lfi -+--a-u

2m.wu -8where G is a unit vector. In a dispersion free state the expectation value of an operator must be one of its eigenvalues, 0 or 1 for projections. Since from A we have that for a dispersion free state either Let & and L be any non- collinear unit vectors and with the signs chosen so that < a > = <b > = 1 requires But with G and 5 non-collinear one readily sees that anb = 0 so that < anb > = 0 So there can be no dispersion free states. The objection to this is the same as before. We are not dealing in B with logical propositions, but with measurements involving, for example

differently oriented magnets. The axiom holds for quantum mechanical states.14 But it is a quite peculiar property of them, in no way a necessity of thought. Only the -9quantum mechanical averages over the dispersion free states need reproduce this property, as in the example of Section II. v. GLEASON The remarkable mathematical work of Gleason4 was not explicitly addressed to the hidden variable problem. It was directed to reducing the axiomatic basis of quantum mechanics. However, as it apparently enables von Netinn's result to be obtained without objectionable assumptions about non-commuting

say, one might measure either P(Q2) z P($2) where (P2 and q2 are orthogonal to 03. but not to one another. These different possibilities require different experimental arrangements

there is no a priori reason to believe that the results for P(G3) should be the same. The result of an observation may reasonably,depend not only on the state of the system (including hidden variables) but also on the complete disposition of the apparatus

see again the quotation from Bohr at the end To illustrate these remarks we construct a very artificial but simple hidden variable decomposition. If we regard all observables as functions of commuting projectors, it will suffice to consider measurements of the latter. Let P1,P2.... be the set of projectors measured by a given apparatus, and for a given quantum mechanical state let their expectation values be )

h2 - hl-,,A. A,,- A, ) - - ' As hidden variable we take a real number 0 < 1, < 1

wespecify that measurement on a state with specified 7, yields the value 1 for Pn if &ml < h < &, and zero otherwise. The quantum mechanical stateis obtained by uniform averaging over is no contradiction with Gleason's corollary, because the result for a given P, depends also on the choice of the others. Of course it would be silly to let the result be affected by a mere permutation of the other P's, so we specify that the same order is taken (however defined) when the Pls are in fact the same set. - 14 I Reflection will deepen the initial impression of artificiality here. However the example suffices to show that the implicit assumption of the impossibility

iam much indebted to Professor Jauch for drawing my attention to this work

in his contribution to the volume cited in Reference 2

130

In particular the analysis of Bohm' seems to lack clarity, or else accuracy. He fully emphasizes the role of the experimental arrangement. However, it seems to be implied (p. 137 of Ref. 6) that the circumvention of the theorem requires the association of hidden variables with the apparatus - 19 as well as with the system observed. The scheme of Section II is a counter example to this. Moreover it will be seen in Section III that if the essential additivity assumption of von Neumann were granted, hidden variables wherever located would not avail. Bohm's further remarks in Reference 17 (p. 95) and Reference 13 (p. 35'3) are also unconvincing. Other critiques of the theorem are cited, and some of them rebutted, by A1bertson.a

Mathematische Grundlagen der Quantenmechanik, (Verlag Julius- Berlin,), [English trans.: Princeton University Press, 19551. All page numbers quoted are those of the English edition. The problem is posed in the preface, and on page 209. The formal proof occupies essentially pages 305-324, and is followed by several pages of commentary. A self-contained exposition of the proof has been presented by J. A1bertson e

This is contained in von Neumann's B-I (p. 3ll),&(p. 313), and 2 (p. 314)

Reference 10, p. 209

Reference 10, p. 325

In the two dimensional case <-a > = < b > = 1 (for some quantum mechanical state) is possible only if the two projectors are identical (2 = 6). Then sn'~ = a = b and < anb > =<a> = < b > = 1. - 20 -

The simplest example for illustrating the discussion would then be a particle of spin 1, postulating a sufficient variety of spinexternal-field interactions to permit arbitrary complete sets of spin states to be spatially separated

There are clearly enough measurements to be interesting that can be made in this way. We will not consider whether there are others

Causality and Chance in Modern Physics,(D Princeton and New York,)

in Quantum Theory, D. R. Bates, Ed Press, New York and London,)

Phys. Rev., lO$, 1070