ON ARTHUR FINE'S INTERPRETATION QUANTUM MECHANICS

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1 ALLEN STAIRS ON ARTHUR FINE'S INTERPRETATION QUANTUM MECHANICS OF The no-hidden-variable proofs of von Neumann, Jauch and Piron and Kochen and Specker have left many workers in the foundations of quantum mechanics unconvinced. Nonetheless, it is probably fair to say that most are convinced by the Bell-Wigner argument that local hidden variables are impossible. A notable exception is Arthur Fine, who insists that the Bell-Wigner result has nothing to do with locality, and who defends a point of view which, while not described as a hidden variable theory, claims that quantum mechanics can be understood as "... an essentially statistical account of a well-defined and localized domain. ''~ By "well-defined", Fine means that every magnitude has a definite value and that classical logic is preserved. In this paper, I attempt to throw cold water on Fine's hopes for a classical understanding of quantum theory by focussing on his treatment of the problem of joint distributions. I will begin with an earlier paper, 'Probability and the Interpretation of Quantum Mechanics '2, which provides important background, and then proceed to 'On the Completeness of Quantum Mechanics' in which the above thesis is defended. The main point of the earlier paper is to argue that no violations of our ordinary concepts of probability and logic are involved in the fact that quantum mechanics does not provide joint distributions for every set of magnitudes. The programme of that paper involves introducing a family L of propositions each of the form quantity Q is confined to Borel set A (abbreviated as o[a]) and forming the language/s generated by L by the usual closure under ( v, ^ - ). Fine then attempts to establish the following two claims. I. /S admits of bivalent valuations consistent with QM. II. /S-plus-valuations admits of just those probability assignements that QM does in fact make. (pp ) Synthese 42 (1979) /79] $ Copyright 1979 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

2 92 ALLEN STAIRS The bivalent valuations are restricted by the following 'regularity requirements'. (a) To each state ~b there corresponds a function ~,+ from i onto {0,1} (b) If P~[A] = 1, then v~[a]= 1 [where P~[A] is the QM probability assigned to Q[A] by ~b] (c) If v~[a] = 1 and A---< B, then v~[b] = 1 (d) If ~. is the complement of A, the v~[a] = 1 only if v~[a] = 0. (e) If P~ is the operator projecting on the closed subspace M then v~[{l}] = 1 iff ~b E M. (p.18) (Condition (e) results from Fine's interpretation of v~[{1}] = 1 as asserting that the state lies in M.) These five requirements do not guarantee that every magnitude has a definite value, and indeed, Fine considers valuations such that for some quantities Q, every proposition Q[A], is false. These will not concern us here, however. Fine also considers valuations satisfying the following condition. (f) To every quantity Q and state ~b there corresponds some real number x such that v~[{x}] = 1. Valuations satisfying (f) are called atomic valuations. Fine maintains that such valuations are possible without contradicting the requirements of the theory. Although it would be worth considering his defence of this claim, we shall not do so here, since our main concern is with probability. Fine notes that if magnitudes are represented as random variables over a common phase space, there will be a joint distribution for each pair of magnitudes. He concludes, therefore, that the phase space is too restrictive a setting for quantum theory and proposes instead the following approach. Simply introduce for each state ~b and quantity Q a probability measure P~ on the Borel sets... It may happen that certain quantities are interpretable as compounds of other quantities. In such cases this way of using probability will assign probability to the corresponding compound events. When this happens, of course, it will be a result of the theory. It will not follow from some extra-theoretical prescription for how to use

3 ON FINE ON QM 93 probability; for example, the prescription that quantities be construed as random variables over a common phase space. (p. 13) It is important to note that, having introduced this general setting for probability and having defined a family of atomic valuations, Fine does not simply assume that "/7,-plus-valuations admits of just those probability assignments that QM does in fact make." Rather, he attempts to justify this claim, beginning with the following obser- vations: If one has in mind the intuitive picture of a valuation as specifying a possible state of the world, then one might try to connect the valuations with probability by requiring that the probability (in $) for an event (= proposition) be measured by the proportion of those states of the world in which the event occurs (i.e. of those valuations v* in which the proposition is true). Since in general there will be too many states to measure by simple proportions, this idea amounts to forming a classical probability space out of the valuations themselves and then transferring the measure on this space to the quantum mechanical probability for the event in question. (pp ) Fine does not regard this programme as feasible for all quantities, since it runs into difficulties with condition (e) on admissible valuations. 3 His solution is to treat projection operators and non-projection operators differently. The probabilities for the former are (roughly speaking) to be obtained directly from Hilbert space, but the latter are associated with a classical probability space. Specifically for the non-idempotents, let V * be the set of valuations associated with the state ~b and F* be the smallest o--algebra of subsets of V s. For Q non-idempotent, define P~{v*Iv~[A] = l} = P~[A] where P(~[A] is the quantum mechanical probability in 4) of Q[A]. p4, can be extended in a variety of ways to a measure on all of F ~. It would seem to be a consequence of this approach that joint distributions could be introduced for arbitrary pairs of Q~, Q: of nonidempotent magnitudes. One would simply use the following rule P(~I,Q2[AxB] = e~[{v~:v~l[a] = v~:b = 1}] However, this would yield more joint distributions than quantum mechanics itself provides. Consequently, Fine requires a principle

4 94 ALLEN STAIRS which will distinguish between those functions P~,o2 which represent genuine quantum mechanical joint distributions and those which do not. Therefore, Fine introduces the following constraint. Let [: RXR~be a Borel function. In order that P~.Q2 be interpretable as a joint distribution in QM we require that there exist a quantity f(q~, Qz) such that for all states 4~ PI(Q,.Q2)[ A] = P~,, Q2[/-!(A)] (jd) (p. 30) A pair of Q1, Q2 quantum mechanical magnitudes will only satisfy jd if Q~ commutes with Q2. Therefore in the context of the phase space, jd instructs us to interpret P~,o2 as a joint distribution only when Q1 and Q2 represent magnitudes where associated operators commute. What Fine did not realize at the time of writing the earlier paper is that there is no measure on the phase space which will reproduce the correct joint distributions even for commuting quantities. 3 It was this realization which lead him to the programme of 'On the Completeness of Quantum Theory'. The aim of the later paper is... to make simultaneous assignments of values to all the quantities of the theory in satisfaction of the" Spectrum Rule. Completeness requires... that these values be the very ones found in measurement, and good sense then directs that they should be distributed according to the probabilities of the theory... (p. 281) Hidden variable theories, as Fine characterizes them, attempt to achieve this aim by introducing a classical phase space and representing quantities as random varibles on the space. A natural constraint on hidden variable theories would seem to be that for all commuting A,B (JD) P~,B = P~A],[B] where the left side is the quantum mechanical joint distribution of A and B and the right side is the phase space joint distribution of the corresponding random variables.the case of two spin-1/2 particles in the singlet state, considered by Bell, shows that JD cannot be satisfied in the phase space (and hence that jd, understood as above, cannot either). Label the two particles S and S'. Let {si} be the spin magnitudes of S and {S~} be the the spin magnitudes of S'. Each Si com-

5 ON FINE ON QM 95 mutes with each SI, and so for all such pairs, the QM joint distribution is well-defined. The Bell-Wigner argument shows that there is a small finite set of such magnitudes whose quantum mechanical joint distributions cannot be reproduced on a phase space. Fine comments on this situation and what he says is interesting. The failure of JD shows...that we cannot fulfill both of these aims. We cannot construe the measured values as merely revealing what is there all along and have the statistics of what is there all along coincide with the probabilities of the theory. (p. 280) In spite of the impression which this passage creates however, Fine is not saying that it is impossible to conceive of quantum mechanical systems as having definite values, revealed by measurement and, roughly speaking, occurring with frequencies corresponding to the quantum mechanical probabilities. As Fine sees it, the problem is really that... the pairs of values that turn up in simultaneous measurements of commuting observables do not distribute according to phase space rules; they are distributed according to the different rules of quantum theory. (p. 280, emphasis mine) In other words, Fine sees the failure of JD merely as indicating a problem with the way in which hidden variable theories represent probabilities (i.e. by means of a phase space). Fine's solution is to drop the phase space and substitute the following approach. For each state ~b, let V ~ be the set of all functions x which assign each magnitude an eigenvalue with non-zero probability. Let Q(x) (x E V 6) be the value which x assigns to Q. To accommodate the statistics, associate with each Q a probability space in the following manner. Let and and V'~(Q) = {Q(x)lx E V 4'} B~ = the Borel subsets of V*(Q) P~ = the probability measure which th associates with Q according to quantum theory. This gives us for each Q a classical probability space S~ = < V*(Q),

6 96 ALLEN STAIRS B~, P~ >. Initially, then, no joint distributions are derivable. They are introduced only when the magnitudes in question commute, thus satisfying jd. If A and B commute, their joint distribution will be a measure on the Borel subsets of V*(A)xV~(B), corresponding to the appropriate quantum mechanical joint distribution. Formally, this approach solves all the problems. The derivation of the Bell-Wigner result involves expressions for the probabilities of simultaneous values of non-commuting quantities. Since these probabilities are not defined in the approach just described, the Bell-Wigner result seems to be circumvented. However, there is no interesting mathematical difference between this way of doing things and the method of representing probabilities in the earlier paper, according to which one simply introduces for each quantity a probability measure on B(R). In that paper, Fine took considerable pains to attempt a justification of the claim that the valuations and probabilities fit together in a suitable way. In the later paper no such attempt is made. What I will now argue is that no satisfactory justification could be given. The best way to see this is to begin with the situation which Wigner 4 considers in his reconstruction of Bell. Let S and S ~ be a pair of systems in the singlet state as above. Let A, B, C be three directions in physical space and, by abuse of notation, let A, B, C be the corresponding S-spin magnitudes and A ~, B ~, C ~ be the corresponding S~-spin magnitudes. For the sake of definiteness, let A, B, C be coplanar directions with OAc = 120, 0AB = OBC = 60 - On Wigner's view the assumption of locality amounts to associating each local spin magnitude, which is a non-maximal magnitude for the composite system represented by H2@H 2, with a unique random variable on the phase space, rather than with a different representative for each maximal magnitude with which it is associated. The joint event 'M = x and N ~ = y' is then simply the intersection of the phase space element which represents 'M = x' and that which represents 'N ~ = y', and this element is an element in the sub-algebra corresponding to M(~)N ~. Fine denies that this has anything to do with locality (for reasons which we will not discuss here) and he rejects the phase space. He does nonetheless associate each of the local magnitudes with a unique 'statistical variable' and thus accepts what is really the crucial step in

7 ON FINE ON QM 97 Wigner's argument. A further requirement which Fine could hardly fail to accept is that the statistics be confirmable in principle. Since all experiments involve observations of only a finite number of systems, however, this means that Fine must allow that the statistics of a finite set of magnitudes be realizable to an arbitrary degree of precision in some possible finite ensemble. Indeed, if this were not possible, then the claim that the statistics are nonetheless correct would seem not even to make sense, let alone be confirmable. Let us inquire, then, if this is possible in the case under consideration. The problem can be seen more clearly if we first ask what it would mean for the statistics to be realized exactly in a finite ensemble. We note that each of the six local magnitudes has two values which we will denote by + and -. Let mr (m~) represent 'M takes value + (-)' and let m~(m~) represent 'M 1 takes value + (-)'. The statistics of the singlet state (which we will denote by ~) are such that for any pair M, M ~ of spin magnitudes corresponding to a single direction, (1) P.(m~ & m~-) = P.(m~ & me) = 0 (2) P.(m~- & m~)= P~,(m~ &m~) = 1/2 In general for M, N ~ (3) and P.(m~ & n~) = P.(m~ & n~) = ½sin E 10MN (4) P.(m~ & n~) = P.(m? & n~) = ~cos 2 10MN For the directions A, B, C, it follows that (5) P,v(a~ & c~) = ½sin260 = 3/8 (6) P.(a~- & 53) = P(b~ & c~) = 1/8 The question which Fine must answer, then, is whether there could exist a finite ensemble of pairs of particles in the singlet state, sucb that (i) each particle has a definite value for spin in each direction A. B, C, (ii) in which no pair consists of two particles with the same value of spin in a single direction, and (iii) in which the probabilities of (5) and (6) are matched exactly by the proportions of pairs in the

8 98 ALLEN STAIRS ensemble which fit the relevant descriptions. In a word, the answer is ~'No '~" To see this, we simply apply an argument virtually identical in ~ structure to Wigner's. If the probabilities of were realized exactly in a finite ensemble, no pair would satisfy nor (mt & mf) (mt & md Let 'N(mT & n~-)' denote the number of pairs of which '(mt & n~)' is true, and let '(+ + - ;- + +)' stand for '(at & btc7 & a~ & b~ & c~)' and so on. From what has just been said, it follows that an expression such as (+ + + ; + + +) is not satisfied by any pair in the ensemble. Therefore (7) N(aT&c~)=N(+ +-;--+)+N(+--;- + +) (8) N(aT & b~-) = N(+ - + ;- + -)+N(+ - - ;- + +) (9) N(bt & c~)=n(+ + -;- + +)+N(- + -;+ - +) Hence, using 'prop? as shorthand for 'proportion of pairs such that..' we have (10) prop(at & c~)-<prop(at & b~)+ prop(b~ & c~-) for any finite ensemble which realizes the statistics exactly. But in the case considered, this is impossible, since 3/8 ~ 1/8 + 1/8. The fact thatqm does not provide a distribution for such expressions as (at & b;... c~) is irrelevant. If quantum mechanical ensembles were as Fine describes them, then in any finite ensemble of pairs in the singlet state there must be a definite number of pairs of which such an expression is true. All that the failure of the existence of a th6oretical distribution could mean is that in any particular case, one could have no

9 ON FINE ON QM 99 well founded expectations about the percentage of the ensemble in which such expressions would be satisfied. So far, we have shown that the statistics could not be satisfied exactly in a finite ensemble. If one dropped the requirement of exactness, then (7) to (9) would become, schematically, (7') (8') prop (a~ & c~) = A + B + e prop (a~- & b~-) = C + B + e' (9') prop (b ~ & c ~ = A + D + e" where each epsilon is the sum of 14 terms. We can no longer derive the inequality, but this changes things very little. It still must be possible, if the probabilities are to be realizable, to describe a finite ensemble in which (7') is arbitrarily close to 3/8, (8') and (9') are each arbitrarily close to 1/8 and each summand of each epsilon (and hence, each epsilon) is arbitrarily close to zero. It is easy to see that this is not possible. I conclude then, that Fine has not succeeded in avoiding Bell's result. The real significance of the Bell-Wigner argument is that no system of the sort which Fine contemplates could possibly exhibit the properties of a quantum mechanical system in the singlet state. The phase space is really just an acknowledgement of the fact that ensembles of such classical systems can be thought of as sets with well-defined subsets corresponding to the various properties countenanced by the theory. Fine could, of course, preserve his belief in the classical nature of quantum mechanical systems by positing a non-local effect to. account for the observed frequencies. But this would be to deny that the quantum mechanical joint distribution of M and N j simply tells us the probabilities of conjunctions of propositions of the forms 'M= x' and 'N ~= y', and consequently to accept a crucial feature of the Bell-Wigner characterization of locality. If what I have said here is correct, there is a sense in which it really amounts to pointing out the obvious. The underlying point, however, is not trivial. There is no merely accidental connection between classical logic and the apparatus of the classical phase space. I

10 100 ALLEN STAIRS suggest in the same vein that it is no mere accident that the classical probability space is not the device used to represent probabilities in quantum mechanics. Dalhousie University and University of Western Ontario NOTES J Fine (1974), p Fine (1973). 3 In particular, if Q is a projection operator onto a one-dimensional subspace spanned by ~b', and if 0 </(~b,~b')12 < 1, then {v~lv~[{1}] = 1} = A and hence has probability = 0. But QM tells us that this probability is non-zero (see p. 28 of Fine [1974]). 4 Wigner (1970). REFERENCES Bell, J. S., 'On the Einstein-Podolsky-Rosen Paradox', Physics 1 (1964), Fine, A., 'On the Completeness of Quantum Theory', In P. Suppes (ed.), Logic and Probability in Quantum Mechanics, Reidel, Dordrecht, Fine, A., 'Probability and the Interpretation of Quantum Mechanics', British Journal for the Philosophy of Science 24 (1973), Wigner, E., 'On Hidden Variables and Quantum Mechanical Probabilities', American Journal of Physics 38 (1970),