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Il Nuovo Cimento A (1965-1970)

, Volume 47, Issue 4, pp 841–859 | Cite as

When can hidden variables be excluded in quantum mechanics?

  • B. Misra
Article

Summary

The problem of hidden variables is examined in the axiomatic formulation of quantum mechanics based on the algebra of observables. After a brief introductory survey of the earlier investigations, we first investigate the structure ofC*-algebras which allow dispersion-free positive linear functionals. The result obtained is a direct generalization of the well-known result of von Neumann concerning the hidden variables. In the next Section, we assume, as before, that the observables form the Hermitian elements of aC*-algebra. But we now relax the requirement on «states» and allow the so-called monotone-positive functionals (which are not necessarily linear) to represent states. It is then shown that even when such generalized states are allowed, a system admits hidden variables only if its algebra of observables is Abelian;i.e., only if all observables are mutually compatible. In another Section, we investigate the question of hidden variables under the assumption that the observables, instead of forming aC*-algebra, have a certain more general algebraic structure.

Keywords

Quantum Mechanic Hide Variable Tacit Assumption Deterministic Description Hermitian Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Riassunto

Si esamina il problema delle variabili nascoste nella formulazione assiomatica della meccanica quantistica basata sull'algebra degli osservabili. Dopo una breve rassegna introduttiva degli studi precedenti, si analizza dapprima la struttura delle algebreC* che consentono funzionali lineari positivi privi di dispersione. Il risultato ottenuto è una diretta generalizzazione del ben noto risultato di von Neumann riguardante le variabili nascoste. Nella successiva Sezione si suppone, come prima, che gli osservabili formino gli elementi hermitiani di un'algebraC*. Ma ora si indeboliscono le condizioni sugli «stati» e si lascia che i cosiddetti funzionali monotoni positivi (che non sono necessariamente lineari) rappresentino gli stati. Si dimostra allora che, anche quando sono ammessi questi stati generalizzati, un sistema ammette variabili nascoste solo se la sua algebra degli osservabili è abeliana, cioè solo se tutti gli osservabili sono mutuamente compatibili. In un'altra Sezione si studia la questione delle variabili nascoste nell'ipotesi che gli osservabili, invece di formare un'algebraC*, abbiano una certa struttura algebrica più generale.

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Copyright information

© Società Italiana di Fisica 1967

Authors and Affiliations

  • B. Misra
    • 1
  1. 1.Institute of Theoretical PhysicsUniversity of GenevaGeneva

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