Axiomatic unsharp quantum theory (From Mackey to Ludwig and Piron) |
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Authors: | Gianpiero Cattaneo Federico Laudisa |
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Institution: | (1) Dipartimento di Scienze dell'Informazione, Università di Milano, Via Comelico 39, I-20135 Milano, Italy;(2) Dipartimento di Filosofia, Università di Firenze, Via Bolognese 52, I-50139 Firenze, Italy |
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Abstract: | On the basis of Mackey's axiomatic approach to quantum physics or, equivalently, of a state-event-probability (SEVP) structure, using a quite standard fuzzification procedure, a set of unsharp events (or effects ) is constructed and the corresponding state-effect-probability (SEFP) structure is introduced. The introduction of some suitable axioms gives rise to a partially ordered structure of quantum Brouwer-Zadeh (BZ) poset; i.e., a poset endowed with two nonusual orthocomplementation mappings, a fuzzy-like orthocomplementation, and an intuitionistic-like orthocomplementation, whose set of sharp elements is an orthomodular complete lattice. As customary, by these orthocomplementations the two modal-like necessity and possibility operators are introduced, and it is shown that Ludwig's and Jauch-Piron's approaches to quantum physics are interpreted in complete SEFP. As a marginal result, a standard procedure to construct a lot of unsharp realizations starting from any sharp realization of a fixed observable is given, and the relationship among sharp and corresponding unsharp realizations is studied. |
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