A unified framework for the algebra of unsharp quantum mechanics. (English) Zbl 0898.03022
The aim of the paper is to provide a common general framework for unsharp algebras for both classical and quantum propositional logics. For this purpose very general structures, so-called sum Brouwer-Zadeh algebras (SBZ-algebras) are introduced. Examples for such algebras are the real unit interval \([0,1]\) with truncated sum, the algebra of all fuzzy subsets of a fixed set, measurable spaces, the algebra of all effect operators on a Hilbert space and the algebra of all closed subspaces of a Hilbert space. Also structures similar to SBZ-algebras as well as observables, states, different kinds of orthogonality, sharp and unsharp elements, connections to logic and quantum and classical SBZ-algebras of effects are investigated. It is shown that any unsharp element can be approximated by a pair of sharp elements, one of them approximating the unsharp element from the bottom and the other from the top.
Reviewer: H.Länger (Wien)
MSC:
| 03G12 | Quantum logic |
| 81P10 | Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) |
| 06C15 | Complemented lattices, orthocomplemented lattices and posets |
Keywords:
sum Brouwer-Zadeh algebra; fuzzy set; measurable space; effect operator; Hilbert space; closed subspace; observable; state; orthogonality; unsharp algebras; unsharp elements; sharp elementsReferences:
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