AXIOMATIC UNSHARP QUANTUM-THEORY (FROM MACKEY TO LUDWIG AND PIRON)

On the basis of Mackeys 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.