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arXiv:quant-ph/0610252 · AIP conference proceedings

On Hysteresis of Value Assignments to Observables during Changes of Context

Satoshi Uchiyama

Published Jan 1, 2007• 0 citations
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Abstract

The functional composition principle is generalized by taking into account of history of context change. Analysis of Peres’ example shows hysteresis of value assignments. It is shown that value assignments depending on the history of context change are possible in the case that the Hilbert space of state vectors is finite dimensional.
DOI
10.1063/1.2713490

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Recently, Schreckenberg investigated symmetries in the context of history quantum theories. In the case that the space of histories is given by the set of projectors on some finite dimensional Hilbert space he obtained, first, a complete characterization of all physical symmetries of history quantum theories — an analog of Wigner’s theorem — and, second, a complete mathematical characterization of symmetries of single decoherence functionals. In this paper we extend Schreckenberg’s results to the case where the underlying space of histories is given by the set of projectors on some infinite dimensional Hilbert space.
In order to deal with Kochen and Specker's no-go result, van Fraassen invoked 'splitting' of non-maximal observable depending on 'context'. In his contextual approach there are different physical quantities corresponding to a non-maximal operator. Such physical quantities are different, so those values are not required to coincide. However, Heywood and Redhead showed the impossibility of local truth-value assignment in contextual approach. Their argument is restricted to finite dimensional and non-relativistic case. In this paper we shall show a similar no-go result in more general framework (von Neumann algebra). Also, our result will be applied to Algebraic quantum field theory.
The semantics underlying the propositional structure of Hilbert space quantum mechanics involves an inherent ambiguity concerning the impossibility of assigning definite truth values to all propositions pertaining to a quantum system without generating a Kochen-Specker contradiction. Although the preceding Kochen-Specker result forbids a global, absolute assignment of truth values to quantum mechanical propositions, it does not exclude ones that are contextual. In this respect, the Bub-Clifton “uniqueness theorem” is utilized for arguing that truth-value definiteness is consistently restored with respect to a determinate sublattice of propositions defined by the state of the quantum system concerned and a suitable “preferred” observable. It is suggested that the most natural choice of the latter, especially for confronting the problem of truth valuation in quantum mechanics, results in the determinateness of the observable to be measured. An account of truth of contextual correspondence is thereby provided that is appropriate to the quantum domain of discourse. The resulting account of truth is compatible with a realist conception of truth. Such an account essentially denies that there can be a universal context of reference or an Archimedean standpoint from which to evaluate logically the totality of facts of nature.
Within the decoherent histories formulation of quantum mechanics, we consider arbitrarily long histories constructed from a fixed projective partition of a finite‐dimensional Hilbert space. We review some of the decoherence properties of such histories including simple necessary decoherence conditions and the dependence of decoherence on the initial state. Here we make a first step towards generalization of our earlier results [Scherer and Soklakov, e‐print: quant‐ph/0405080, (2004) and Scherer et al., Phys. Lett. A 326, 307, (2004)] to the case of approximate decoherence.