Perspectives

Abstract

PerspectivesIsomorphic representation of Birkhoff-von Neumann quantum logics, and therefore also of orthomodular lattices L(H) of (orthogonal projections onto) closed subspaces of Hilbert spaces by families of fuzzy sets endowed with Łukasiewicz operations opens new opportunities for solving at least two long-standing problems, namely the development of quantum probability calculus in a way completely analogous to the orthodox Kolmogorovian probability theory, and the construction of a phase space representation of quantum mechanics not plagued by the appearance of negative probabilities. However, it should be stressed that what we present here is only a brief prospect for future studies which will certainly require much further investigation. Fuzzy Set Models of Quantum ProbabilityIn some experiments on quantum systems the relative frequencies of obtaining various results, interpreted as probabilities, do not fulfil the numerical constraints imposed by classical (Kolmogorovian) probability theory. Such instances, usually connected with the violation of Bell's inequalities, strongly indicate the necessity of modification of the probability calculus used in quantum mechanics.There are several approaches to the subject that can generally be termed "quantum probability" and even the brief review of all of these would lead us far beyond the scope of this section. Therefore, we shall concentrate on the quantum-logical treatment of this subject.In the quantum logic approach to the foundations of quantum mechanics the Kolmogorovian triple ( , F, P) consisting of a space of elementary events , a Boolean σ-algebra F of selected subsets of (random events), and a probability measure P, is replaced by a couple (L , p) consisting of a σ-orthocomplete orthomodular poset (i.e. quantum logic) L and a probability measure (state) p defined on L. It follows from the very definition (see Sect. 6.1) that probability measures on quantum logics satisfy all numerical constraints imposed on Kolmogorovian probability measures: they are nonnegative, normalized, and σ-additive on families of