statistical interpret a it on of qm

8/2/2019 Statistical Interpret a It On of QM

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REVIEWS OF MODERN PHYSIC S VOLUME 42, NUMHER 4 OCTOH El~ 1970

'..".~e Statistica. . '. .nterpretation of Quantum'V!'.ec.~anics

L. E. BALLKNTINEDepartment of Physics, Simon Fraser Unieer'sity, Bnrnaby, J3.C., Canada

The Statistical Interpretation of quantum theory is formulated for the purpose of providing a sound interpretationusing a minimum of assumptions. Several arguments are advanced in favor of considering the quantum state descriptionto apply only to an ensemble of similarily prepared systems, rather than supposing, as is often done, that it exhaustivelyrepresents an individual physical system. Most of the problems associated with the quantum theory of measurementare artifacts of the attempt to maintain the latter interpretation. The introduction of hidden variables to determine theoutcome of individual events is fully compatible with the statistical predictions of quantum theory. However, a theoremdue 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 spaciallyseparated, systems.

CONTENTS pretive 3,ttempts is promised for a later volume.~ ~ ~ LBut] even the cautious word 'attempts' may betoo positive a description for what are only pro-grams ~ ~ ~ that are not yet clear even in outline."

. . 358. . . 358. . . 359. . . 360. . . 360. . . 362. . . 362. . . 364. . . 364. . . ,364. . . 365. . . 367. . . 367. . . 368. . ~ 368

370. . . 371. . . 372. . . 374. . . 374. . . 376

. . . 377. . . 378. . . 378

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1. Introduction. . . . . . , . . . , . . . . . ~. . . . . . . . . . . . . , . . . . . .1.0 Preface and Outline. . . . . . . . . . . . . . . . . . . . . . . . . . .1.1 Mathematical Formalism of Quantum Theory. . . .1.2 Correspondence Rules. . . . . . . . . . . . . . . . . . . . . . . . .1.3 Statistical Interpretation. . . . . . . . . . . . . . . . . . . . . .

2. The Theorem ofEinstein, Podolsky, and Rosen. . . . .2.1 A Thought Experiment and the Theorem. . . . . . . .2.2 Discussion of the EPR Theorem. . . . . . . . . . . . . . . .

3. The Uncertainty Principle. . . . . . . . . . . . . . . . . . . . . . . . .3.1 Derivation. . . . . . . , . . . . . . . . . . . . . . . . , . . . . . . . . . .3.2 Relation to Experiments. . . . . . . . . . . . . . . . . . . . . . .3.3 Angular and Energyime Relations. . . . . . . . . . . .

4. The Theory of Measurement. . . . . . . . . . . . . . . ~. . . . . .4.1 Analysis of the Measurement Process. . . . . . . . . . .4.2 The Difficulties of the "Orthodox" Interpretation4.3 Other Approaches to the Problem ofMeasurement4.4 Conclusionheory ofMeasurement. . . . . . . . . . .

5. Joint Probability Distributions. . .~. . . . . . . . . . . . . . . . .6. Hidden Variables6.1 Von Neumann's Theorem. . . . . . . . . . . . . . . . . . . . . .6.2 Bell's Rebuttal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.3 Bell's

Theorem. . . . . . . . . . . . . . . . ... .

.. .

. .. . . . . .6.4 Suggested Experiments. . . . . . . . . . . . . . . . . . . . . . . .

7. Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . .References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

INTRODUCTION

1.0 Preface and OutlineThis article is not a historical review of how the

quantum theory and its statistical interpretation cameto be. That task has been admirably carried out byMax Jammer (1966) in his book The ConceptualDetetopntertt of Quarttunz Mechartics, and also by vander Waerden (1967). Our point of departure canconveniently be introduced by considering the follow-

ing statement made by Peierls (1967) in a review ofJammer's book:"Chapter 7 ~ ~ ~ is headed 'The Copenhagen Inter-

pretation. ' ~ ~ ~ the phrase suggests that this is only oneof several conceivable interpretations of the sametheory, whereas most physicists are today convincedthat the uncertainty relations and the ideas of comple-mentarity are essential parts of the structure of quan-tum mechanics ~ ~ . A discussion of alternative inter-

If, as appears to be the case, the latter remarks byPeierls refer to the models known as hidden-variabletheories (see Sec. 6), we agree that these should betreated as new theories, and that they are not newinterpretations of quantum mechanics "any more thanquantum mechanics is a new interpretation of classicalphysics."However we shall show, contrary to the viewexpressed by Peierls, that the Copenhagen interpreta-tion contains assumptions which are not "essentialparts of the structure of quantum mechanics, "and thatone such assumption is at the root of most of thecontroversy surrounding "the interpretation of quantummechanics. "

Itis the assumption that the quantum

state description is the most, complete possible descrip-tion of an individual physical system.An interpretation which ismore nearly minimal in the

sense of including all verifiable predictions of quantumtheory, but without the contestable features of theCopenhagen interpretation, we shall call the StutisticalInterpretation The distinc. tion between these inter-pretations (which share many features in common)will be made in the following sections. SuKce it to say,for now, that if we identify the Copenhagen Inter-pretation with the opinions of Bohr, then the StatisticalInterpretation is rather like those ofEinstein. Contraryto what seems to be a widespread misunderstanding,Einstein's interpretation corresponds very closely withthe one which is almost universally used by physicistsin practice; the additional assumptions of the Copen-hagen interpretation playing no real role in the applica-tions of quantum theory.The outline of this paper is as follows. I'irst we give a

brief summary of the mathematical formalism ofquantum theory in order to distinguish the formalism,which we accept, from the physical interpretation,which we shall examine critically. %'e then consider

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L. E. BALLENrrNE Statistical Interpretation ofQuantunt Mechanics 359

di8erent interpretations, and expound the StatisticalInterpretation in detail.The Secs. 2, 3, and 4 could be prefaced by Appendix

*xi of Popper (1959) on the proper use of imaginaryexperiments, in which he points out that Gedankenexperiments can be used to criticize a theory but not tojustify or prove a theory. Our discussions of the Gedankenexperiment of Einstein, Podolsky, and Rosen, of theuncertainty principle, and of the measurement processare undertaken to criticize the assumption that a statevector provides a complete description of an individualsystem. Although arguments of this type can refute thehypothesis being criticized, they cannot, of course,"prove" the Statistical Interpretation but can onlyillustrate its advantages.Sections 5 and 6 deal with two concepts (joint

probability distributions for position and momentum,and hidden variables) which have often been thoughtto be incompatible with quantum theory. That belief,however, was based in an essential way upon the abovehypothesis which we criticize and reject. It turns outthat, within certain limits, the formalism of quantumtheory can be extended (Not modified) to include theseconcepts within the Statistical Interpretation.Finally we summarize our conclusions.

1.1 Mathematical Formalism ofQuantum TheoryQuantum theory, and indeed any theory, can be

divided Lsee Prugovecki (1967), or Tisza (1963)$into:

(a) A mathematical formalism consisting of a set ofprimitive concepts, relations between these concepts(either postula, ted or obtainable

bygiven rules of

deduction), and a dynamical law.(b) Correspondence rules which relate the theoretical

concepts of (a) to the world of experience.This division is not absolutelearly one must have

a formalism in order to make correspondence rules, butunless one has at least some partial idea of corre-spondence rules, one would not know what one wastalking about while constructing the formalismnevertheless it is convenient for the present task.The mathematical formalism of quantum theory is

well known and can be abstracted from any of severaltextbooks (Dirac, 1958;Messiah, 1964).The primitiveconcepts are those of state and of observable.

F1 An observable is represented by a self-adjointoperator on a Hilbert space. It has a spectral repre-sentation,

R=QrF', (1.1)where the Pare orthogonal proj ection operatorsrelated to the orthonormal eigenvectors of E by

withP&p&f (1.3b)

Zp-=1 (1.3c)

This state operator formalism is reviewed by Pano(1957).F3 A pure state can be defined by the condition p'= p.It follows that for a pure state there is exactly onenonzero eigenvalue of p, say,

p&1=0 for sQ's .In this case we have .= I ~-&(~- I, (1.5)and so a pure state may be represented by a vector inthe Hilbert space. A general state which is not pure iscommonly called a mixed state.

F4 The average value of an observable E in the statep is given by(R)=Tr (pR), (1.6)

where Trmeans the trace of the operator in parentheses.For a pure state represented by the normalized vector~ P&, (1.6) reduces to (R)=(P ~ R ~ f&.By introducingthe characteristic function (e'&'~&, we can obtain theentire statistical distribution of the observable E in thestate p. It follows that:FS The only values which an observable may take onare its eigenvalues, and the probabilities of each of theeigenvalues can be calculated. In the case of a purestate represented by the normalized vector ~ P&, the

probabilityof eigenvalue

rof Ris

g~

Q~

tt,

r.)~'.

This is a generalization of Born's (1926) famouspostulate that the square of a wave function representsa probability density.Sofar we have only given necessary, but not sufhcient,

conditions for the mathematical representations ofobservables andof states. To complete the specification,the following is usually postulated:

Here the numbers r are the eigenvalues of E., and theparameter a labels the degenerate eigenvectors whichbelong to the same eigenvalue of E.The sums becomeintegrals in the case of continuous spectra. Equation(1.1) is equivalent to the statement that an observablemust possess a complete orthogonal set of eigenvectors.F2 A state is represented by a state operator (alsocalled a statistical operator or density matrix) whichmust be self-adjoint, nonnegative definite, and of unittrace. This implies that any state operator may bediagonalized in terms of its eigenvalues and eigenvectors,

P=ZP I 0-&(4.I,

I'=P ~ a, r&(a,r~. F6 The Hilbert space is a direct 'sum of coherent1 23


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360 RKvlE&s OP MoDERN PHYsIcs ~ OcTQBER 1970

subspaces, within each of which (almost) every vectormay represent a pure state. This is a formal statementof the superposition princip/e with allowance beingmade for superselection rules. ' The set of all mixedstates can be constructed from the set of all pure statesusing (1.3).F7 Any self-adjoint operator which commutes withthe generators of superselection rules, or equivalently,all of whose eigenvectors lie within coherent subspacesof F6, represents an observable. This postulate may becriticized on the grounds that it is dificult to imagine aprocedure for observing a quantity like (x'p,'+p,'x'),which should be an observable according to thispostulate. The general form of F7 is unnecessaryin most, if not all practical applications, but it is usedin Von Neumann's theorem (Sec.6.1).FS The dynamical law or equation of motion dependsin detail upon the physical system under consideration

(i.e., number of degrees of freedom, whether relativisticor nonrelativistic), but in every case it can be writtenin the form,

to arise in practice.We can, for example, unambiguouslydetermine the Hamiltonian operator for any 6nitenumber of particles interacting through velocity-independent potentials in the presence of an arbitraryexternal electromagnetic 6eld.The diferent interpretations of quantum theory are

most sharply distinguished by their interpretations ofthe concept of state. Although there are many shades ofinterpretation (Bunge, 1956),we wish to distinguishonly two:

(I) The Statistical Interpretation, according to whicha pure state (and. hence also a general state) providesa description of certain statistical properties of anensemble of similarily prepared systems, but need notprovide a complete description of an individual system.

This interpretation is upheld, for example, by Einstein(1949),by Popper (1967),and by Blokhintsev (1968).Throughout this paper the capitalized name "StatisticalInterpretation" refers to this specific interpretation(described in detail in Sec.1.3).

in general, orp(t) = Up(to) U '

I4(~)&=UI4(~0)&

(1.7a)

(1.7b)

(II) Interpretations which assert that a pure stateprovides a complete aud exhaustive descriptiou of auindkviduat system (e.g., an electron).

1.2 Correspondence RulesThe correspondence rules must relate the primitive

concepts of state and observable to empirical reality.Xn so doing they will provide a more specific inter-pretation for the averages and probabilities introducedin F4 and F5.The natural requirement placed upon an observable

is that we should be able to observe it. More precisely,an observable is a dynamical variable whose value can,in principle, be measured. For canonically conjugateva,riables the corresponding operators are obtainedthrough Dirac's canonical commutation relation,

qP Pq=fii. (1.8)There is no general rule for constructing a uniqueoperator to represent an arbitrary function f(q, p)because of the noncommutability of q and p (Shewell,1959). Fortunately the most general case does not seem' A superselection rule (Wick et ul. , 1952;Galindo et a/. , 1962)

is a restriction on the superposition principle. For example, avector which is a linear combination of integer and half-integerangular momentum' eigenvectors cannot represent a physicalstate. A coherent space is one in which the superposition principlehas unrestricted validity.

for a pure state, where U=U(t, to) is a unitary operator.The above is not intended to be an axiomatization of

quantum theory, but merely a compact summary ofthe mathematical formalism of the theory as it exists atpresent and in practice. Except for the reservationnoted in F7 it should be noncontroversial. Such is notthe case with the correspondence rules.

This class contains a great variety of members, fromSchrodinger's original attempt to identify the electronwith a wave packet solution of his equation to theseveral versions of the Copenhagen Interpreta, tion.'Indeed many physicists implicitly make assumption IIwithout apparently being aware that it is an additionalassumption with peculiar consequences. It is a majoraim of this

paperto point out that the hypothesis

IIis

unnecessary for quantum theory, and moreover that itleads to serious di6iculties.

1.3 The Statistical InterpretationThe term, Statistical Interpretation, will be used

throughout this paper in the specific sense here ex-pounded, and should not be confused with the lessspeci6c usage of this term by other writers (e.g.,Messiah, 1964, Chap. 1V) who do not distinguish itfrom the Copenhagen interpretation.Of primary importance is the assertion that a quantum

' Bohr (1935,1949) argued against Einstein, who rejected II.Heisenberg's position is somewhat unclear. One reference (1930,p. 33) contains a statement fully in accord with the StatisticalInterpretation; however, in a later writing specifically in defenseof the Copenhagen interpretation (1955, p. 26) he states "anindividual atomic system can be represented by a wave func-tion. . .".His interpretation of probability (1958,Chap. 3) wasbased on the Aristotelian notion of "potentia", which is quitedifferent from the Statistical Interpretation. Messiah (1964) isquite explicit (p. 152, 158) in favoring II over I.Although bothclaim orthodoxy, there now seems to be a difference of opinionbetween what may be called the Copenhagen school representedby Rosenfeld, and the Princeton school represented by Wigner(see Rosenfeld, 1968and references therein). But since both fac-tions appear to accept hypothesis II, our criticism will apply toboth.


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L. E. BALLzNriNz Statistecal IaterPretateoa of Qaaltaas 3Eechalecs 361

state (pure or otherwise) represents an ertsemble ofsimilarly prepared systems. For example, the systemmay be a single electron. Then the ensemble will be theconceptual (infinite) set of all single electrons which

have been subjected to some state preparation tech-nique (to be specified for each state), generally byinteraction with a suitable apparatus. Thus a rnomen-tum eigenstate (plane wave in configuration space)represents the ensemble whose members are singleelectrons each having the same momentum, but dis-tributed uniformly over all positions. A more realisticexample which occurs in scattering problems is a finitewave train with an approximately well-defined wave-length. It represents the ensemble of single electronswhich result from the following schematically describedprocedur- acceleration in a machine, the output fromwhich can take place in only some finite time interval(due to a "chopper"), and collimation which rejectsany particle whose momentum is outside certain limits,We see that a quantum state is a mathematical repre-sentation of the result of a certain state preparationprocedure. Physical systems which have been subjectedto the same state prepa,ration will be simila, r in some oftheir properties, but not in all of them (similar inmomentum but not position in the first example).Indeed the physical implication of the uncertaintyprinciple (discussed in detail in Sec.3) is that no statepreparation procedure is possible which would yield anensemble of systems identical in all of their observableproperties. Thus it is most natural to assert that aquantum state represents an ensemble of similarilyprepa, red systems, but does not provide a completedescription of an individual system.When the physical system is a single particle, as in

the above examples, one must not confuse the ensemble,which is a conceptual set of replicas of one particle inits experimental surroundings, with a beam of particles,which is another kind of (many-particle) system. Abeam may simulate an ensemble of single-particlesystems if the intensity of the beam is so low that onlyone particle is present at a time.The ensembles contemplated here are different in

principle from those used in statistical thermodynamics,where we employ a, representative ensemble for cal-culations, but the result of a calculation may be com-pared with a measurement on a single system. Alsothere is some arbitrariness in the choice of a repre-

sentative ensemble (microca,nonical, canonical, grandcanonical). But, in general, quantum theory predictsnothing which is relevant to a single measurement(excluding strict conservation laws like those of charge,energy, or momentum), and the result of a calculationpertains directly to an ensemble of similar measure-ments. For example, a single scattering experimentconsists in shooting a single particle at a target andmeasuring its angle of scatter. Quantum theory doesnot deal with such an experiment, but rather with thestatistical clistribution (the differential cross section)

of the results of an ensemble of similar experiments.Because this ensemble is not merely a representative orcalculational device, but rather it can and must berealized experimentally, it does not inject into quantum

theory the same conceptual problems posed in statisticalthermodynamics.In general quantum theory will not predict the result

of a measurement of some observable E. But theprobability of each possible result r, ca,lculated ac-cording to F5, may be verified by repeating the statepreparation and the measurement many times, andthen constructing the statistical distribution of theresults. As pointed out by Popper (see Korner, 1957,p. 65, p. 88), one should distinguish between theprobability, which is the relative frequency (ormeasure)of the various eigenva, lues of the observable in theconceptual infinite ensemble of all possible outcomes ofidentical experiments (the sample space), and thestatistical fretIuency of results in an actual sequence ofexperiments. The probabilities are properties of thestate preparation method a,nd are logically independentof the subsequent measurement, although the statisticalfrequencies of a long sequence of similar measurements(each preceded by state preparation) may be expectedto approximate the probability distribution. If hy-pothesis I is adopted, we may say simpIy that theprobabilities are properties of the state.The various interpretations of a quantum state are

related to differences in the interpretation of proba,bility (see Popper, 1967 for a good exposition of thispoint). In contrast to the Statistical Interpretation,some matherna, ticians and physicists regard probabilityas a measure of knowledge, and assert that the use ofprobability is necessitated only by the incompletenessof one's knowledge. This interpretation can legitimatelybe applied to games like bridge or poker, where one' sbest strategy will indeed be inQuenced by any knowledge(accidentally or illegally obtained) about an opponent'scards. But physics is not such a game, and as Popperhas emphasized, one cannot logically deduce new andverifiable knowledg- statistical knowledg- literallyfrom a lack of knowledge.Heisenberg (1958), Chap. 3, combined this "sub-

jective" interpretation with the Aristotelian notion of"potentia."He considers a particle to be "potentiallypresent" over all regions for which the wave functionP(r) is nonzero, in some "intermediate kind of reality, "until an act of observation induces a "transition fromthe possible to the actual." In contrast, the StatisticalInterpretation considers a particle to always be at someposition in space, each position being realized withrelative frequency

~ P(r) ~' in an ensemble of similarilyprepared experiments. The "subjective"and Aristotelianideas are primarily responsible for the suggestion tha,tthe observer plays a peculiar and essential role inqua,ntum theory. Whether or not they can be con-sistently developed, the existence of the StatisticalInterpretation demonstrates that they are rot necessary,


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and in my opinion they bring with them no advantagesto compensate for the additional metaphysical com-plication.If the expression "waveparticle duality" is to be

used at all, it must not be interpreted literally. In theabove-mentioned scattering experiment, the scatteredportion of the wave function may be equally dis-tributed in all directions (as for an isotropic scatterer),but any one particle will not spread itself isotropically;rather it will be scattered in some particular direction.Clearly the wave function describes not a single scat-tered particle but an ensemble ofsimilarily acceleratedand scattered particles. At this point the reader maywonder whether a statistical particle theory can accountfor interference or diffraction phenomena. But there isno diS.culty. As in any scattering experiment, quantumtheory predicts the statistical frequencies of the variousangles through which a particle may be scattered. For acrystal or di6raction grating there is only a discrete setof possible scattering angles because momentumtransfer to and from a periodic object is quantized by amultiple of hp=h/d, where Ap is the component ofmomentum transfer parallel to the direction of theperiodic displacement d. This result, which is obviousfrom a solution of the problem in momentum repre-sentation, was first discovered by Duane (1923),although this early paper had been much neglecteduntil its revival by Lande (1955, 1965).There is noneed to assume that an electron spreads itself, wavelike,over a large region of space in order to explain diffrac-tion scattering. Rather it is the crystal which is spreadout, and the electron interacts with the crystal as awhole through the laws of quantum mechanics. For alonger discussion of this and related problems such asthe two-slit experiment, see Lande (1965). In everycase a diffraction pattern consists of a statisticaldistribution of discrete particle events which areseparately observable if one looks in fine enough detail.In the words of Mott (1964,p. 409), "Students shouldnot be taught to doubt that electrons, protons and thelike are particles ~ ~ ~ The wave cannot be observed inany way than by observing particles."Although we sh'dl discuss his work in greater detail

below, we should emphasize here the great contributionof Einstein to this subject. His "Reply to Criticisms"(Einstein, 1949),expressed very clearly his reasons foraccepting a purely statistical (ensemble) interpretationof

quantum theory, and rejectingthe

assumption thatthe state vector provided an exhaustive description ofthe individual physical system.

' I note in passing that I.ande's ambitious program to deriveall of quantum theory from a few simple principles has not yetbeen completely successful. Reviewers of his book (Shimony,1966;Mitten, 1966) have pointed out additional assumptionsimplicit in his argument which are virtually equivalent to assum-ing some of the results he wishes to derive. However, these criti-cisms do not detract froni his discussion of diffraction scattering.

2. THE THEOREM OF EINSTEIN, PODOLSKY,AND ROSEN

The tenability of hypothesis II, Sec. 1.2, was chal-lenged in a paper by Einstein, Podolsky, and Rosen

(1935) (abbreviated EPR) entitled "Can Quantum-Mechanical Description of Reality Be ConsideredComplete?" Their argument is often referred to as theParadox of EPR, as though it ought to be capable ofresolution as, say, Zeno's paradox. LRosenfeld (1968)has scurrilously referred to it as the EPR fallacy.]Weshall show however that, properly interpreted, it is awell dined theorem, paradoxical only to the extentthat the reader may not have expected the conclusion.In order to precisely answer the question posed in

their title, EPR introduce the following definitions:D1 A necessary condition for a complete theory is that"every element of physical reality must have a counterpartin the physical theory."D2 A sufficient condition for identifying an. element ofreality is, "If, without in any way disturbing a system,we can predict with certainty (i.e., with probability equalto unity)' the value of a physical quantity, then thereexists an element of physical reality corresponding to thisphysical quantity. "Their argument then proceeds by showing, through

consideration of a thought experiment, that two non-commuting observables should, under suitable condi-tions, both be considered elements of reality. Since nostate vector can provide the value of both of theseobservables, they conclude that the quantum statevector cannot completely describe an individualsystem, but only an ensemble of similarily preparedsystems.

2.1A Thought Exyeriment and the TheoremThe experiment described below was introduced by

Bohm (1951,p. 614ff).By considering measurements ofspins rather than of particle positions and momenta(as in the original EPR experiment), we avoid anyunnecessary complications with the positionmomentumuncertainty principle which may arise from the mini-mum degree of particle trajectory definition needed toperform an experiment.Two particles of spin one-half are prepared in an

unstable initial state of total spin zero. The pairseparates, conserving total spin, and one of the par-ticles (which are chosen to be distinguishable, forconvenience) passes through the inhomogeneous inag-netic field of a Stern-Gerlach apparatus (see Fig. 1).

4 As a nearly pedantic refinement, we would prefer to replacethis phrase by with probubility 1, cohere e muy be mude urbi-trurity smul/. This allows for arbitrarily accurate approximationsto problems which may not be formally solvable, and it allowsone to Ivoid certain igrejevant critIcisms. (See Footnote 5).


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L. E. BALLRNTrNz Statzstzea/ Interpretation ofQuantum 3fechanzees

=0, for t(0 or t)r, (2.2)where r is the transit time of the particle through themagnetic field which is of gradient II' and is directed

alongthe s axis.

Omitting the position-dependent factors, we canwrite the initial state vector, a spin singlet, as

f(0)= Iu+(1)u (2) (1)u+(2)}/K2. (2.3)Here I+ and I are the two-component spinors corre-sponding to the eigenvalues ~1 of o.and the argu-ments I or 2 refer to the two particles. Since

t

r(t)= ~xp ( e(&') r(o), ('24)

the eGect of the interparticle interaction V~2 will benegligible if

(2 5)

a condition which can be realized by making the transittime r short enough. s By { V ~ we mean the modulusof the relevant matrix elements of Vi~ in (2.4) .

The unceitainty princip].e is trivially satisfied sincewerequire only that p,=p,=0 approximately, and that.dy be small enough so we know which particle hasentered themagnetic field.The translational motion of the particles can be

treated classically, and the spin can be described by aspin Hamiltonian in which the magnetic field at theposition of the particle is treated as a function of time,

H= Viz+ Vrr (2.1)V12 V(ras, (ri, trz) is the interaction between the

two particles, and V~ is the interaction with themagnetic field.

V&(t)=sH'~, for 0&t(r,

At time t=7-, after the particle has interacted withthe magnetic field, the state vector will be

P(r) = {exp (H're/5)u+(1)u (2)

exp(iH'rzi/A,

)zt

(1)u+(2) }/W2.

(2.6)The result of the interaction is to produce a correlationbetween s components of momentum and spin ofparticle 1, and spin of particle 2. If pi,= H'r, theno.i,+1,and (rg,1;or if pi,=H'r then (ri, and(rz,+1. It is only necessary to make the magneticfield gradient H' large enough so that the two values ofpi, are unambiguously separated.Ke have thus shown that the s component of the

spin of particle 2 can be determined to an arbitrarilyhigh degree of accuracy by a measurement which,because of the spacial separation and negligible effectof V, does not in any way disturb particle 2. Thusaccording to definition D2, o.~, is an elenzerzt of reaHty.However the initial singlet state is invariant underrotation, and can equally be expressed as

4 (0)= {~+(1)~-(2)~-(1)~+(2)}/v2 (2 &)where the spinors v+ and v are eigenvectors of 0- . Ifthe SternGerlach magnet were rotated so that thefield was directed along the x axis, then by an identicalargument we would conclude that 0.~ was an elementof reality. Since no state vector can provide a value forboth of the noncommuting observables o.2~ and a-2EPR conclude, in accordance with D1, that the statevector does not provide a complete description of anindividual system.One might try to avoid this conclusion by adopting

an extreme positivist philosophy, denying the realityof both cT&, and o-&, until the measurement has actuallybeen performed. But this entails the unreasonable,essentially solipsist position that the reality of particle2 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 followingtheorem':

Q2))) (l(Q&

FIG. 1.Schematic illustration of the apparatus for theEinstein, Podolsky, and Rosen experiment.

~ This overcomes the objection of Sharp (1961)that any inter-

actionn

(even gravitational) would prevent us from writing P(t)as a product of two factors; one for particle 1, and one for par-ticle 2. Such an objection is quite irrelevant because any statevector can be expressed as a linear combination of product-typebasis vectors, and the interaction only afFects the time depend-ence of the expansion coefficientswhich we treat to an arbitrarydegree of precision.

The following two statements are incompatable:

(1) The state vector provides a complete andexhaustive description of an individual system;(2) The real physical conditions of spatially separated

(noninteracting) objects are independent.Of course one is logically free to accept either one ofthese statements (or neither). Einstein clearly acceptedthe second, while Bohr apparently favored the first.The importance of the EPR argument is that it provedfor the first time that assuming the first statementabove densmds rejection of the second, and vice versa,a fact that was not at all obvious before 1935,and whichmay not be universally realized today.


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2,2 Discussion of the EPR TheoremIt should be emphasized that the result of the EPR

experiment is in no way paradoxical. The implicationof (2.6) in the Statistical Interpretation is that if theexperiment is repeated many times we should obtainthe result p~,=H'r, Oq,=+1, 02,= in aboutone-half of the cases, and the opposite result in theother half of the cases. This correlation between thespins of the two particles will be the same no matterwhich components are measured.Another essential point, which has not always been

realized, is that the EPR theorem in no way contradictsthe mathematical formalism of quantum theory. It wasonly intended to exhibit the difBculties which followfrom the assumption that the state vector completelydescribes an individual system; an interpretive assump-tion which is not an integral part of the theory. In hisfamous reply to EPR, Bohr (1935) said, "Such an

argumentation Las that of EPR], however, wouldhardly seem suited to aGect the soundness of quantum-mechanical description, which is based on a coherentmathematical formalism ~ ~ ~ ." It would appear thatBohr failed to distinguish between the mathematicalformalism of quantum theory, and the Copenhageninterpretation of that theory. Einstein's criticism wasdirected only at the latter, not at the former. Con-versely the self-consistency and empirical success ofthe former are no defense against specific crit icism ofthe latter. Such an erroneous perception of the KPRpaper as being an attack on quantum theory itself mayexplain (psychologically) the often repeated statementthat Bohr had successfully refuted their argument. Infact Bohr's paper offers noreal challenge to the validityof the EPR theorem as stated above, nor does the EPRtheorem pose any "paradox" or threat to the validity ofquantum theory.Bohr's reply to EPR is really a criticism of their

definitions of completeness and physical reality, but it isof a rather imprecise character. A satisfactory pursuitof this line of attack should first admit the validity ofthe EPR theorem with their definitions. Second itshould propose alternative definitions with arguments infavor of the superiority of the new definitions. Finallyit should show that the state vector provides a completedescription of physical reality in terms of the newdefinitions. Bohr has done none of these.In the succeeding years numerous comments on the

EPR paper have been published, many of which areless than satisfactory. A discussion of a paper by Furry(1936) is given in Footnote 13, Sec.4.3 of this paper.Breitenberger (1965) has criticized many authors forcontributing to confusion, including Bohr for suggestingthat the observation ofparticle 1 "creates" the physi-cally real state of particle 2position approachingthe absurdity of solipsism. ' Breitenberger correctly' I have not found an explicit statement to this e6ect in Bohr's

writings, although essentially this position has been taken bysome followers of Bohr.

emphasizes that there is nothing paradoxical about theEPR experiment, which is a prototype for manycoincidence experiments, but he seems to forget theoriginal purpose of the EPR argument.Several people have proposed experiments similar inprinciple to that of EPR (Day, 1961; Inglis, 1961;

Bohm and Aharonov, 1957).While these experimentsare interesting in their own right, it should be em-phasized that their results will not distinguish betweenEinstein's and Bohr's interpretations of quantumtheory. If the results of experiments were to di6erfrom the theoretical predictions, this would contradictthe formalism of quantum theory itself, not just one ofthe interpretations.

3. THE UNCER'MINTY PRINCIPLE3.1 Derivation

where AA is known as the standard deviation of thedistribution. Although states exist (at least as mathe-matical idealizations) for which the variance of thedistribution for any one observable is arbitrarily small,it can easily be shown, as was first done by Robertson(1929) and Schrodinger (1930), that the product ofthe variances of the distributions of two observablesA and 8 has a lower bound,

where ABBA=iC.The first term may vanish, butfor canonically conjugate observables )see Eq. (1.8)]we must always have

hqAp& h/2. (3.3)The meaning of these results is unambiguous. The

averages of quantum theory (postulate F4) are realizedby performing the same measurement on many similarlyprepared systems (or equivalently by performing themeasurement many times on the same system whichmust be resubmitted to the same state preparationbefore each measurement). In order to measure AA

(or AB) one must measure A (or8) on many similarilyprepared systems, construct the statistical distributionof the results, and determine its standard deviation.The results (3.2) and (3.3) assert tha,t for any par-ticular state (i.e., state preparation) the product of thewidths of the distributions of A measurements and of 8measurements may not be less than some lower limit.A term such as the statistical dispersion principle wouldreally be more appropriate for these results than thetraditional name, uncertainty principle Adiscussion.similar to this is given by Margenau (1963).

To a given state there will, in general, correspond astatistical distribution of values for each observable.A suitable measure of the width of the distribution foran observable A is the variance,

(3.1)


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L. E. BALLENTiNE Statiistica/ Iuterpretatioii ofQuautuiu Mechauecs 363

3.2 Relation to ExperimentsThere exist many widespread statements of the

uncertainty principle which are diGerent from the onethat is derived from statistical quantum theory. Verycommon is the statement that orIe carlrlot measure thetwo quantities q and p simultaneously without errorswhose product is at least as large as fij2.This statementis often supported by one or both of the followingarguments:

(i) A measurement of q causes an unpredictable anduncontrollable disturbance of p, and vice versa. LThiswas first proposed by Heisenberg (1927) and is widelyrepeated in text booksj.(ii) The position and momentum of a particle do not

even exist with simultaneously and perfectly welldefined. (though perhaps unknown) values (Bohm,1951,p. 100).

Clearly the statistical dispersion principle and thecommon statement of the uncertainty principle are notequivalent or even closely related. The latter refers toerrors of simultaneous measurements of q and p on onesystem, and it is plausible that one of these measure-ments could cause an error in the other. On the otherhand, the former refers to statistical spreads in ensemblesof measurements on similarily prepared systems. Butonly one quantity (either q or p) is measured on anyone system, so there is no question of one measurementinterfering with the other. Furthermore the standarddeviations hq and tIp of the statistical distributionscannot be determined unless the errors of the individualmeasurements, 8q and 8p are much smaller than the

standard deviations (seeFig. 2) .These points have beenemphasized by Margenau (1963) and by Popper(1967).Prugovecki (1967) has pointed out that, farfrom restricting simultaneous measurements of non-commuting observables, quantum theory does not dealwith them at all; its formalism being capable only ofstatistically predicting the results of measurements ofone observable (or a commutative set of observables).In order to deal with simultaneous measurements heproposes an extension of the mathematical formalism,to which we shall return in Sec. 5 of this paper.We now consider to what extent the common state-

ment of the uncertainty principle may be true, even

=i,h,q+ C/

L

q I Q 5pFrG. 2. Illustration of the uncertainty principle. The two histo-

grams represent the frequency distributions of independent meas-urements of q and p on similarly prepared systems. There are, inprinciple, no restrictions on the precisions of individual measure-ments Bq and Bp, but the standard deviations will always satisfydgnp) ti/2.

Ji, 5y

X) Xg

FIG. 3. An experimental arrangement designed to simultane-.ously measure y and p, such that the error product Bybpcanbe arbitrarily small.

though it is not derivable from quantum theory.Argument (ii) is easily seen to be unjustified. It isbased on the obvious fact that a wave function with awell de6ned wavelength must have a large spacialextension, and conversely a wave function which islocalized in a small region of space must be a Fouriersynthesis of components with a wide range of wave-lengths. Using de Broglie's relation between momentumand wavelength, p=h/X, it is then asserted that aparticle cannot have definite values of both positionand momentum at any instant. But this conclusionrests on the almost literal identification of the particlewith the wave packet (or what amounts to the samething, the assumption that the wave function providesan exhaustive description of the properties of the

particle) .The untenable nature of such an identificationis shown by the example of a particle incident upon asemitransparent mirror with detectors on either side.The particle will either be re.ected or transmittedwithout loss of energy, whereas the wave packet isdivided, half its amplitude being transmitted and halfreQected. A consistent application of the StatisticalInterpretation yields the correct conclusion that thedivision of the wavepacket yields the relative proba-bilities for transmission and reRection of particles. Butthere is no justification for assertion (ii) .Argument (i) must be considered more seriously

since Heisenberg (1930) and Bohr (1949) have givenseveral examples for which it is apparently true. How-

ever, Fig. 3 shows a simple experiment for which it isnot valid. A particle with known initial momentump passes through a narrow slit in a rigid massive screen.After passing through the hole, the momentum of theparticle will be changed due to diGraction eGects, butits energy will remain unchanged. When the particlestrikes one of the distant detectors, its y coordinate isthereby measured with an error by. Simultaneouslythis same event serves to measure the y component ofmomentum, p=p sin 8, with an error 8pwhich may bemade arbitrarily small by making the distance L,


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L. E. BAx.LzNrrNz Statesteca/1nterpretation ofQuantum 3feclzanecs 367

and thus it limits the precision which future predictionsfor any system can be made. But it does not imposeany restriction on the accuracy to which an event canbe reconstructed from the data of both state preparationand measurement in the time interval between thesetwo operations. Heisenberg (1930,p. 20) made nearlythis distinction in his statement, " ~ ~ the uncertaintyrelation does not refer to the past." His subsequentremark, "It is a matter of personal belief ~hether ~ - ~the past history of the electron I as inferred from anexperiment like that of Fig. 3$ can be ascribed anyphysical reality or not,"ased on the fact that itcannot be reconfirmed by any other future measure-ment, seems unduely cautious. In fact, the majority ofreal physical measurement are of just this type. Indeedfor every future oriented experimental operation whichyields verifiable information, there must be anotherpast oriented operation whose function is not to beverifiable but to verify (Popper, 1967, pp. 2528). Itis just this essential fact which is emphasized by thedistinction between state preparation and measurement

D&phL,&A/2 (wrong) . (3.7)But (3.7) is obviously wrong since AL, can be arbitrarilysmall, and y has a meaningful range of only 2m radians;hence, dy&2x.The resolution of the apparent paradox, as pointed

out by Judge and Lewis (1963) and by Susskind andGlogower (1964), lies in the observation that theoperator iS(B/Bp) is Hermitian only on the space offunctions of y which have period 2x, and multiplicationby p destroys this periodicity property. In order todeduce a correct uncertainty relation, one must replacethe coordinate y by some periodic function. This can bedone in many ways but the most convenient seems to bethe use of cos rp and sin p (Carruthers and Nieto,1968),from which it can be shown that

(DL,)'L(h cosp)'+ (6 sin q)'j& (fc/2) '( (sin y)'+ (cosj )') . (3.8)

The derivation of the energyime uncertaintyrelation cannot follow the standard form (3.2) becausetime is not usually represented by an operator. In factone can show (Susskind and Glogower, 1964) that, if theenergy spectrum has a lower bound, then there doesnot exist a Hermitian operator which is canonicallyconjugate to the Hamiltonian in the sense of (1.8).However, the following result can be deduced from

3.3 Angular and Energy-Time RelationsAn apparent contradiction of the uncertainty prin-

ciple, which is frequently rediscovered by brightstudents, concerns the polar angle q and the s componentof angular momentum I., If L, is represented byikey/Bp, thn. it would appear that I &p, L,]=i6, from

which it would follow that

(3.2) (Messiah, 1964,p. 319),aA~z& I I d(A)/di I, (3.9)

where A is an arbitrary observable, and the averages

are calculated for an arbitrary time-dependent state.If one defines a characteristic time for the system andthe state as

T= min& I~a I (~)/l I-'I, (3.10)then one may write

7-hE&-',h. (3.11)Clearly this result does not imply that one cannotmeasure the energy of a system exactly at an instant oftime, as is sometimes stated, but rather that the spreadof energies associated with any state is related to thecharacteristic rate of change associated with the same

4. THE THEORY OFMEASUREMENT

8After completion of this manuscript, a paper by Allcock (1969)containing a fuller discussion of energy-time relations appeared.

The desirability of an analysis of the measurementprocess by means of quantum theory was perhaps erstindicated through Bohr's insistence that the "wholeexperimental arrangement, " (Bohr, 1949, p. 222),object plus all apparatus, must be taken into accountin order to specify well-defined conditions for anapplication of the theory. However he never carriedthis program to its logical conclusion; the description ofboth the object and the measuring apparatus by theformalism of quantum theory. Such an analysis isuseful for several reasons.It has frequently been asserted that the acts of

measurement or observation play a different role inquantum theory than in the rest of physics. Suchclaims should be critically analyzed.Because the measurement apparatus usually

(always?) contains a final stage to which classicalmechanics is applicable, the measurement of a quantal(i.e., essentially nonclassical) object by such anapparatus involves the union ofmicro and macrophysicsin an essential way. Any proposed revisions at themicroscopic level, of the concept of physical real.ity orof the role of the observer will meet a critical test here.It is widely believed that Ehrenfest's theorem (Messiah,p. 216) proves the consistency of quantum theory withthe classical limit, but this is only partly true. Ehrenfest'stheorem demonstrates that the average values ofobservables obey the classical equations of motionprovided the quantum state function is such thatsta,tistical deviations of the basic observables (co-ordinates and momenta) from their averages arenegligible. But this leaves open the question of v hetheror not the state of a macroscopic object coupled tomicroscopic objects, as calculated from the quantalequation of motion for realistic but general initial


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REvIEws oP MoDERN PHYsIcs OGTQBER 1970

conditions, will necessarily possess this property. Aswill be shown, this appears not to be so in the caseofmeasurement.It is not the compatibility of the classical and quantal

equations of motion (in the appropriate limit) whichconcerns us but rather the compatibility of the classicaland quantal concepts of state. Thus an analysis ofmeasurement will be very helpful in deciding the ques-tion raised in Sec. 1.2, that is whether a quantum statedescribes an individual system (object plus apparatusin this case), or whether it must refer only to anensemble of systems.

4.1 Analysis of the Measurement ProcessThe essence of a measurement is an interaction

between the object to be measured and a suitableapparatus so that a correspondence is set up betweenthe initial state of the object and the Anal state of theapparatus. This interaction may or may not change thevalue of the observable being measured, the final stateof the object being of no significance to the success ofthe measurement.Suppose we wish to measure the observable R of the

object I, for which there must be a complete set ofeigenvectors,

zl r;r&=r If;r).Denote a set of states for the apparatus II by I II;n&,where the eigenvalue a is the appropriate "pointerposition" of the apparatus, and I II;0) is the initialpremeasurement state. The interaction between theobject and the apparatus ideally should, if it is tofunction as a measurement, set up a unique corre-

spondence between the initial value of r and the Analvalue of o., o., for example. That is, if the initial statefor the system I+II is I I; r) I II;0), then the equationofmotion must lead to the Anal state

vlf;r&ln;0&= lf;r&lff;~' (4)if the observable R is not changed by the interaction' ';or in the general case

U I I; r& I II;0)= I I;P, ) I II;n), (4.1b)where U is the time-development operator for theduration of the interaction between I and II.Note thatthe vectors I 1;P) need not be orthogonal since or-

'Many authors give the unwarranted impression that thishighly special case is general.'OAraki and Yanase (1960) have shown that unless R com-

mutes with all universal additively conserved quantities, thenno interaction existswhich could satisfy (4.1a).However, theyalso show that this equation can nevertheless be satished to anarbitrary degree of accuracy, provided the apparatus is madearbitrarily large. This fact is illustrated in the measurement ofthe s component of spin by a Stern-Gerlach apparatus (Sec.2.1) which used an external magnetic Geld to break the conser-vation of S and S.The source of this magnetic Geld is rigidlyGxed to the earth, and so the apparatus is electively inGnite, asit must be to satisfy the theorem of Araki and Yanase. See alsoYanase (1961).

thogonality of the 6nal states corresponding to differentr values is guaranteed by the fact that n, &n, forr'Nr.Suppose now that the initial state of the object is

not an eigenvector of R, but rather some linear com-bination

I f;4&=Z I f; &(r I 0). (4.2)From (4.1) and the linearity of the equation of motion,the Anal state for the system must be

v I 1;y& I zf; o&=Z(&I p& I 1;y ) I II; n&r

= If+fr;f&, say.A specific illustration of this general analysis is providedby the EPR experiment discussed in Sec.2.1.We mayconsider the measured quantity R to be the spin ofparticle 2, and the "pointer" of the apparatus to be themomentum of particle 1 (assuming that two detectorsare placed so as to determine its sign) .According to the Statistical Interpretation and F5of the formalism, the probability of the apparatus"pointer" being uat the end of an experiment isI (r I P& I'. That is to say, if the experiment wererepeated many times, always preceded by the same statepreparation for both object and apparatus, the relativefrequency of the value n, would be I (r I P) I'. That thisshould be identical with the probability of the objecthaving had the value r for the dynamical variable Rimmediately before interacting with the apparatus II,is just the criterion that II functions as a measuringapparatus for R.4.2 The Difhculties of the "Orthodox" InterpretationThe discussion of the analysis of Ineasurement

according to the Statistical Interpretation was so simpleand natural that further comment almost seemsredundant. But if instead of the basic assumption of theStatistical Interpretation that a state vector char-acterizes an ensemble of similarily prepared systems,one assumes that a state provides a complete descriptionof an iedividlul system, then the situation is quitediGerent. Unfortunately this assumption has beenmade so frequently, more often tacitly than not, thatit and the resulting point of view have come to beconsidered "orthodox".The "orthodox" quantum theory of measurement

was formalized byJ.

Von Neurnann (1955) and hasbeen clearly reviewed by Wigner (1963).The firstdiKculty encountered is that if one tries to interpret thefinal-state vector (4.3) as completely describing onesystem (object plus apparatus), then one is led to saythat the pointer of the apparatus has no definite posi-tion, since (4.3) is a coherent superposition of vectorscorresponding to diGerent values of n. But the pointercan be a quite classical object and it mat.es no sense tosay that it has no position at any instant of time,especially since the diGerent values of o.are macro-


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L. E. BALLzNrtwz Slafisticat Interpretation of Qaantnnz Mechanics 369

scopically distinguishable. Thus if we attribute thestate vector to an individual system, we are inevitablyled to classically meaningless states for a classicalobject."To circumvent this difhculty, the "orthodox" theory

assumes that, in addition to the continuous evolutionof the state vector according to the equation of motion,there is also an unpredictable discontinuous "reductionof the state vector" upon "measurement, " from (4.3) to

II @")III; " (4 4)where O.. is the pointer position which is actuallyobserved. Proponents of this point of view also usuallyassume that the value of the observable R is notchanged by measurement, so that in pla,ce of (4.4)they would obtain

lI;r') lrr;n, ), (4.4')but this point is not essential. The word "measure-

ment" is enclosed in quotes because its use in thispeculiar context of the "orthodox" theory is notequivalent to the more physical definition of jiieasure-ment given in Sec.3.It should be emphasized that this reduction cannot

arise from the equation of motion. The form of (4.3),involving a superposition of macroscopically distinctpointer positions, depends only upon the linearity of theequation of motion and not upon any of our simplifyingassumptions. For example, one might object that thepointer position eigenvalue n is not sufhcient to label aunique state vector for the apparatus, since there are ahuge number of commuting observables. But thismeans only that each vector l II;n) must be replacedby a set of vectors, all the sets being mutually or-thogonal. Komar (1962) has shown that even withthese extra degrees of freedom, a 6nal state of the form(4.4) is not possible."If the reductiors of the state vectoris to enter the theory at all, it must be introduced as aspecial postulate (often called the proj ectiors postulate)If one merely applied the projection postulate to the

object I, one would say that upon "measurement" thestate changes discontinuously from (4.2) to l I; r'),where r' is the "measured" value of the observable E..On the other hand, the analysis of the interactionbetween the object I and the apparatus II leads to thestate (4.3). To avoid a direct contradiction, Von"This was first pointed out by Schrodinger (1935),who pro-

posed the following hypothetical experiment. A cat is placed ina chamber together with a bottle ofcyanide, a radioactive atom,and a device which will break the bottle when the atom decays.One half-life later the state vector of the system will be a super-position containing equal parts of the living and dead cat, butany time we look into the chamber we will see either a live cator a dead one. A literal translation of "Schrodinger's cat paradox",as this is called, has been given by Jauch (1968,p. 185),but hisdiscussion of the paradox is not adequate, and his volume andpage reference is incorrect."His conclusion that no theory consistent with quantum me-

chanics can account for the occurrence ofevents innature, how-ever, is not true if one drops the assumption that a state vectordescribes completely an individual system.

Neumann introduced the observer III, who supposedlyperforms a "measurement" on I+II (this "measure-ment" however is merely an observation), and thusreduces the state vector to (4.4) or (4.4'). He thenshows that itmakes no difference whether the apparatusII is considered to be part of the observer or part of theobject being observed.Von Neumann's theory of measurement is very

unsatisfactory. It suggests that the passive act ofobservation by a conscious observer is essential to theunderstanding of quantum theory. Such a conclusion iswithout foundation, for we have seen that the StatisticalInterpretation does not require any such considerations.Moreover, the atoms of the observer's body may, inprinciple, be treated by quantum theory, and theresulting state vector will be similar to (4.3) (withI replaced by I+II, and II replaced by III). VonNeumann's theory would require another hypotheticalobserver IV to observe III, and so reduce the statevector. But this can only lead to an infinite regression,which, in an earlier era might have been taken as aproof of the existence ofGod (the Ultimate Observer)!No one seems to have drawn this unfounded conclusion,but the equally unfounded conclusion that the con-sciousness of an observer should be essential to thetheory has, unfortuna tely, bee@ taken seriously(Heitler, 1949,p. 194;Wigner, 1962).Because of the diAiculties inherent in the projection

postulate, we must critically examine the reasons givenin support of its introduction. Dirac (1958, p. 36)argues that a second measurement of the same ob-servable immediately after the first measurementmust always yield the same result. Clearly this could

be true at most for the very special class of measure-ments that do not change the quantity being measured.A statement of such limited applicability is hardlysuitable to play any fundamental role in the foundationsof qua,ntum theory. In fact this argument is based onthe implicit (and incorrect) assumption that measure-ment is equivalent to state preparatiors, the contrastingdefinitions of which were given in Sec.3, The necessityfor distinguishing between these two concepts has beenpointed out by Margenau (1958;1963).For example,a polaroid filter placed in the path of a photon beamconstitutes a state preparation with respect to thepolarization of any transmitted photons. A secondpolaroid at the same angle has no further effect. Butneither of these processes constitutes a meusuremeet.To measure the polarization of a photon one must alsodetect whether or not the photon was transmittedthrough the polaroid 61ter. Since the detector willabsorb the photon, no second measurement is possible.It has also been claimed (Messiah, 1964,p. 140) that

a discontinuous reduction of the state vector is aconsequence of an alleged uncontrollable perturbationof the object by the measuring apparatus. A counterexample to this claim is provided by the EPR experi-ment (Sec. 2.1), in which the spin of particle 2 is


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370 REvIEws oz MQDERN PHYslcs ' OGTQBER 1970

measured indirectly, without any disturbance of thatparticle whatsoever. In general the apparatus willinteract with the object, but the eGect of this is notsuch as to bring about the hypothetical redlctioe. Kedo not consider as credible the occasional statement(Dirac, 1958,p. 110) that a mere observation (i.e., anexperimenter looking at his apparatus) could affect thestate of the system in a manner incompatible with theequation of motion.Although it is not essential to the theory, Popper

(1967) has pointed out that the redlced state vector(4.4) can be interpreted in a natural way. Accordingto the Statistical Interpretation, the state vector (4.3)represents, not an individual system, but the ensembleof all possible systems (object plus apparatus), each ofwhich has been prepared in a certain way and thenallowed to interact. The state vector (4.4) representsthe subensemble whose definition includes the additionalspecification that the result of themeasurement (pointerreading) be ee, . There is no question of reduction of thestate vector being a physical process.Margenau (1963, p. 4/6) has taken a more sym-

pathetic attitude toward Von Neumann's theory ofmeasurement on the grounds (gathered from personalconversation) that he regarded his projection postulateas convenient but not necessary. In reply I would saythat I And no evidence of this interpretation in VonNeumann's writings. Moreover, the great amount ofconfusion generated, as well as the theoretical eGortexpended trying to explain the redgctiors of the stateveclor, suggest that the projection postulate has beenmuch more of an inconvenience than a convenience.

interaction between the object I and the apparatus-II,the final state operator for the combiried system I+IIwill be the pure state

(4.5)

(4.3')

where c,= (r [P) in the nota.tion of (4.3). Now themixed state formed from reduced vectors of the type (4.4),

p,=P [ c, [ [ I; $, ) [ II; er, )(I;4, [ (II;n, [ (4.6)is not equivalent to p~. Even though they both yield thesame probability distribution for the position of theapparatus "pointer" n, their predictions for otherobservables may disagree."Because time developmentis effected by a unitary transformation (1.7a), whichpreserves the pure state condition p'= p, it is impossiblefor the pure state (4.5) ever to evolve into the mixedstate (4.6).Realizing this fact, many authors (Feyerabend, 1957;

Wakita, 1960, 1962;Daneri et a/. , 1962; Jauch, 1964)have proposed that, nevertheless, (4.5) and (4.6) maybe equivalent for all practical purposes. Specifically,they suggest that if the macroscopic nature of theapparatus and its relevant observables are taken intoaccount, then the average of any macroscopic ob-servable M will be the same for these two states. Withthe abbreviated notation [ Pn)= I;g) [ II;n), theseaverages are, respectively,

4.3 Other Approaches to the Problem ofMeasurement

An enormous number of papers have been written onthe problem of measurement in quantum theory.Margenau (1963),lists over 60 separate articles, andpromises a list of about 200on request. The reader canshorten his task greatly by ignoring all papers whichtry, without modifying quantum theory, to accornodatea reductioN of the state vector, and which also assume thestate vector to describe an individual system. Thepreceding arguments have demonstrated the impossi-bility of such programs.Bohm and Bub (1966)propose a nonlinear modifica-

tion of the Schrodinger equation of motion which issupposed to be effective only during the measurementprocess, and which causes the reducfioe. However wedo not consider that the postulation of a new interactionpeculiar to measurement should be taken seriously.Moreover, the simplicity of the account of measure-rnent provided by the Statistical Interpretation under-mines the motivation for any drastic modification ofthe mathematical formalism.To discuss the next approach it is convenient to

restate the results of our analysis in the language ofstate operators (or statistical operators). After the

(4.8)The problem is then whether these expressions are equalunder suffi.ciently general circumstances.Now if M commutes with A (the pointer position,

whose eigenvalues are er,), it will be diagonalizable inthis representation, and the interference terms {rAr')will be absent from (4.7). Hence the two expressionswill be equal. However, (contrary to Jauch's assump-tion) the set of macroscopic observables is n.ot Abelian(consider the pointer momentum, which does notcommute with A), so the macroscopic nature of cM isnot sufhcient to guarantee equality of (4.7) with (4.8) .In the case where the observable 3f belongs entirely

"A proof of this fact is contained in a paper by Furry (1936).But his comments on the work-of Einstein, Podolsky, and Rosen(Sec. 2) are based upon a misconception. Furry interprets theEPR conclusion that-certain dynamical variables such as 0-2, inour Eq. i2.6l are elemerits oj reality, as meaning that the stateof the system after interaction between the object and the appa-ratus is a mixture of eigenstates of the element of reality; i.e, amixed state related to (2.6) in the same way as (4.6) is relatedto (4.5). But EPR do not claim that the equation of motionyields the wrong quantum state, as Furry has misinterpretedthem, but rather that no quantum state, pure or mixed, canprovide a complete description of an individual system. Thismakes Furry's argument irrelevant.


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L. E. BALLENrzNE Statistical Imterpretatiort ofQNarttaas 3IecharN'cs 371

to the measurement apparatus II (i.e., ifM= 1Mzz), zethen

=(I;P lI;f~

)(II;er lMzzlII;n ). (4.9)Thus if the states l I;P, ) are mutually orthogonal,which would be true, in particular, if the measurementofR did not change the value ofR, (l I;P)then becomesl I; r)), then there would be no difference between(4.7) and (4.9). Clearly a similar result holds forobservables belonging entirely to the object I, but ofcourse no such result is valid for observables whichbelong to I and II jointly. 'Daneri, Loinger, and Prosperi (1962, 1966) (DLP)

consider the pure state (4.5) or equivalently (4.3) tobe the final state of the combined system immediatelyafter interaction between the object and the apparatus.They then concentrate on the amplification aspect ofthe measurement necessary for a microscopic object totrigger a macroscopic response in the apparatus, andinvoke the ergodic theory of quantum statisticalmechanics in order to treat the transition of the appa-ratus from its initial "metastable"" condition toequilibrium. They claim, in essence, that the equilib-rium state can be represented by the mixed stateoperator (4.6) .The attitude of DLP is well expressed in the words of

Rosenfeld (1965,p. 230),who endorses their approach:"The reduction of the initial state of the atomic systemhas nothing to do with the interaction between thissystem and the measuring apparatus: in fact, it isrelated to a process taking place in the latter apparatus

after all interaction with the atomic system has ceased."Rosenfeld also emphasizes the view that the presence orabsence of a conscious human observer is irrelevant tothe problem. '7The above remarks show that DLP and Rosenfeld

explicitly reject the "orthodox" interpretation discussedin Sec.4.2 (as does the present author), but they do sowithout actually confronting the question of whetherthe state operator (or vector) represents asingle systemor an ensemble of similarily prepared systems.The analysis of generalized quantum amplification

devices by DLP is interesting and correct, at least tothe extent that their ergodic hypothesis is valid. But itdoes not constitute the essence of measurement in

general. To paraphrasean example

given by jauch' The tensor product notation A8 means that A operates

on the ( I l vectors, and 8 operates on the ( II l vectors."This illustrates the fact that the state of a two-componentsystem cannot be uniquely determined by measurements on eachcomponent separately. Jauch (1968,Secs. 11-7 and 8') gives agood discussion of the mathematical aspects of this problem, andin particular of the tensor product formalism."As pointed out in DLP (1966), this is not necessarily thethermodynamic definition of metastability."On this point Rosenfeld seems to have modi6ed his earlier

views, as expressed in a discussion exchange with Viger (seeKorner, 19/7, pp. ].8$-{j).

(1968,p. 169),if the measuring device is a photographicplate, then the essence of measurement may be per-formed by a single silver-halide complex. The am-plification process does not take place until the plate isdeveloped, which may be months later. Althoughinteresting in its own right, the DLP theory does notanswer the questions which lead us to study the'theoryofmeasurement.

4.4 Conclusionheory of MeasurementThis section on measurement theory is lengthy

because of the great length of the literature on thissubject. However, the conclusions are quite simple. Theformal analysis of a measurement has been undertakennot for its own sake, but only to test the consistency ofalternative interpretations of quantum theory. Usingonly the linearity of the equation of motion and thedefinition of measurement, we see that: the interactionbetween the object and the measuring apparatus leads,in general, to a quantum state which is a coherentsuperposition of macroscopically distinct "pointerpositions." In the Statistical Interpretation this dis-persion of pointer positions merely represents thefrequency distribution of the possible measurementresults for a given state preparation.But if one assumes that the state vector completely

describes an individual system, then the dispersionroust somehow be a property of the individual system,but it is nonsensical to suppose that a macroscopicpointer has no de6nite position. None of the attemptsto solve this problem using some form of reductt'ore ofthe state vector are satisfactory. No such problem arisesin the Statistical Interpretation, in which the statevector represents an ensemble of similarily preparedsystems, so it must be considered superior to its rivals.We have neglected the frankly subjective inter-

pretation (Sussmann, 1957; Heisenberg, 1958), ac-cording to which the quantum state description isnotsupposed to express the properties of a physical systemor ensemble of systems but our krlomledge of theseproperties, and changes of the state (such as the sup-posed reduction of state during measurement) areidenti6ed with changes of knowledge, not so muchbecause it is wrong as because it is irrelevant ' One can,of course, study a person's knowledge of physics ratherthan studying physics itself, but such a study is notgermane to this paper. For example, someone's knowl-edge of a certain system can change discontinuously asa result of a blow to the head which causes amnesia,as well as through the receipt of new information from a~8Another area in which the subjective interpretation causes

confusion is the relationship between entropy and information.It is true that the aquisition of information requires a corre-sponding increase of entropy (Brillouin, 1962);hence the associa-tion of information with negative entropy is useful. But the con-verse proposition that entropy is nothing but a measure of our].ack of information is a fallacy. Entropy also has a thermody-namic meaning, dS=dQ/T, which is valid regardless of the exist-ence or nonexistence of Maxwell's demon.


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372 REVIEWS OP MODERN PHYSICS ~ OCTOBER 1970

measurement, although the subjectivists tend to ignorethe former while stressing the latter.

~(~;O)=(-p (W))=f exp (i]A)P(A;iP)dA, (5.3)

hence E(A;P) is equal to the inverse Fourier transformofM((;f) .By expanding the exponential

(5 4)0 Q 0

we see that a knowledge of the characteristic function isequivalent to a knowledge of all the moments of thedistribution. According to the formalism (see F4),these are given by"

(5.5)By analogy we may introduce a characteristic

function for the joint probability distriubtion(8)"(ix)mm(g X y) = P &q"P"), (5.6)

e ~ SImt"We shall consider only pure states since these illustrate the

essential problems. The generalization to mixed states is straight-forward."In this section it is necessary to distinguish between a phys-ical observable and the mathematical operator which representsit. The operators will be distinguished by a circumQex.

5. JOINT PROBABILITYDISTRIBUTIONSIn contrast to the previous sections, which were

concerned entirely with interpretation of the existingformalism (summarized in Sec. 1.1), this section andthe next consider possible extensions of the formalism.These will not be modifications of the establishedformalism, but additions to it which are compatiblewith both the established formalism and the StatisticalInterpretation.As discussed in Sec.3 with regard to the uncertainty

principle, quantum theory is not inconsistent with thesupposition that a particle has at any instant both ade6nite position and a definite momentum, althoughthere is a widespread folklore to the contrary. Of coursewe are not compelled either to accept or to reject thissupposition, but it is of interest to explore it on atentative basis.Our first problem is to construct, for the noncom-

muting observables q and p, a joint probability dis-tribution I'(q, p; f) for any state I $),i9 such that thesingle variable distributions constructed from it agreewith the established formalism, i.e.,

P'(q, P;0) dP=I'(q; 0)=I 8 I q) I', (5 1)fI'(q, P; 0) dq= I'(P;&)=(& I P) I' (5 2)

Additional conditions will be considered later.Let us first consider the method by which the single

observable probability distribution function can beconstructed. The characteristic function for the dis-tribution ofan arbitrary observable A is given

by

(e'I qnpm~2 np I qmiIimqii=o E&)

then one obtains (Moyal, 1949)

(5.8b)

-P(q, P; 4)

= (2m.)-'fQ I q', riri) exp (rp) (q+-',rh I p) dr,(5.9)

which was first introduced by Wigner (1932).Becausethis expression may become negative it cannot beinterpreted as a genuine probability distribution, andWigner proposed it only as a calculational device.Another distribution which suGers from the same defecthas been derived by Margenau and Hill (1961)usingthe correspondence

q"P" l (q"p +I"q"). (5.10)Since every correspondence rule which associates an

operator with a classical function ofqand

p,g(q, P) G(q, I), (5.11)

provides a joint probability distribution, it is possible toformulate the most general form of P(q, p; p) whichwill satisfy (5.1) and (5.2).We refer to the originalpapers for details (Cohen, 1966;Margenau and Cohen,1967).These authors investigated the possibility ofconstructing a joint probability distribution such that

Q IO(q, p) I4)=ffg(q, p)&(q,p;4) dqdP, (512)

from which, by a Fourier inversion, we obtain,

&(q P 4)= (2ir) 'ffM(e, X;f) exp P iHq+Xp) ) dedx. (5.7)

As long as the moments (q") and (p ) are given by(5.5), then (5.7) will automatically satisfy (5.1) and(5.2).The difficulty now evident is that that there is no

unique way in which to take products of noncommutingoperators. For example q'p' may be represented byl (qv'+7'q'), :(q7+Iq)', l (qA71+fqA), any ofseveral other forms. In these examples we have alreadyused the additional plausible restrictions that theproduct be a self-adjoint operator, and that it be sym-metric under interchange of q and p. Shewell (1959)has considered several correspondence rules which havebeen proposed, and found none to be fully satisfactory.Some of the rules do not yield a unique operator, whileothers yield an operator for a power of the Hamiltonian,[H(q, p))", which is not equal to (H)". Each suchpossible choice will yield a different joint probabilitydistribution which satisfies (5.1) and (5.2).If one chooses

exp Li(ttq+XP))exp (ioq+imp), (5.8a)or equivalently


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L. E. Bxzr.ENrrNE Statcsttca/Interpretationof Quantuntechanscs 373

and also such that for any function E(4 I &(&(q,p)) I4)=ff&(c(q,P))I'(q,P;4) dqdP.

(5.13)

In other words, they asked whether there exists a jointprobability distribution such that the averages of allobservables can be calculated by a phase-space averageas in classical statistical mechanics."Their answer was negative. Although it is always

possible to satisfy (5.12), (5.13) could be simul-taneously satisfied only if the correspondence

&(g(q,P) ) ~&(t'(q,7) ) (5 14)can be satisfied for all functions E with the samecorrespondence rule as tha, t leading to (5.11). Thisthey- prove to be impossible."With the wisdom ofhindsight, we should not be surprised that (5.12) and(5.13) cannot be satisfied, for if it were otherwise, thenquantum mechanics could be expressed as a specialcase of classical mechanics.To deal with simultaneous values of position and

momentum variables in a logically consistent fashionwe need only require that a joint probability dis-tribution exists which satisfies (5.1) and (5.2), and

I'(q P; 4')&o (5.15)for all quantum states. These conditions are obviouslysatisfied by the function

I'(q p p) = I (y I q) Is I (y I p) Is (5.16)but this form is probably not unique. Indeed, it cannotbe applicable to scattering experiments (see Fig. 3),where, merely by geometry, there must be a closecorrelation between position and momentum at largedistances from the scattering center. The investigationof possible joint probability distributions is clearly notcomplete.Bopp (1956, 1957) has undertaken not only to

represent quantum mechanics in terms of ensembles inphase space, but also to derive the statistical equationsfrom simple principlesn a priori derivationotmerely a translation from the vector-operator formalismof quantum theory. Bopp's work divers from the abovein that his joint probability distribution does not satisfy(5.1) and (5.2) .Nevertheless, there is a definite relationbetween the averages calculated in Hopp's theory andthose of quantum theory (denoted by subscripts 8 andQ). For example,(q&o=(q)q, (P)o= (P&o,

).=).+i/4,(p &.=(p &.+~V1

"Because (5.1) and (5.2) are satisfied, this will clearly betrue for any function of q only, ,or of p only."Itwould seem inevitable that arbitrary functional relationsinvolving q and p cannot be preserved by any quantal operatorcorrespondence rule, since the equation qpq=0 for classicalvariables becomes jpq=i7i in quantum theory. Clearly onecannot map hiero onto ih in a consistent fashion.

I,= Iz:q,&z&q:},Ii= [t '.pi&ts&p2}, (5.17)

and the imaginary part of P represents the uncertaintyassociated with the previous statement. Certainconditions must be satisfied in order for this inter-pretation to be sensible. Since q and p are certain to besomewhere in their spectra, we must have Rer'1,and Im I'0, as I~ and I~ are enlarged to include theentire spectra from ~ to+~ . If Ii and I2 are smallerthan the irreducible errors in a single measurement,

"The naive interpretation (Matthews, 1968,Chap. 3) of theoperator product AB as corresponding to a measurement of 8followed by a measurement of A is refuted by the example of thePauli spin operators for which 0;0-=io;.A measurement of afollowed by a measurement of 0 is in no way equivalent to ameasurement af 0,

Here / is some constant with dimensions of length(interpreted by Bopp as the finest accuracy of aposition measurement) .The problem of defining a satisfactory joint proba-

bility distribution for position and momentum may notbe entirely mathematical, for we must also decide on"the empirical definition of the concept 'simultaneousvalues of position and momentum, ' " according toPrugovecki (1967).To this end he studies variousoperational methods ofmeasuring both these observablessimultaeeously on the same individual system, notmerely on different individual systems representative ofthe same state preparation. He suggests that theirreducible errors in individual measurements ought tobe taken into account in a generalized formalism (inaddition to the statistical fluctuations in an ensemble Ofsimilar measurements, which are treated in the estab-lished formalism) . It should be emphasized thatsimultaneous measurement of noncommuting ob-servables, even with a finite precision, has no counter-part in the established formalism which treats onlymeasurement of a single observable (or a commutingset of observables)."Now to experimentally determinea probability distribution, one must perform a numberof measurements and construct a histogram of theresults (see Fig. 2) . Each point on the curve will have avertical uncertainty due to the statistical uncertaintyof using a finite sample, and a horizontal uncertaintydue to the error involved in a single measurement.(In the case of simultaneous measurement of q and p,a single measurement would be a pair of ni.mbers(q, p), and the error would be an area in the qp plane) .If the latter cannot be reduced to zero, we cannotdetermine a probability distribution exactly, even if weeliminate statistical uncertainties by repeating themeasurement an arbitrarily large number of times.To deal with such a circumstance Prugoveckidefines

a complex probability distribtttiort, Pea "(Ii)&I2),withthe following properties: the real part of P representsthe probability that q will be in the interval, I& andsimultaneously p will be in the interval I2, where


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374 REvIEws QP MQDERN PHYsIcs ' OGTOBER 1970

then Im 8 shouM be very large because one cannotdetermine whether or not a measured value of (q, P)lies within such a small region of phase space, andtherefore one cannot determine the relative frequency(probability) of such an event. Of course the complexprobability distribution must reduce to the real proba-bility distribution of the established formalism in thecase of commuting operators.The specific expression proposed by Prugovecki for

the complex probability distribution is

where Ej and E2 are projection operators which projectonto the subspaces spanned by the eigenvectors of jcorresponding to the spectral interval I~, and of jcorresponding to I2, respectively This form, althoughperhaps the most obvious choice, may not be satis-factory because he shows that for any nontrivialintervals

Ijand I-. there exist states for which

ReE(0,This would be tolerable if simultaneously we had alarge value for Im P, so that we could say that theprobability was not well defined in such a case. But infact one obtains Im P=O for that state which makesReEmost negative. Since a "well defined" but negativeprobability makes no sense, some modification of either(5.18) or the interpretation seems in order (such as,perhaps, a restriction on the states P which are to beconsidered physically realizable). This objection doesnot necessarily invalidate the concept of a complexprobability distribution, but the usefulness of such adistribution is yet to be determined.

6. HIDDEN VARIABLESQuantum theory predicts the statistical distribution

of events (i.e., the results of similar measurementspreceded by a certain state preparation). But if theprepared. state does not correspond to an eigenvector ofthe particular observable being measured, then theoutcome of any individual event is not determined byquantum theory. Thus one is led to the conjecture thatthe outcome ofan individual event may be determinedby some variables which are not described by quantumtheory, and which are not controllable in the statepreparation procedure. The statistical distributions ofquantum theory wouM then be averages over these"hidden variables."The entirely reasonable question, "Are there hidden-

variable theories consistent with quantum theory, andif so, what are their characteristics?," has been un-fortunately clouded by emotionalism. A discussion ofthe historical and psychological origins of this attitudewould not be useful here. We shall only quote one ex-ample of an argument which is in no way extreme(Inglis, 1961,p. 4), "Quantum mechanics is so broadlysuccessful and convincing that the quest [for hiddenvariables j does not seem hopeful."The vacuous charac-ter of this argument should be apparent, for the success

of quantum theory within its domain of definition (i.e.,the calculation of statistical distributions of events)has no bearing on the existence of a broader theory(i.e., one which could predict individual events).The question of hidden variables has also been

subject to a genuine confusion (i.e., not merely due tothe conflicting personalities or metaphysical ideas of theprincipal characters in the debate), for Von Neumannproved a theorem from which he concluded that nohidden-variable theory could reproduce all of thestatistical predictions of quantum theory. Although hismathematical theorem is correct, his physical conclusionis not, and the first example of a hidden-variable modelwas published by Bohm (1952).However, a clearanalysis of the nature of Von Neumann's theorem andits relation to hidden variables was not achieved untilmuch later (Bell, 1966), and the 14 year intervalbetween these papers yielded many inconclusive dis-cussions pro and con, as well as some "improvedproofs" of the impossibility of hidden variables, Weshall expound briefly the content of Von Neumann'stheorem, and the reason why it does not rule out hiddenvariables in quantum theory. Other relevant theoremswill also be discussed.The Statistical Interpretation, which regards quan-

tum states as being descriptive of ensembles of similarilyprepared systems, is completely open with respect tohidden variables. It does not demand them, but itmakes the search for them entirely reasonable [thiswas the attitude of Einstein (1949)g.On the otherhand, the Copenhagen Interpretation, which regards astate vector as an exhaustive description of an in-dividual system, is antipathetic to the idea of hiddenvariables, since a more complete description than thatprovided by a state vector would contradict thatinterpretation.

6.1 Von Neumann's TheoremVon Neumann's mathematical theorem concerns the

average of an observable, represented by a Hermitianoperator, in a general ensemble subject only to a fewconditions. The original derivation (Von Neumann,1955,pp. 305-325) is lengthier than necessary, and. weshall follow the more conciseversion given by Albertson(1961).Von Neumann makes the following essentialassumptions which in his book are identified by thesymbols in brackets:

(i) [Ij If an observable is represented by theoperator R, then a function f of that observable isrepresented by f(E).(ii) [II) The sum of several observables represented

by R, 8, ~ ~ ~ is represented by the operator 8+5+ ~ ~ ~,regardless of whether these operators are mutuallycommutative.(iii) [p. 313, not identifiedf The correspondence

between Hermitian operators and observables is oneto one,


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L. E. BwLLzNrrNz Stats'stica/ Interpretats'on ofQuantnnt Mechanics 375

+ Q IV" ' ReR,+IF'"' ImR I, (6.2)m,n;m&n

where R =R *=(rt ~ R ~ nt) is a number, andU-'=I &( I,Vt=n&(nt i+ i m&(n i,IF&" &=(l n&(rn ) [ nt&(rt )) (63&

are Hermitian operators. According to assi,mption(iii) these operators are all to be regarded as observables,and hence (ii) and (v) may be applied to obtain

(R&=Z(U- )R,n

+ Q I (V'""')ReR+(IF'""')Im R . (6.4)m,n;m&n

This result may be rewritten as(R)= Z p~ = Tr (pR), (65)

ml +

if we define the matrix elements of the statisticaloperator p to be

p (U(n) )p =,(V&" ~&

statistical interpret a it on of qm

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