By F. J. Belinfante
A Survey of Hidden-Variables Theories is a three-part ebook at the hidden-variable theories, referred during this publication as ""theories of the 1st kind"". half I experiences the explanations in constructing forms of hidden-variables theories. the hunt for determinism ended in theories of the 1st style; the search for theories that seem like causal theories while utilized to spatially separated platforms that interacted some time past ended in theories of the second one type. components II and III extra describe the theories of the 1st type and moment style, respectively.
This booklet is written to make the literature on hidden variables understandable to those that are burdened through the unique papers with their controversies, and to regular reader of physics papers.
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Extra resources for A Survey of Hidden-Variables Theories
Therefore, before they are added, each V(\p, l) op should be multiplied by the probability wß with which the eigenvalues of observables described by l0^ occur in the ensemble^ described by ϋ(ψ)ορ. By (10), this is ina theory of the first kind the probability pp with which the ξ(β) occur in an equilibrium distribution. Thus, instead of (52), we shall try whether we can write Uop in the form £/(V)op = fpPV(ip9$e>U· iß) (53) This improvement, of course, does not make it possible to write Uop really as a super position of density matrices for dispersionfree states (i/;, £) of which each could be used to calculate expectation values of all observables in such a dispersionfree state.
Independent of this result, for any realistic hidden-variables theory for our two-dimensional case with ψ = φι, we see from (45a, b) that there is some probability Pi that ξ is such that (β) ξ = Bi and that s2 v12+vl2 = — = tan 0, (46a) and a probability P 2 that ξ is such that (Β)ψ v12 + v*2 = ζ = B2, and that - c2 = - c o t 0. (46b) It does not matter that vi2 + v^9 as a function off, is necessarily discontinuous where we pass from the f values for which (46a) is valid, to the ξ values for which (46b) is valid, In hidden-variables theories, such discontinuities must be expected, 29 VON NEUMANN'S CLAIM OF IMPOSSIBILITY What is disturbing, however, is that v12 + v*12 depends on 0.
In Appendix A we show that in this case (53) takes the form (A-12), with probability 1 for the state C = Ci with VJf\Ci) = Uft = ( and probability 0 for the state C = C2 with ^ } ( C 2 ) = I ] . Here, w[C) V/Ci) J, and H>ijC) Vß(C2) add up to 1/β, and both are this time proportional to Uji9 as required by von Neumann, but the latter is proportional to Un only in a trivial way, by M4C) = 0, because Vji(C*) certainly is not proportional to Uß. We thus find no serious fault with von Neumann's mathematical results, but we must object against the conclusion he draws from these results.