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Journal of Mathematical Physics

Weak dispersion-free states and the hidden variables hypothesis

Pavel Pták

Published Apr 1, 1983• 14 citations
Publisher

Abstract

We investigate dispersion-free states which are additive only on the pairs containing a central element (central-absolutely compatible). We show that any logic possesses plenty of such states, in fact, as many as a certain Boolean algebra. The latter result matches the hidden variables conjecture.
DOI
10.1063/1.525758

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The question of the possible existence of hidden variables is re-examined. It is shown that hidden variables can exist only if every proposition (yes-no experiment) is compatible with every other one. It is further shown that this property is in contradiction with empirical facts. The theorem which leads to this conclusion is a strengthening of the theorem of von Neumann on the same subject. The question is raised whether there exist perhaps quantum mechanical systems which admit approximate dispersion-free states.
Introduction To know the quantum mechanical state of a system implies, in general, only statistical restrictions on the results of measurements. It seems interesting to ask if this statistical element be thought of as arising, as in classical statistical mechanics, because the states in question are averages over better defined states for which individually the results would be quite determined. These hypothetical ‘dispersion free’ states would be specified not only by the quantum mechanical state vector but also by additional ‘hidden variables’ – ‘hidden’ because if states with prescribed values of these variables could actually be prepared, quantum mechanics would be observably inadequate. Whether this question is indeed interesting has been the subject of debate. The present paper does not contribute to that debate. It is addressed to those who do find the question interesting, and more particularly to those among them who believe that ‘the question concerning the existence of such hidden variables received an early and rather decisive answer in the form of von Neumann's proof on the mathematical impossibility of such variables in quantum theory.’ An attempt will be made to clarify what von Neumann and his successors actually demonstrated. This will cover, as well as von Neumann's treatment, the recent version of the argument by Jauch and Piron, and the stronger result consequent on the work of Gleason. It will be urged that these analyses leave the real question untouched.