Description of Reality Be Considered portion of the wave function may be equally dis­ Complete?” Their argument is often referred to as the tributed in all directions (as for an isotropic scatterer), Paradox of EPR, as though it ought to be capable of but any one particle will not spread itself isotropically; resolution as, say, Zeno’s paradox. £Rosenfeld (1968) rather it will be scattered in some particular direction. has scurrilously referred to it as the EPR fallacy.] We Clearly the wave function describes not a single scat­ shall show however that, properly interpreted, it is a tered particle but an ensemble of similarily accelerated well defined theorem, paradoxical only to the extent and scattered particles. At this point the reader may that the reader may not have expected the conclusion. wonder whether a statistical particle theory can account In order to precisely answer the question posed in for interference or diffraction phenomena. But there is their title, EPR introduce the following definitions: no difficulty. As in any scattering experiment, quantum theory predicts the statistical frequencies of the various D1 A necessary condition for a complete theory is that angles through which a particle may be scattered. For a “every element of physical reality must have a counterpart crystal or diffraction grating there is only a discrete set in the physical theory.” of possible scattering angles because momentum D2 A sufficient condition for identifying an element of transfer to and from a periodic object is quantized by a reality is, “//, without in any way disturbing a system, multiple of Ap = h/d, where Ap is the component of we can predict with certainty (i.e., with probability equal momentum transfer parallel to the direction of the periodic displacement d. This result, which is obvious to unity)4 the value of a physical quantity, then there exists an element of physical reality corresponding to this from a solution of the problem in momentum repre­ physical quantity.” sentation, was first discovered by Duane (1923), although this early paper ha,d been much neglected Their argument then proceeds by showing, through until its revival by Lande (1955, 1965). There is no consideration of a thought experiment, that two non­ need to assume that an electron spreads itself, wavelike, commuting observables should, under suitable condi­ over a large region of space in order to explain diffrac­ tions, both be considered elements of reality. Since no tion scattering. Rather it is the crystal which is spread state vector can provide the value of both of these out, and the electron interacts with the crystal as a observables, they conclude that the quantum state whole through the laws of quantum mechanics. For a vector cannot completely describe an individual longer discussion of this and related problems such as system, but only an ensemble of similarily prepared the two-slit experiment, see Lande (1965).3 In every systems. case a diffraction pattern consists of a statistical distribution of discrete particle events which are 2.1 A Thought Experiment and the Theorem separately observable if one looks in fine enough detail. The experiment described below was introduced by In the words of Mott (1964, p. 409), “Students should Bohm (1951, p. 614ff). By considering measurements of not be taught to doubt that electrons, protons and the spins rather than of particle positions and momenta like are particles • • • The wave cannot be observed in (as in the original EPR experiment), we avoid any any way than by observing particles.” unnecessary complications with the position-momentum Although we shall discuss his work in greater detail uncertainty principle which may arise from the mini­ below, we should emphasize here the great contribution mum degree of particle trajectory definition needed to of Einstein to this subject. His “Reply to Criticisms” perform an experiment. (Einstein, 1949), expressed very clearly his reasons for Two particles of spin one-half are prepared in an accepting a purely statistical (ensemble) interpretation unstable initial state of total spin zero. The pair of quantum theory, and rejecting the assumption that separates, conserving total spin, and one of the par­ the state vector provided an exhaustive description of ticles (which are chosen to be distinguishable, for the individual physical system. convenience) passes through the inhomogeneous mag­ netic field of a Stern-Gerlach apparatus (see Fig. 1). 31 note in passing that Lande’s ambitious program to derive all of quantum theory from a few simple principles has not yet been completely successful. Reviewers of his book (Shimony, 4 As a nearly pedantic refinement, we would prefer to replace 1966; Witten, 1966) have pointed out additional assumptions this phrase by with probability 1—e, where e may be made arbi­ implicit in his argument which are virtually equivalent to assum­ trarily small. This allows for arbitrarily accurate approximations ing some of the results he wishes to derive. However, these criti­ to problems which may not be formally solvable, and it allows cisms do not detract from his discussion of diffraction scattering. one to avoid certain irrelevant criticisms. (See Footnote 5). L. E. BallEntine Statistical Interpretation of Quantum Mechanics 363

The uncertainty principle is trivially satisfied since we At time t=r, after the particle has interacted with require only that px—pz — 0 approximately, and that the magnetic field, the state vector will be Ay be small enough so we know which particle has p(r) — {exp ( — iH,rz1/fb)u+(l)u-.(2) entered the magnetic field. The translational motion of the particles can be — exp (iH'rZi/fi)u^(l)u+(2) }/V2. (2.6) treated classically, and the spin can be described by a spin Hamiltonian in which the magnetic field at the The result of the interaction is to produce a correlation position of the particle is treated as a function of time, between s components of momentum and spin of particle 1, and spin of particle 2. If p\z— — H'r, then H= Viz+ V h- (2.1) d\z = +1, and cf2z — — 1; or if pu = H'r then criz = — 1 and Here V\2 — V (r12,Oi, or2) is the interaction between the (j2z — + 1. It is only necessary to make the magnetic two particles, and Vh is the interaction with the field gradient H' large enough so that the two values of magnetic field. piz are unambiguously separated. We have thus shown that the 2 component of the Vn{t)=zH'(jiz, for 0 < t< ry spin of particle 2 can be determined to an arbitrarily = 0, for /<0 or t>r, (2.2) high degree of accuracy by a measurement which, because of the spacial separation and negligible effect where t is the transit time of the particle through the of V12, does not in any way disturb particle 2. Thus magnetic field which is of gradient H' and is directed according to definition D2, a2z is an element of reality. along the z axis. However the initial singlet state is invariant under Omitting the position-dependent factors, we can rotation, and can equally be expressed as write the initial state vector, a spin singlet, as P(0) = {*+(1)^(2) —«u.(l)«+(2) }/v2. (2.3) *(0) = M l> - ( 2 ) —fl_(l)»+(2) }/V2, (2.7) Here u+ and u._ are the two-component spinors corre­ where the spinors v+ and vu are eigenvectors of ax. If sponding to the eigenvalues ±1 of crz, and the argu­ the Stern-Gerlach magnet were rotated so that the ments 1 or 2 refer to the two particles. Since field was directed along the x axis, then by an identical argument we would conclude that a2x was an element m = exp { ( - -) jT* iZ'(O), (2.4)of reality. Since no state vector can provide a value for both of the noncommuting observables a2x and a2z, the effect of the interparticle interaction V\2 will be EPR conclude, in accordance with Dl, that the state negligible if vector does not provide a complete description of an I V1 2 1 1, (2.5) individual system. One might try to avoid this conclusion by adopting a condition which can be realized by making the transit an extreme positivist philosophy, denying the reality time r short enough.5 By | V12 | we mean the modulus of both (j\z and a2z until the measurement has actually of the relevant matrix elements of Vi2 in (2.4). been performed. But this entails the unreasonable, essentially solipsist position that the reality of particle ....^ 2 depends upon some measurement which is not con­ nected to it by any physical interaction. In any case, the conclusion can be stated, as was first done by Einstein (1949, p. 682), as the following u ) theorem: The following two statements are incompatable: (1) The state vector provides a complete and exhaustive description of an individual system; (2) The real physical conditions of spatially separated (noninteracting) objects are independent. Fig. 1. Schematic illustration of the apparatus for the Einstein, Podolsky, and Rosen experiment. Of course one is logically free to accept either one of these statements (or neither). Einstein clearly accepted 8 This overcomes the objection of Sharp (1961) that any inter- the second, while Bohr apparently favored the first. action (even gravitational) would prevent us from writing p(t) as a product of two factors; one for particle 1, and one for par­ The importance of the EPR argument is that it proved ticle 2. Such an objection is quite irrelevant because any state for the first time that assuming the first statement vector can be expressed as a linear combination of product-type above demands rejection of the second, and vice versa, basis vectors, and the interaction only affects the time depend­ ence of the expansion coefficients which we treat to an arbitrary a fact that was not at all obvious before 1935, and which degree of precision. may not be universally realized today. 364 Reviews of M odern Physics • October 1970

2,2 Discussion of the EPR Theorem emphasizes that there is nothing paradoxical about the It should be emphasized that the result of the EPR EPR experiment, which is a prototype for many experiment is in no way paradoxical. The implication coincidence experiments, but he seems to forget the of (2.6) in the Statistical Interpretation is that if the original purpose of the EPR argument. experiment is repeated many times we should obtain Several people have proposed experiments similar in the result pu — —H't, ori2= + l, 0 -2z= —1 in about principle to that of EPR (Day, 1961; Inglis, 1961; one-half of the cases, and the opposite result in the Bohm and Aharonov, 1957). While these experiments other half of the cases. This correlation between the are interesting in their own right, it should be em­ spins of the two particles will be the same no matter phasized that their results will not distinguish between which components are measured. Einstein’s and Bohr’s interpretations of quantum Another essential point, which has not always been theory. If the results of experiments were to differ realized, is that the EPR theorem in no way contradicts from the theoretical predictions, this would contradict the mathematical formalism of quantum theory. It was the formalism of quantum theory itself, not just one of only intended to exhibit the difficulties which follow the interpretations. from the assumption that the state vector completely describes an individual system; an interpretive assump­ 3. THE UNCERTAINTY PRINCIPLE tion which is not an integral part of the theory. In his famous reply to EPR, Bohr (1935) said, “Such an 3.1 Derivation argumentation [as that of EPR], however, would To a given state there will, in general, correspond a hardly seem suited to affect the soundness of quantum- statistical distribution of values for each observable. mechanical description, which is based on a coherent A suitable measure of the width of the distribution for mathematical formalism • • •.” It would appear that an observable A is the variance, Bohr failed to distinguish between the mathematical formalism of quantum theory, and the Copenhagen (2L4)*=<64-, (3.1) interpretation of that theory. Einstein’s criticism was where A A is known as the standard deviation of the directed only at the latter, not at the former. Con­ distribution. Although states exist (at least as mathe­ versely the self-consistency and empirical success of matical idealizations) for which the variance of the the former are no defense against specific criticism of distribution for any one observable is arbitrarily small, the latter. Such an erroneous perception of the EPR it can easily be shown, as was first done by Robertson paper as being an attack on quantum theory itself may (1929) and Schrodinger (1930), that the product of explain (psychologically) the often repeated statement the variances of the distributions of two observables that Bohr had successfully refuted their argument. In A and B has a lower bound, fact Bohr’s paper offers no real challenge to the validity of the EPR theorem as stated above, nor does the EPR (A A)\U3Y>{h{AB+BA)-{A){B)y+\{C)\ (3.2) theorem pose any “paradox” or threat to the validity of quantum theory. where AB—BA — iC. The first term may vanish, but Bohr’s reply to EPR is really a criticism of their for canonically conjugate observables [see Eq. (1.8)] definitions of completeness and physical reality, but it is we must always have of a rather imprecise character. A satisfactory pursuit AqAp>h/2. (3.3) of this line of attack should first admit the validity of the EPR theorem with their definitions. Second it The meaning of these results is unambiguous. The should propose alternative definitions with arguments in averages of quantum theory (postulate F4) are realized favor of the superiority of the new definitions. Finally by performing the same measurement on many similarly it should show that the state vector provides a complete prepared systems (or equivalently by performing the description of physical reality in terms of the new measurement many times on the same system which definitions. Bohr has done none of these. must be resubmitted to the same state preparation In the succeeding years numerous comments on the before each measurement). In order to measure A A EPR paper have been published, many of which are (or AB) one must measure A (or B) on many similarily less than satisfactory. A discussion of a paper by Furry prepared systems, construct the statistical distribution (1936) is given in Footnote 13, Sec. 4.3 of this paper. of the results, and determine its standard deviation. Breitenberger (1965) has criticized many authors for The results (3.2) and (3.3) assert that for any par­ contributing to confusion, including Bohr for suggesting ticular state (i.e., state preparation) the product of the that the observation of particle 1 “creates” the physi­ widths of the distributions of A measurements and of B cally real state of particle 2—a position approaching measurements may not be less than some lower limit. the absurdity of solipsism.6 Breitenberger correctly A term such as the statistical dispersion principle would really be more appropriate for these results than the 61 have not found an explicit statement to this effect in Bohr’s writings, although essentially this position has been taken by traditional name, uncertainty principle. A discussion some followers of Bohr. similar to this is given by Margenau (1963). L. E. Ballentine Statistical Interpretation of Quantum Mechanics 365

3.2 Relation to Experiments There exist many widespread statements of the uncertainty principle which are different from the one that is derived from statistical quantum theory. Very common is the statement that one cannot measure the two quantities q and p simultaneously without errors whose product is at least as large as fi/2. This statement is often supported by one or both of the following arguments: (i) A measurement of q causes an unpredictable and uncontrollable disturbance of p, and vice versa. [This was first proposed by Heisenberg (1927) and is widely repeated in text books]. X1 x? (ii) The position and momentum of a particle do not Fig. 3. An experimental arrangement designed to simultane­ even exist with simultaneously and perfectly well ously measure y and py, such that the error product 8y8pv can defined (though perhaps unknown) values (Bohm, be arbitrarily small. 1951, p. 100). though it is not derivable from quantum theory. Clearly the statistical dispersion principle and the Argument (ii) is easily seen to be unjustified. It is common statement of the uncertainty principle are not based on the obvious fact that a wave function with a equivalent or even closely related. The latter refers to well defined wavelength must have a large spacial errors of simultaneous measurements of q and p on one extension, and conversely a wave function which is system, and it is plausible that one of these measure­ localized in a small region of space must be a Fourier ments could cause an error in the other. On the other synthesis of components with a wide range of wave­ hand, the former refers to statistical spreads in ensembles lengths. Using de Broglie’s relation between momentum of measurements on similarily prepared systems. But and wavelength, p = h/ \ , it is then asserted that a only one quantity (either q or p) is measured on any particle cannot have definite values of both position one system, so there is no question of one measurement and momentum at any instant. But this conclusion interfering with the other. Furthermore the standard rests on the almost literal identification of the particle deviations Aq and Ap of the statistical distributions with the wave packet (or what amounts to the same cannot be determined unless the errors of the individual thing, the assumption that the wave function provides measurements, 8q and 8p are much smaller than the an exhaustive description of the properties of the standard deviations (see Fig. 2). These points have been particle). The untenable nature of such an identification emphasized by Margenau (1963) and by Popper is shown by the example of a particle incident upon a (1967). Prugovecki (1967) has pointed out that, far semitransparent mirror with detectors on either side. from restricting simultaneous measurements of non­ The particle will either be reflected or transmitted commuting observables, quantum theory does not deal without loss of energy, whereas the wave packet is with them at all; its formalism being capable only of divided, half its amplitude being transmitted and half statistically predicting the results of measurements of reflected. A consistent application of the Statistical one observable (or a commutative set of observables). Interpretation yields the correct conclusion that the In order to deal with simultaneous measurements he division of the wavepacket yields the relative proba­ proposes an extension of the mathematical formalism, bilities for transmission and reflection of particles. But to which we shall return in Sec. 5 of this paper. there is no justification for assertion (ii). We now consider to what extent the common state­ Argument (i) must be considered more seriously ment of the uncertainty principle may be true, even since Heisenberg (1930) and Bohr (1949) have given several examples for which it is apparently true. How­ A q — - * ■ A p ever, Fig. 3 shows a simple experiment for which it is r~h \ not valid. A particle with known initial momentum ( A, /T 1 r / l\ p passes through a narrow slit in a rigid massive screen. After passing through the hole, the momentum of the J ! t k particle will be changed due to diffraction effects, but k - 8 q —►] Us— 5 p its energy will remain unchanged. When the particle strikes one of the distant detectors, its y coordinate is Fig. 2. Illustration of the uncertainty principle. The two histo­ grams represent the frequency distributions of independent meas­ thereby measured with an error 8y. Simultaneously urements of q and p on similarly prepared systems. There are, in this same event serves to measure the y component of principle, no restrictions on the precisions of individual measure­ ments 8q and 8p, but the standard deviations will always satisfy momentum, py—p sin 6, with an error 8py which may be AqAp>h/2. made arbitrarily small by making the distance L arbitrarily large. Clearly the product of the errors 8py famous microscope for measuring the position of an need not have any lower bound, and so the common electron. If the angular aperture of the microscope is statement of the uncertainty principle given above e and the wavelength of light used is X, then the accuracy cannot be literally true. One may raise the objection of the position measurement will be limited to that py has not been measured, but only defined in the above equation. Plowever this method of measuring 8x= \/sine, (3.4) momentum by means of geometrical inference from a by the resolving power of the instrument. (We use the position measurement is universally employed in symbol 8 to indicate the uncertainty or error in an scattering experiments. It rests upon the assumption of individual measurement, while A refers to the standard linear motion in a field-free region (Newton’s First deviation of an ensemble of similar measurements). Law), which remains valid in quantum mechanics Because the direction of the scattered photon is un­ (at least for L much greater than a de Broglie wave­ known within a cone of angle €, the x component of length) . momentum of the electron will be changed by some The distinction between experiments which satisfy unknown increment in the range d=sine(V^)- Note and which violate the uncertainty principle can be that this experiment is not an example of simultaneous clarified with the help of the concepts of state preparation measurement of x and px. Only x is measured here. and measurement; the distinction between these having Suppose now that we have a state preparation been emphasized by Margenau (1958, 1963) and by apparatus which will produce an ensemble of electrons Progovecki (1967). with a negligible spread in momentum (i.e., Apxtt0 ). State preparation refers to any procedure which will Suppose that we try to select from this ensemble a yield a statistically reproducible ensemble of systems. sub ensemble which has as small as possible a spread The concept of state in quantum theory (see Sec. 1.3) Ax. We do this by measuring the x coordinate of each can be considered operationally as an abbreviation for a particle and selecting those which fall within the description of the state preparation procedure.7 Of minimum resolvable range 8x of a particular value, say, course there may be more than one experimental xr. Thus our sub ensemble will have a spread procedure which yields the same statistical ensemble, i.e., the same state. An important special case (which is Ax=X/sine. (3.5) sometimes incorrectly identified with measurement) is a filtering operation, which ensures that if a system Because the recoil momentum absorbed by an electron passes through the filter it must immediately afterward may vary, the spread in momentum among the members have a value of some particular observable within a of this sub ensemble will be restricted range of its eigenvalue spectrum. Apxtt sin e(h/\). (3.6) On the other hand, measurement of some quantity R for an individual system means an interaction between Hence the ensemble which we have selected will have the system and a suitable apparatus, so that we may statistical dispersion of magnitude AxApxtth. infer the value of R (within some finite limits of Contrast the above result with the experiment accuracy) which the system had immediately before illustrated in Fig. 3. In this case we are able to determine the interaction (or the value of R which the system the values of the position and momentum of a particle would have had if it had not interacted, allowing for the to an accuracy 8y8py<£h. But this information refers to possibility that the interaction will disturb the system). the motion of the particle during a certain interval The essential distinctions between the two concepts before the measurement. We cannot use it to generate are that state preparation refers to the future, whereas an ensemble of particles with statistical spreads measurement refers to the past; and equally important, AyApy<8py. If we try to avoid this scatter by information about a system if it passes through the removing one of the detectors from the array at the apparatus. plane x=x2, and inferring from a negative response of The statistical dispersion principle (3.2, 3.3), which all the remaining detectors that the particle passed follows from the formalism of quantum theory, is a through the hole, then the momentum spread of the statement about the minimum dispersion possible in ensemble formed in this way will not be 8py (the error of any state preparation. It is significant that experiments measurement described above). Rather it will be which satisfy the uncertainty principle can be employed Apytt8y/% (due to diffraction effects). The spread of as state preparations, whereas experiments which particle positions in the ensemble will be Ay£^<5y (the violate the uncertainty principle cannot. size of the hole), and so the statistical dispersion Consider, for example, Heisenberg’s (1930, p. 21) principle will be satisfied. In conclusion, the uncertainty principle restricts the 7 Lamb (1969) has described an idealized method of preparing degree of statistical homogeneity which it is possible to an arbitrary single-particle state. achieve in an ensemble of similarly prepared systems, L. E. B allentine Statistical Interpretation of Quantum Mechanics 367 and thus it limits the precision which future predictions (3.2) (Messiah, 1964, p. 319), for any system can be made. But it does not impose A A AE>^% | d(A)/dl |, (3.9) any restriction on the accuracy to which an event can be reconstructed from the data of both state preparation where A is an arbitrary observable, and the averages and measurement in the time interval between these are calculated for an arbitrary time-dependent state. two operations. Heisenberg (1930, p. 20) made nearly If one defines a characteristic time for the system and this distinction in his statement, “ • • • the uncertainty the state as relation does not refer to the past.” His subsequent t — min^) {AA | d(A)/dt |'_1}, (3.10) remark, “It is a matter of personal belief whether • • • the past history of the electron [as inferred from an then one may write experiment like that of Fig. 3] can be ascribed any rAE>yh. (3.11) physical reality or not,”—based on the fact that it cannot be reconfirmed by any other future measure­ Clearly this result does not imply that one cannot ment, seems unduely cautious. In fact, the majority of measure the energy of a system exactly at an instant of real physical measurement are of just this type. Indeed time, as is sometimes stated, but rather that the spread for every future oriented experimental operation which of energies associated with any state is related to the yields verifiable information, there must be another characteristic rate of change associated with the same past oriented operation whose function is not to be state.8 verifiable but to verify (Popper, 1967, pp. 25-28). It is just this essential fact which is emphasized by the 4. THE THEORY OF MEASUREMENT distinction between state preparation and measurement. The desirability of an analysis of the measurement process by means of quantum theory was perhaps first 3.3 Angular and Energy-Time Relations indicated through Bohr’s insistence that the “whole An apparent contradiction of the uncertainty prin­ experimental arrangement,” (Bohr, 1949, p. 222), ciple, which is frequently rediscovered by bright object plus all apparatus, must be taken into account students, concerns the polar angle ip and the z component in order to specify well-defined conditions for an of angular momentum Lz. If Lz is represented by application of the theory. However he never carried — ihd/dipy then it would appear that \jp, Lz] = ifty from this program to its logical conclusion; the description of which it would follow that both the object and the measuring apparatus by the formalism of quantum theory. Such an analysis is AipALz>%/2 (wrong). (3.7) useful for several reasons. It has frequently been asserted that the acts of But (3.7) is obviously wrong since ALz can be arbitrarily measurement or observation play a different role in small, and

(ft/2)2((sin = E|I;r>. (4.2) measurement will be very helpful in deciding the ques­ r tion raised in Sec. 1.2, that is whether a quantum state From (4.1) and the linearity of the equation of motion, describes an individual system (object plus apparatus the final state for the system must be in this case), or whether it must refer only to an U\I;t)\II; 0 > = X > | *> 11; 4>r)| II; a,) ensemble of systems. r 4.1 Analysis of the Measurement Process = | I+ II;/), say. (4.3) A specific illustration of this general analysis is provided The essence of a measurement is an interaction by the EPR experiment discussed in Sec. 2.1. We may between the object to be measured and a suitable consider the measured quantity R to be the spin of apparatus so that a correspondence is set up between particle 2, and the “pointer” of the apparatus to be the the initial state of the object and the final state of the momentum of particle 1 (assuming that two detectors apparatus. This interaction may or may not change the are placed so as to determine its sign). value of the observable being measured, the final state According to the Statistical Interpretation and F5 of the object being of no significance to the success of of the formalism, the probability of the apparatus the measurement. “pointer” being ar at the end of an experiment is Suppose we wish to measure the observable R of the j (r | \p) |2. That is to say, if the experiment were object I, for which there must be a complete set of repeated many times, always preceded by the same state eigenvectors, preparation for both object and apparatus, the relative R 11;r) = r | I;r>. frequency of the value ar would be | {r | \//) |2. That this Denote a set of states for the apparatus II by | II; a), should be identical with the probability of the object where the eigenvalue a is the appropriate “pointer having had the value r for the dynamical variable R position” of the apparatus, and | II; 0) is the initial immediately before interacting with the apparatus II, premeasurement state. The interaction between the is just the criterion that II functions as a measuring object and the apparatus ideally should, if it is to apparatus for R. function as a measurement, set up a unique corre­ spondence between the initial value of r and the final 4.2 The Difficulties of the “Orthodox” Interpretation value of a, ar, for example. That is, if the initial state The discussion of the analysis of measurement for the system I+ II is | I; r) | II; 0), then the equation according to the Statistical Interpretation was so simple of motion must lead to the final state and natural that further comment almost seems redundant. But if instead of the basic assumption of the U | I; r> | II; 0>= | I; r) | II; ar\ (4.1a) Statistical Interpretation that a state vector char­ if the observable R is not changed by the interaction9-10; acterizes an ensemble of similarily prepared systems, or in the general case one assumes that a state provides a complete description of an individual system, then the situation is quite J7 11; r> | II; 0>= | I; fr) | II; ar), (4.1b) different. Unfortunately this assumption has been made so frequently, more often tacitly than not, that where U is the time-development operator for the it and the resulting point of view have come to be duration of the interaction between I and II. Note that considered “orthodox”. the vectors |I;<£r) need not be orthogonal since or- The “orthodox” quantum theory of measurement was formalized by J. Von Neumann (1955) and has 9 Many authors give the unwarranted impression that this highly special case is general. been clearly reviewed by Wigner (1963). The first 10Araki and Yanase (1960) have shown that unless R com­ difficulty encountered is that if one tries to interpret the mutes with all universal additively conserved quantities, then final-state vector (4.3) as completely describing one no interaction exists which could satisfy (4.1a). However, they also show that this equation can nevertheless be satisfied to an system (object plus apparatus), then one is led to say arbitrary degree of accuracy, provided the apparatus is made that the pointer of the apparatus has no definite posi­ arbitrarily large. This fact is illustrated in the measurement of tion, since (4.3) is a coherent superposition of vectors the z component of spin by a Stern-Gerlach apparatus (Sec. 2.1), which used an external magnetic field to break the conser­ corresponding to different values of a. But the pointer vation of Sx and Sy. The source of this magnetic field is rigidly can be a quite classical object and it makes no sense to fixed to the earth, and so the apparatus is effectively infinite, as it must be to satisfy the theorem of Araki and Yanase. See also say that it has no position at any instant of time, Yanase (1961). especially since the different values of ar are macro- L. E. B allentine Statistical Interpretation of Quantum Mechanics 369 scopically distinguishable. Thus if we attribute the Neumann introduced the observer III, who supposedly state vector to an individual system, we are inevitably performs a “measurement” on I+ II (this “measure­ led to classically meaningless states for a classical ment” however is merely an observation), and thus object.11 reduces the state vector to (4.4) or (4.4'). He then To circumvent this difficulty, the “orthodox” theory shows that it makes no difference whether the apparatus assumes that, in addition to the continuous evolution II is considered to be part of the observer or part of the of the state vector according to the equation of motion, object being observed. there is also an unpredictable discontinuous “reduction Von Neumann's theory of measurement is very of the state vector” upon “measurement,” from (4.3) to unsatisfactory. It suggests that the passive act of observation by a conscious observer is essential to the I !;<£>•') I II ; oLr'),(4.4) understanding of quantum theory. Such a conclusion is where aT> is the pointer position which is actually without foundation, for we have seen that the Statistical observed. Proponents of this point of view also usually Interpretation does not require any such considerations. assume that the value of the observable R is not Moreover, the atoms of the observer's body may, in changed by measurement, so that in place of (4.4) principle, be treated by quantum theory, and the they would obtain resulting state vector will be similar to (4.3) (with I replaced by I+ II, and II replaced by III). Von 11; O I ii; <*,')> (4.4') Neumann's theory would require another hypothetical but this point is not essential. The word “measure­ observer IV to observe III, and so reduce the state ment” is enclosed in quotes because its use in this vector. But this can only lead to an infinite regression, peculiar context of the “orthodox” theory is not which, in an earlier era might have been taken as a equivalent to the more physical definition of measure­ proof of the existence of God (the Ultimate Observer) I ment given in Sec. 3. No one seems to have drawn this unfounded conclusion, It should be emphasized that this reduction cannot but the equally unfounded conclusion that the con­ arise from the equation of motion. The form of (4.3), sciousness of an observer should be essential to the involving a superposition of macroscopically distinct theory has, unfortunately, been taken seriously pointer positions, depends only upon the linearity of the (Heitler, 1949, p. 194; Wigner, 1962). equation of motion and not upon any of our simplifying Because of the difficulties inherent in the projection assumptions. For example, one might object that the postulate, we must critically examine the reasons given pointer position eigenvalue a is not sufficient to label a in support of its introduction. Dirac (1958, p. 36) unique state vector for the apparatus, since there are a argues that a second measurement of the same ob­ huge number of commuting observables. But this servable immediately after the first measurement means only that each vector | II; a) must be replaced must always yield the same result. Clearly this could by a set of vectors, all the sets being mutually or­ be true at most for the very special class of measure­ thogonal. Komar (1962) has shown that even with ments that do not change the quantity being measured. these extra degrees of freedom, a final state of the form A statement of such limited applicability is hardly (4.4) is not possible.11 12 If the reduction of the state vector suitable to play any fundamental role in the foundations is to enter the theory at all, it must be introduced as a of quantum theory. In fact this argument is based on special postulate (often called the projection postulate). the implicit (and incorrect) assumption that measure­ If one merely applied the projection postulate to the ment is equivalent to state preparation, the contrasting object I, one would say that upon “measurement” the definitions of which were given in Sec. 3. The necessity state changes discontinuously from (4.2) to 11; r')y for distinguishing between these two concepts has been where rf is the “measured” value of the observable R. pointed out by Margenau (1958; 1963). For example, On the other hand, the analysis of the interaction a polaroid filter placed in the path of a photon beam between the object I and the apparatus II leads to the constitutes a state preparation with respect to the state (4.3). To avoid a direct contradiction, Von polarization of any transmitted photons. A second Polaroid at the same angle has no further effect. But 11 This was first pointed out by Schrodinger (1935), who pro­ neither of these processes constitutes a measurement. posed the following hypothetical experiment. A cat is placed in a chamber together with a bottle of cyanide, a radioactive atom, To measure the polarization of a photon one must also and a device which will break the bottle when the atom decays. detect whether or not the photon was transmitted One half-life later the state vector of the system will be a super­ through the polaroid filter. Since the detector will position containing equal parts of the living and dead cat, but any time we look into the chamber we will see either a live cat absorb the photon, no second measurement is possible. or a dead one. A literal translation of “Schrodinger’s cat paradox”, It has also been claimed (Messiah, 1964, p. 140) that as this is called, has been given by Jauch (1968, p. 185), but his a discontinuous reduction of the state vector is a discussion of the paradox is not adequate, and his volume and page reference is incorrect. consequence of an alleged uncontrollable perturbation 12 His conclusion that no theory consistent with quantum me­ of the object by the measuring apparatus. A counter chanics can account for the occurrence of events in nature, how­ ever, is not true if one drops the assumption that a state vector example to this claim is provided by the EPR experi­ describes completely an individual system. ment (Sec. 2.1), in which the spin of particle 2 is measured indirectly, without any disturbance of that interaction between the object I and .the apparatus II, particle whatsoever. In general the apparatus will the final state operator for the combined system I+ II interact with the object, but the effect of this is not will be the pure state such as to bring about the hypothetical reduction. We do not consider as credible the occasional statement p/= I i+H;/)(i+ii;/l> (4.5) (Dirac, 1958, p. 110) that a mere observation (i.e., an I i+ H ;/ ) = Z a 11; <£r> | II; oir),(4.3') experimenter looking at his apparatus) could affect the r state of the system in a manner incompatible with the equation of motion. where cr=(r\\p) in the notation of (4.3). Now the Although it is not essential to the theory, Popper mixed state formed fromvectors of the type (4.4), (1967) has pointed out that the reduced state vector k |211; ,) | II; ar)(I; r | <11; «r I (4.6) (4.4) can be interpreted in a natural way. According r to the Statistical Interpretation, the state vector (4.3) is not equivalent to p/. Even though they both yield the represents, not an individual system, but the ensemble same probability distribution for the position of the of all possible systems (object plus apparatus), each of apparatus “pointer” a, their predictions for other which has been prepared in a certain way and then observables may disagree.13 Because time development allowed to interact. The state vector (4.4) represents is effected by a unitary transformation (1.7a), which the subensemble whose definition includes the additional preserves the pure state condition p2 = p, it is impossible specification that the result of the measurement (pointer for the pure state (4.5) ever to evolve into the mixed reading) be ar>. There is no question of reduction of the state (4.6). state vector being a physical process. Realizing this fact, many authors (Feyerabend, 1957; Margenau (1963, p. 476) has taken a more sym­ Wakita, 1960, 1962; Daneri et al., 1962; Jauch, 1964) pathetic attitude toward Von Neumann’s theory of have proposed that, nevertheless, (4.5) and (4.6) may measurement on the grounds (gathered from personal be equivalent for all practical purposes. Specifically, conversation) that he regarded his projection postulate they suggest that if the macroscopic nature of the as convenient but not necessary. In reply I would say apparatus and its relevant observables are taken into that I find no evidence of this interpretation in Von account, then the average of any macroscopic ob­ Neumann’s writings. Moreover, the great amount of servable M will be the same for these two states. With confusion generated, as well as the theoretical effort the abbreviated notation | r, ck)= 11; r) | II; a), these expended trying to explain the reduction of the state averages are, respectively, vector, suggest that the projection postulate has been much more of an inconvenience than a convenience. Tr (p/M) = cT-cr*{fa r, Oir I M | 4>r', 0Lr') plification process does not take place until the plate is = (I; <#>r 11; r) then becomes | I; r)), then there would be no difference between 4.4 Conclusion—Theory of Measurement (4.7) and (4.9). Clearly a similar result holds for This section on measurement theory is lengthy observables belonging entirely to the object I, but of because of the great length of the literature on this course no such result is valid for observables which subject. However, the conclusions are quite simple. The belong to I and II jointly.15 formal analysis of a measurement has been undertaken Daneri, Loinger, and Prosperi (1962, 1966) (DLP) not for its own sake, but only to test the consistency of consider the pure state (4.5) or equivalently (4.3) to alternative interpretations of quantum theory. Using be the final state of the combined system immediately only the linearity of the equation of motion and the after interaction between the object and the apparatus. definition of measurement, we see that the interaction They then concentrate on the amplification aspect of between the object and the measuring apparatus leads, the measurement necessary for a microscopic object to in general, to a quantum state which is a coherent trigger a macroscopic response in the apparatus, and superposition of macroscopically distinct “pointer invoke the ergodic theory of quantum statistical positions.” In the Statistical Interpretation this dis­ mechanics in order to treat the transition of the appa­ persion of pointer positions merely represents the ratus from its initial “metastable”16 condition to frequency distribution of the possible measurement equilibrium. They claim, in essence, that the equilib­ results for a given state preparation. rium state can be represented by the mixed state But if one assumes that the state vector completely operator (4.6). describes an individual system, then the dispersion The attitude of DLP is well expressed in the words of must somehow be a property of the individual system, Rosenfeld (1965, p. 230), who endorses their approach: but it is nonsensical to suppose that a macroscopic “The reduction of the initial state of the atomic system pointer has no definite position. None of the attempts has nothing to do with the interaction between this to solve this problem using some form of reduction of system and the measuring apparatus: in fact, it is the state vector are satisfactory. No such problem arises related to a process taking place in the latter apparatus in the Statistical Interpretation, in which the state after all interaction with the atomic system has ceased.” vector represents an ensemble of similarily prepared Rosenfeld also emphasizes the view that the presence or systems, so it must be considered superior to its rivals. absence of a conscious human observer is irrelevant to We have neglected the frankly subjective inter­ the problem.17 pretation (Sussmann, 1957; Heisenberg, 1958), ac­ The above remarks show that DLP and Rosenfeld cording to which the quantum state description is not explicitly reject the “orthodox” interpretation discussed supposed to express the properties of a physical system in Sec. 4.2 (as does the present author), but they do so or ensemble of systems but our knowledge of these without actually confronting the question of whether properties, and changes of the state (such as the sup­ the state operator (or vector) represents a single system posed reduction of state during measurement) are or an ensemble of similarily prepared systems. identified with changes of knowledge, not so much The analysis of generalized quantum amplification because it is wrong as because it is irrelevant.18 One can, devices by DLP is interesting and correct, at least to of course, study a person’s knowledge of physics rather the extent that their ergodic hypothesis is valid. But it than studying physics itself, but such a study is not does not constitute the essence of measurement in germane to this paper. For example, someone’s knowl­ general. To paraphrase an example given by Jauch edge of a certain system can change discontinuously as a result of a blow to the head which causes amnesia, 14The tensor product notation A ® B means that A operates on the 11) vectors, and B operates on the | I I ) vectors. as well as through the receipt of new information from a 15 This illustrates the fact that the state of a two-component system cannot be uniquely determined by measurements on each 18 Another area in which the subjective interpretation causes component separately. Jauch (1968, Secs. 11-7 and 8) gives a confusion is the relationship between entropy and information. good discussion of the mathematical aspects of this problem, and It is true that the aquisition of information requires a corre­ in particular of the tensor product formalism. sponding increase of entropy (Brillouin, 1962); hence the associa­ 16 As pointed out in DLP (1966), this is not necessarily the tion of information with negative entropy is useful. But the con­ thermodynamic definition of metastability. verse proposition that entropy is nothing but a measure of our 17 On this point Rosenfeld seems to have modified his earlier lack of information is a fallacy. Entropy also has a thermody­ views, as expressed in a discussion exchange with Viger (see namic meaning, d S = d Q /T , which is valid regardless of the exist­ Korner, 1957, pp. 183—6). ence or nonexistence of Maxwell’s demon. 372 Reviews of M odern Physics • October 1970

measurement, although the subjectivists tend to ignore from which, by a Fourier inversion, we obtain, the former while stressing the latter. P( exp (idq-\-iXft), (5.8a) SF (l> P'> P) dp=P(q; = | or | equivalently |2, (5.1) fP(q,P; *) dq=P(p; = | |2. (5.2) n (n\ qnpm_^2- n I )qn~ lftmq\ (5.8b) Additional conditions will be considered later. z=o \l 1 Let us first consider the method by which the single observable probability distribution function can be then one obtains (Moyal, 1949) constructed. The characteristic function for the dis­ p(?, p a ) tribution of an arbitrary observable A is given by = (2-n-)-1/(V' I q—hr%) exp (— {q+\rh \ p) dr, M(£-,p) = (exp (i£A)) (5.9) = /exp atA)P (A (5.3) which was first introduced by Wigner (1932). Because hence P (A ; xp) is equal to the inverse Fourier transform this expression may become negative it cannot be of M (£; p) . By expanding the exponential interpreted as a genuine probability distribution, and 00 (iP)n Wigner proposed it only as a calculational device. M ( f ,P ) = j : — (5.4) » - 0 n\ Another distribution which suffers from the same defect has been derived by Margenau and Hill (1961) using we see that a knowledge of the characteristic function is the correspondence equivalent to a knowledge of all the moments of the distribution. According to the formalism (see F4), qnpm~*\ (qnftm+ftmqn) • (5.10) these are given by20 * Since every correspondence rule which associates an {An)= (p| A n| p).(5.5) operator with a classical function of q and p, By analogy we may introduce a characteristic g(q,p)->6(q,p), (5.11) function for the joint probability distriubtion provides a joint probability distribution, it is possible to M{e,\-,p) = <, (5.6) formulate the most general form of P(q> p;p) which n,m=0 n\m\ will satisfy (5.1) and (5.2). We refer to the original 19 We shall consider only pure states since these illustrate the papers for details (Cohen, 1966; Margenau and Cohen, essential problems. The generalization to mixed states is straight­ 1967). These authors investigated the possibility of forward. 20 In this section it is necessary to distinguish between a phys­ constructing a joint probability distribution such that ical observable and the mathematical operator which represents it. The operators will be distinguished by a circumflex. (P\G(q,p) \P)=ffg(q,p)P(q,p;P) dqdp, (5.12) L. E. Balt.entine Statistical Interpretation of Quantum Mechanics 373 and also such that for any function K Here I is some constant with dimensions of length (* I p) ) | *>= //* ( ? ( ? ,p) fi; +) dqdp. (interpreted by Bopp as the finest accuracy of a position measurement). (5.13) The problem of defining a satisfactory joint proba­ In other words, they asked whether there exists a joint bility distribution for position and momentum may not probability distribution such that the averages of all be entirely mathematiced, for we must also decide on observables can be calculated by a phase-space average “the empirical definition of the concept ‘simultaneous as in classical statistical mechanics.21 values of position and momentum/ ” according to Their answer was negative. Although it is always Prugovecki (1967). To this end he studies various possible to satisfy (5.12), (5.13) could be simul­ operational methods of measuring both these observables taneously satisfied only if the correspondence simultaneously on the same individual system, not merely on different individual systems representative of K(g(q,p))-+K(.G(q,p))(5.14) the same state preparation. He suggests that the can be satisfied for all functions K with the same irreducible errors in individual measurements ought to correspondence rule as that leading to (5.11). This be taken into account in a generalized formalism (in they prove to be impossible.22 With the wisdom of addition to the statistical fluctuations in an ensemble of hindsight, we should not be surprised that (5.12) and similar measurements, which are treated in the estab­ (5.13) cannot be satisfied, for if it were otherwise, then lished formalism). It should be emphasized that quantum mechanics could be expressed as a special simultaneous measurement of noncommuting ob­ case of classical mechanics. servables, even with a finite precision, has no counter­ To deal with simultaneous values of position and part in the established formalism which treats only momentum variables in a logically consistent fashion measurement of a single observable (or a commuting we need only require that a joint probability dis­ set of observables) .23 Now to experimentally determine tribution exists which satisfies (5.1) and (5.2), and a probability distribution, one must perform a number of measurements and construct a histogram of the P(q,p-,t)>0 (5.15) results (see Fig. 2). Each point on the curve will have a for all quantum states. These conditions are obviously vertical uncertainty due to the statistical uncertainty satisfied by the function of using a finite sample, and a horizontal uncertainty due to the error involved in a single measurement. P(q,P;t) = I <* I ?> I21 <>H |2, (5.16)(In the case of simultaneous measurement of q and p, but this form is probably not unique. Indeed, it cannot a single measurement would be a pair of numbers be applicable to scattering experiments (see Fig. 3), (q, p), and the error would be an area in the qp plane). where, merely by geometry, there must be a close If the latter cannot be reduced to zero, we cannot correlation between position and momentum at large determine a probability distribution exactly, even if we distances from the scattering center. The investigation eliminate statistical uncertainties by repeating the of possible joint probability distributions is clearly not measurement an arbitrarily large number of times. complete. To deal with such a circumstance Prugovecki defines Bopp (1956, 1957) has undertaken not only to a complex probability distribution, P^q'v{I{Kh), with represent quantum mechanics in terms of ensembles in the following properties: the real part of P represents phase space, but also to derive the statistical equations the probability that q will be in the interval, h and from simple principles—an a priori derivation—not simultaneously p will be in the interval 12, where merely a translation from the vector-operator formalism of quantum theory. Bopp’s work differs from the above /i={X:gi'-pi0, as h and / 2 are enlarged to include the entire spectra from — 00 to + 00 . If / 1 and / 2 are smaller ______(P2)b =(P*)q+W/P. than the irreducible errors in a single measurement, 21 Because (5.1) and (5.2) are satisfied, this will clearly be true for any function of q only, or of p only. 23 The naive interpretation (Matthews, 1968, Chap. 3) of the 22 It would seem inevitable that arbitrary functional relations operator product as corresponding to a measurement of B involving q and p cannot be preserved by any quantal operator followed by a measurement of A is refuted by the example of the correspondence rule, since the equation qp—pq = 0 for classical Pauli spin operators for which

(iv) [A'] If the observable R is nonnegative, then ensembles of systems with different values of the hidden <*>>o. variables, but if all hidden variables were fixed, the (v) [B'J For arbitrary observables R, S, • • • and resultant subquantal ensemble would be dispersionless. arbitrary real numbers a, b* • •, we must have That is, <(22-<*»*> = 0, (6.7a) (aR+bS+ • • • ) = a(R)+b(S)+ • • • (6.1) or equivalently for all possible ensembles (or states) in which the (R2)=(R)2, (6.7b) averages may be calculated. for all observables R, since by hypothesis every quantity An arbitrary Hermitian operator R may be written as would have a unique value. Applying the result (6.5) to (6.7b) for the case of R— | )(<£ |, where (<£ | <£)= 1, R= X I n)Rnm(m | yields m,n ()= IPI <£)2 (6.8) = Z U ^ R nn n for all normalized vectors | 4>). This implies (4> | p j <£)= 1 or 0 for all | <£), and since this expression varies con­ + £ {V^R.eR„m+ W ^Im R mn}, (6.2) m,n;m<.n tinuously with ! 4>), the constant—T or 0—must be the same for all | <£). Hence we must have either where Rnm=Rmn* = (n | R | m) is a number, and p = l or p = 0 (6.9) EA»>= |»)<»|, for dispersion-free ensembles. F(nm) = | n)(m | + | |, The case p = 0 is ruled out because it would imply W(nm)= —i(\n)(m\ — | m)(n |) (6.3) (R) = 0 for all R in any subquantal ensemble, and so averages of these could never yield the correct averages are Hermitian operators. According to assumption in quantum ensembles. The case p = 1 does not in fact (iii) these operators are all to be regarded as observables, yield a dispersionless ensemble if the vector space is and hence (ii) and (v) may be applied to obtain greater than one dimensional. In this case (R)=Z(U^)Rnn n (1)= Tr (6.10) + £ { (y&m)> Re Rmn-\-) Im (6.4) d being the dimensionality of the space (usually oo), and the left-hand side of (6.7a) becomes This result may be rewritten as (R2)—2(R)2-\- (R)2(l) (6.11) (R)= £ pnJRmn= Tr (pi?), (6.5) m,n which is not zero. Thus Von Neumann concludes that, provided his assumptions (i)-(v) are accepted, there if we define the matrix elements of the statistical are no dispersion-free ensembles, and hence there can operator p to be be no hidden variables. Since these assumptions are Pnn=(U^), generally considered to be a part of quantum theory, he states “It is therefore not, as is often assumed, a Pmn = h (v(nm))+ hi(W<»"> ) question of a reinterpretation of quantum mechanics,—• Pnm=UV(nm)) - i i ( W ^ ) (m

6.4 Suggested Experiments through two nearly crossed polaroid filters. According to the theory, the transmitted beam should then have As emphasized by Bohm, the postulates of quantum almost unique values for the hidden variables. These theory can be tested only if we consider what it would are presumed to relax once more to a uniform dis­ mean to contradict them. Therefore, at the very least, tribution within some relaxation time r, but if the beam the study of hidden variables will have served a useful is incident on a third rotatable polarizer (whose axis is purpose if it leads to the suggestion of interesting at a variable angle 6 to the axis of the second), before experiments, regardless of whether or not the outcome this relaxation takes place the intensity transmitted of those experiments is favorable to the hidden-variable through it is predicted to deviate from the cos2 6 law in a hypothesis. definite fashion. In the experiment, the third polarizer The additivity of averages of noncommuting ob­ was placed as close to the second as possible, but no servables in quantum ensembles (6.1) is a very powerful deviations from the cos2 6 law were observed. Papaliolios assumption, as Von Neumann’s theorem shows, since concludes that the hidden variables, if they exist, must it and a few other postulates allow one to derive (6.5) relax to a uniform distribution in a time r<2.4X10~14 from which all the statistical predictions of quantum sec, which is two orders of magnitude smaller than the theory flow. Von Neumann (1955, p. 309, Footnote value suggested by Bohm and Bub. 164) suggested an experiment based upon the fact that the Hamiltonian operator is the sum of two non­ 7. CONCLUDING REMARKS commuting terms, the kinetic and potential energies. The central theme connecting all of the topics in this For some suitable state, one should measure the average paper has been the superiority of a purely statistical kinetic energy by measurements of the momentum, interpretation of quantum theory, in which a state measure the average potential energy by measurements vector represents an ensemble of similarly prepared of position, and measure the average energy by spec­ systems, as opposed to the stronger assumption that a troscopic methods (each measurement being preceded state vector provides a complete description of an by the same state preparation). Then the additivity of individual system. By the criterion of logical economy, the quantum averages (expectation values) could be the Statistical Interpretation is preferable because it directly tested. A similar experiment which would makes fewer assumptions. The stronger assumption probably be easier to perform could be based on the plays no role in any application of quantum theory, spin components trx, 0.10, so a clear cut experimental objects. test is possible. It used to be argued that the peculiarities consequent A measurement of the correlation in the polarizations upon the assumption that a system is completely of photons emitted from the decay of singlet posi- characterized by a wave function were confined to the tronium, which is similar in principle to the experiment microscopic domain, and that these peculiarities were above, has been performed by Wu and Shaknov (1950), justified by “the unavoidable disturbance of the system but only for two angles. As Bell (1964) has pointed out, by the measuring apparatus.” The latter of these con­ it is easy to make a hidden-variable theory agree with tentions is refuted by the example due to EPR, in quantum theory at certain fixed angles, but not at all which it is possible to measure some observable of an angles. elementary particle, without the apparatus interacting The hidden-variable theory of Bohm and Bub with that particle in any way. It appears (see Witmer, (1966) has been tested by Papaliolios (1967). Ac­ 1967, p. 45) that Bohr recognized this implication of the cording to the theory, when a photon from an arbitrarily EPR argument, for he subsequently cautioned against linearly polarized beam is incident on a Polaroid filter such phrases as “disturbing of phenomena by observa­ (whose polarization axis is not exactly parallel to the tion” (Bohr, 1949, p. 237), but this lesson has been polarization vector of the beam), the photon is or is not forgotten by some modern authors (e.g., Messiah, transmitted depending upon the value of some hidden 1964, p. 140; Matthews, 1968, p. 27). variable. In a normal beam, the hidden variables are The former contention is refuted by an analysis of a uniformly distributed, yielding the usual transmission measurement of a microscopic object by a macroscopic probability. In the experiment, the beam is first passed apparatus (Sec. 4). It follows from very simple con- L. E. B allentine Statistical Interpretation of Quantum Mechanics 379 siderations that the final state of the system (object to Criticisms. In support of his thesis that a wave plus apparatus) must, in general, be a coherent super­ function must be considered to represent an ensemble position of macroscopically distinct “pointer positions.” of systems, and cannot reasonably be assumed to If the state vector is assumed to completely describe the provide a complete description of an individual system, individual system, i.e., if one assumes that the system he considers the decay of a radioactive atom, with an simply does not have a value for any observable except automatic recording detector to register the decay those for which the state is an eigenvector (or perhaps time.26 This example is a prototype of the one used in those whose dispersion is very small), then one will be the Theory of Measurement, and it embodies all the forced to the absurd conclusion that the “pointer” of essential physical content of the more general argument. the apparatus (a macroscopic object) has no definite In this Reply he also restated the conclusion of the EPR position. The supposed reduction of the state vector, and argument in the form which we have called the Theorem all the difficulties and complications associated with it, of EPR, and gave a brief discussion of the nontrivial are only artifacts of the vain attempt to retain the aspect of the classical limit of the quantal state descrip­ above assumption. But to what purpose? Under the tion. more modest assumption that a state vector represents A serious reading of Einstein’s Reply should clear up an ensemble of similarily prepared systems, the measure­ any misconceptions to the effect that he rejected ment process poses no particular problem. quantum theory or misunderstood its foundations. In The criticisms above also apply to certain hidden- fact, he understood the essentially statistical nature of variable theories in which a p field is associated with quantum theory as well as any of his contemporaries, each individual system (Bohm, 1952; Bohm and Bub, and better than many. His only objection was against 1966). Like Von Neumann’s theory, these theories the assumption that a wave function or state vector must also invoke a reduction of the p field upon meas­ could exhaustively describe an individual system, which urement, which in the example of the correlated but we have seen to be an unwarranted and troublesome spacially separated systems considered by EPR, would assumption. This fact, and the fact that Einstein necessitate a signal passing instantaneously across the advocated a fully viable interpretation of quantum macroscopic distances between the systems. Since, as theory (essentially the Statistical Interpretation of this we have seen, such a drastic assumption is not necessary paper although he expressed himself more briefly), do for a satisfactory interpretation of quantum theory, not seem to have been appreciated by his critics.27 we see no reason for considering it further. The foundations of quantum theory are subject to The Uncertainty Principle (Sec. 3) finds its natural continuous discussion, and two almost coincident interpretation as a lower bound on the statistical dis­ papers in the American Journal of Physics, taking persion among similarily prepared system (this inter­ nearly opposite points of view, deserve comment. pretation being deduced, not introduced ad hoc), and Hartle (1968) has made a novel attempt to derive the is not in any real sense related to the possible dis­ statistical assertions of quantum theory from a quantal turbance of a system by a measurement. The dis­ description of individual systems. The conceptual basis tinction between measurement and state preparation of this attempt is questionable, for, as Hartle admits, a is essential for clarity. It is possible to extend the quantum state is not an objective property of an formalism of quantum theory by the introduction of individual system. That is to say, in the words of joint probability distributions for position and momentum Blokhintsev (1968, p. 50), (Sec. 5). This demonstrates that there is no conflict with “If [the wave function were a characteristic of a quantum theory in thinking of a particle as having single particle^ it would be of interest to perform such a definite (but, in general, unknown) values of both measurement on a single particle (say an electron) position and momentum, contrary to an earlier inter­ which would allow us to determine its own individual pretation of the uncertainty principle. wave function. No such measurement is possible.” The Statistical Interpretation does not prejudice the But whereas Blokhintsev (and the present author) possibility of introducing hidden variables which would regards the quantum state as describing an ensemble of determine (in principle) the outcome of each individual similarily prepared systems, Hartle considers it to measurement (Sec. 6). Although such models exist, describe the information possessed by some observer. Bell (1964) showed that they cannot, in general, satisfy The irrelevance of that interpretation has already been the requirement that measurements on spacially commented upon (Sec. 4.4). separated noninteracting objects should be independent. It is ironic that this requirement which Einstein, 26 Messiah’s (1964,p. 158) reply to this argument is essentially that there are experimental arrangements from which it is im­ Podolsky, and Rosen first used to refute the belief that a possible to determine a definite decay time. (His wording to the wave function could completely describe an individual effect that they are incompatible with the existence of a time of system (a belief which stood opposed to hidden varia­ decay is an unjustified overstatement). But this is irrelevant. The point is that it is sometimes (i.e., with suitable apparatus) bles), should also prove disasterous to at least the possible to measure the decay time of a single atom, but this simplest idea of hidden variables. decay time is never predicted by quantum theory. 27 See the articles by Pauli, Born, Heitler, Bohr, and Margenau Many of the ideas expounded in this paper were in the volume edited by Schilpp (1949). However, Margenau’s expressed, in essence, by Einstein (1949) in his Reply recent papers are more compatible with Einstein’s views. On the other hand Park (1968) has carefully studied Duane, W., 1923, Proc. Natl. Acad. Sci. U.S. 9, 158. Einstein, A., B. Podolsky, and N. Rosen, 1935, Phys. Rev. 47, the relations between ensembles and individual systems 777. in statistical theories. He shows how one may proceed ------, 1949, in Albert Einstein: Philosopher-Scientist, edited by logically from the description of ensembles to individual P. A. Schilpp (Library of the Living Philosophers, Evanston, 111.), reprinted by Harper and Row, New York, p. 665. systems in classical statistical mechanics, and how the Fano, U., 1957, Rev. Mod. Phys. 29, 74. attempt to proceed similarily in quantum statistical Feyerabend, P. K., 1957, in Observation and Interpretation, edited by S. Korner (Butterworths, London), p. 121. mechanics to an identification of a pure state operator Freistadt, H., 1957, Nuovo Cimento Suppl. 5, 1-70. with an individual system fails. The present paper is Furry, W. H. 1936, Phys. Rev. 49, 393. consistent with his conclusion; moreover we have shown Galindo, A., A. Morales, and R. Nunez-Lagos, 1962, J. Math. Phys. 3, 324. that such an identification would lead to serious Gleason, A. M., 1957, J. Math & Mech. 6, 885. difficulties in principle. Hartle, J. B., 1968, Am. J. Phys. 36, 704. Although this article has been concerned primarily Heisenberg, W., 1927, Z. Physik 43, 172. ------, 1930, The Physical Principles of Quantum Theory (Univer­ with a consistent interpretation of quantum theory as sity of Chicago Press, Chicago, 111.), reprinted by Dover, New it is presently formulated, one is naturally led to ask for York, 1950. ------, 1955, in Neils Bohr and the Development of Physics, edited hints of future developments. Except for eliminating by W. Pauli (Pergamon, London), p. 12. some false starts, such as theories of wave function ----- , 1958, Physics and Philosophy (Harper & Brothers, New reduction, such hints are hard to find. Recognition that York), pp. 54-55. Heitler, W., 1949, in Albert Einstein: Philosopher-Scientist, edited quantum states should refer to ensembles of similarily by P. A. Schilpp (Library of the Living Philosophers, Evanston, prepared systems would seem to open the door for 111.), reprinted by Harper and Row, New York, p. 179. Inglis, D. R., 1961, Rev. Mod. Phys. 33, 1. hidden variables to control individual events. On the Jammer, M., 1966, The Conceptual Development of Quantum other hand, Bell’s theorem seems to present a severe Mechanics (McGraw-Hill, New York). obstacle for any hidden-variable theory which repro­ Jauch, J. M., 1964, Helv. Phys. Acta 37, 293. ------, 1968, Foundations of Quantum Mechanics (Addison-Wesley, duces exactly all of the predictions of quantum theory, Reading, Mass.). but this need not be so for a theory which departs from ------, and C. Piron, 1968, Rev. Mod. Phys. 40, 228. the formalism of quantum theory and recovers it only Jordan, T. F., 1969, Linear Operators for Quantum Mechanics (Wiley, New York). in some limiting case. Presumably the next step must Judge, D., and J. T. Lewis, 1963, Phys. Letters 5, 190. be a bold departure from the familiar formalism, as Komar, A., 1962, Phys. Rev. 126, 365. Korner, S., Ed., 1957, Observation and Interpretation (Butter­ Einstein’s theory of gravitation departed from that worths, London). of Newton. Lamb, W. E., 1969, Phys. Today 22, No. 4, 23. Lande, A., 1955, Foundations of Quantum Theory (Yale U. P., REFERENCES New Haven, Conn.). ------, 1965, New Foundations of Quantum Theory (Cambridge Albertson, J., 1961, Am. J. Phys. 29, 478. U. P., Cambridge, England). ------, 1962, Brit. J. Phil. Sci. 13, 229. Margenau, H., 1958, Phil. Sci. 25, 23. Allcock, G. R., 1969, Ann. Phys. (N.Y.) 53, 253. ------, 1963, Ann. Phys. (N.Y.) 23, 469. Araki, H., and M. M. Yanase, 1960, Phys. Rev. 120, 622. ------, and L. Cohen, 1967, in Quantum Theory and Reality, edited Bell, J. S., 1964, Physics (N.Y.) 1, 195. by M. Bunge (Springer, New York), p. 71. ------., 1966, Rev. Mod. Phys. 38, 447 ------, and R. N. Hill, 1961, Progr. Theoret. Phys. (Kyoto) 26, Blokhintsev, D. I., 1968, The Philosophy of Quantum Mechanics, 722. (Reidal, Dordrecht, Holland). Original Russian edition, 1965. Matthews, P. T., 1968, Introduction to Quantum Mechanics Bohm, D., 1951, Quantum Theory (Prentice-Hall, Englewood (McGraw-Hill, London). Cliffs, N.J.). Messiah, A., 1964, Quantum Mechanics (North-Holland, Amster­ ------, 1952, Phys. Rev. 85, 166, 180. dam). ------, 1953, Phys. Rev. 89, 458. Misra, B., 1967, Nuovo Cimento 47A, 841. ------, 1962, in Quantum Theory, edited by D. R. Bates (Academic, Mott, N. F., 1964, Contemporary Phys. 5, 401. New York) Vol. 3. Moyal, J. E., 1949, Proc. Cambridge Phil. Soc. 45, 99. ------, and Y. Aharonov, 1957, Phys. Rev. 108, 1070. Papaliolios, C., 1967, Phys. Rev. Letters 18, 622. ------, and J. Bub, 1966, Rev. Mod. Phys. 38, 453. Park, J. L., 1968, Am. J. Phys. 36, 211. Bohr, N., 1935, Phys. Rev. 48, 696. Peierls, R. E., 1967, Sci. Am. 216, No. 1, 137. ------, 1949, in Albert Einstein: Philosopher-Scientist, edited by Popper, K. R., 1959, The Logic of Scientific Discovery (Basic P. A. Schilpp (Library of Living Philosphers, Evanston, 111.), Books, New York). reprinted by Harper and Row, New York, p. 199. ------, 1967, in Quantum Theory and Reality, edited by M. Bunge Bopp, F., 1956, Ann. Inst. Henri Poincard, 15, 81. (Springer, New York), p. 7. ------, 1957, in Observation and Interpretation, edited by S. Korner Prugovecki, E., 1967, Can J. Phys. 45, 2173. (Butterworths, London), p. 189. Robertson, H. P., 1929, Phys. Rev. 34, 163. Born, M., 1926, Z. Physik 37, 863. Rosenfeld, L., 1965, Progr. Theoret. Phys. (Kyoto) Suppl., extra Brillouin, L.. 1962, Science and Information Theory (Academic, number, p. 222 (Yukawa commemorative issue). New York). ------, 1968, Nucl. Phys. A108, 241. Breitenberger, E., 1965, Nuovo Cimento 38, 356. Schilpp, P. A., 1949, Ed. Albert Einstein: Philosopher-Scientist Bunge, M., 1956, Am. J. Phys. 24, 272. (Library of the Living Philosophers, Evanston, 111.), reprinted Carruthers, P., and M. M. Nieto, 1968, Rev. Mod. Phys. 40, 411. by Harper and Row, New York, 1959. Cohen, L., 1966, J. Math. Phys. 7, 781. Schrodinger, E., 1930, Preuss. Akad. Wiss. Berlin, Ber. 19, 296. Daneri, A., A. Loinger, and G. M. Prosperi, 1962, Nucl. Phys. -——, 1935, Naturwiss, 23, 807. 33, 297. Sharp, D. H., 1961, Phil. Sci. 28, 225. ------, A. Loinger, and G. M. Prosperi, 1966, Nuovo Cimento Shewell, J. R. 1959, Am. J. Phys. 27, 16. 44B, 119. Shimony, A., 1966, Phys. Today 19, No. 9, 85. Day, T. B., 1961, Phys. Rev. 121, 1204. Susskind, L., and J. Glogower, 1964, Physics (N.Y.) 1, 49. Dirac, P. A. M., 1958, The Principles of Quantum Mechanics Sussman, G., 1957 in Observation and Interpretation, edited by (Clarendon, Oxford), 4th edition. S. Korner (Butterworths, London),p. 131. L. E. Ballentine Statistical Interpretation of Quantum Mechanics 381

Tisza, L., 1963, Rev. Mod. Phys. 3$, 1S1. Wick, G. C., A. S. Wightman, and E. P. Wigner, 1952, Phys. Van der Waerden, B. L., 1967, Sources of Quantum Mechanics Rev. 88, 101- (North-Holland, Amsterdam). Wigner, E. P., 1932, Phys. Rev. 40, 749, Von Neumann, J., 1955, Mathematical Foundations of Quantum ------, 1962, in The Scientist Speculates, edited by I. J. Good Mechanics (Princeton U. P., Princeton, N.J.). This is a trans­ (Basic Books, New York), p. 284. lation of Mathematische Grundlagen der Quantenimechanik ------, 1963, Am. J. Phys. 31, 6. (Springer, Berlin, 1932). ------, 1970, Am. J. Phys. 38, 1005. Wakita, H., 1960, Progr. Theoret. Phys. (Kyoto) 23, 32. Witmer, E. E., 1967, Am. J. Phys. 35, 40. ------, 1962, Progr. Theoret. Phys. (Kyoto) 27, 139, 1156. Wu, C. S., and I. Shaknov, 1950, Phys. Rev. 77, 136. Whitten, R. C., 1966, Am. J. Phys. 34, 1203. Yanase, M. M., 1961, Phys. Rev. 123, 666.

REVIEWS OF MODERN PHYSICS VOLUME 42, NUMBER 4 OCTOBER 1970 Charges and Generators of Symmetry Transformations in Quantum Field Theory*

CLAUDIO A. ORZALESIf Department of Physics, Columbia University, New York, New York

Within the Wightman approach to quantum field theory, we review and clarify the properties of formal charges, defined as space integrals for the fourth component of a local current. The conditions for a formal charge to determine an operator (generator) are discussed, in connection with the well-known theorems of Goldstone and of Coleman. The symmetry transformations generated by this operator—given its existence—are also studied in some detail. For generators in a scattering theory, we prove their additivity and thus provide a simple way to characterize them from their matrix elements between one-particle states. This characterization allows an immediate construction of the unitary operators implementing the symmetry transformations, and implies that all internal symmetry groups are necessarily compact. We also indicate how to construct interacting fields having definite internal quantum numbers. The present status of the proof of Noether’s theorem and of its converse is discussed in the light the rather delicate properties of formal charges.

CONTENTS of the zeroth component of a local four-vector current: 1. Introduction...... 381 Q(x<>) = S dxjo(x). (1.1) 2. Preliminaries...... 382 A. Some Preliminary Remarks on the Quantal Noether Quantities of this kind appear in the discussion of Theorem...... 382 B. Some Basic Definitions and Theorems...... 383 symmetries and broken symmetries in quantum field 3. Properties of Formal Charges...... 385 theory, and are one of the basic tools in the modern A. Generalities...... 385 B. The Main Theorems...... 385 “ current-algebraic” approach to elementary particle C. What It All M eans...... 387 physics (Gell-Mann, 1962; Adler and Dashen, 1968). D. The Proofs...... 388 It has been repeatedly emphasized (Kastler, E. Additional Remarks...... 392 4. Charges as Generators of Symmetry Transformations. ... 393 Robinson, and Swieca, 1966; Schroer and Stichel, 1966; A. Commutators Involving Charges and Definition of Dell’Antonio, 1967; Swieca, 1966; Katz, 1966; Fabri and Generators...... 394 Picasso, 1966; Fabri, Picasso, and Strocchi, 1967; and B. Symmetries and Spontaneously Broken Symmetries. . . 395 C. Symmetry Transformations Generated by a Charge. .. 397 De Mottoni, 1967) that equations of the type (1.1) 5. Characterization of Charges and Symmetries in Scatter­ have rather delicate convergence properties, and that a ing Theory...... 398 A. Characterization of Charges...... 399 certain care has to be exercised when considering such B. Generalizations and Applications...... 402 expressions. This fact limits the extent to which Q can C. Characterization of Internal Symmetries...... 403 be thought of as a generator of symmetry or broken D. Interpolating Helds With Definite Internal Quantum Numbers...... 404 symmetry transformations.1 The same convergence 6. Summary and Conclusions...... 405 properties are at the basis of Goldstone’s theorem Appendix...... 406 References...... 408 (Goldstone, 1961; Goldstone, Salam and Weinberg, 1962; Kastler, Robinson and Swieca, 1966), and of 1. INTRODUCTION Coleman’s theorems (Coleman, 1965 and 1966; Following the results of Goldstone et al. (Goldstone, Pohlmeyer, 1966; Schroer and Stichel, 1966; and 1961; Goldstone, Salam, and Weinberg, 1962) and of 1 The nomenclature as well as the mentioned restrictions will Coleman (1966), in recent years there has been a be clarified later on. For present purposes, a generator of sym­ metry transformations is to be identified with a self-adjoint continual interest in the properties of formal charges. operator which commutes with P, the momentum operator, and A formal charge Q is defined here as the space integral commutes with the 5 matrix. A conserved current leads to an exact symmetry if the associated charge is a generator of symmetry * Research supported in part by the U.S. Atomic Energy Com­ transformations. Spontaneously broken symmetries occur when mission. current conservation does not imply the existence of a symmetry. t Present address: Department of Physics, New York Uni­ Intrinsically broken symmetries arise when the current is not versity, New York, N.Y. 10012. conserved.