A study of measures on the splitting subspaces of an inner product space

Abstract

A.N. Kolmogorov's 1933 paper, Foundations of the Theory of Probability l25] provided an axiomatic formulation of what is today considered classical measure theory. This formulation is based on a O"-algebra of subsets representing the set of verifiable assertions of a physical system. If implication is represented by set-inclusion, then the O"-algebra is a Boolean logic with conjunction and disjunction represented by intersection and union respectively, while negation b iep1esenteJ by set complementation. Probabilities are assigned to the assertions of the system via a probability measure on the O"-algebra. With the aim of providing a theoretical framework for quantum mechanics, in 1936, Birkhoff and von Neumann published The logic of quantum mechanics [2]. Von Neumann observed that {O, 1}-valued observables can be regarded as the assertions about the physical system. Each such assertion would thus correspond to a self djoint operator on a Hilbert space H having the two-point spectrum {O, 1 }, and so would be a projection operator. Due to the one-to-one correspondence between projection operators and closed subspaces of H, the set of assertions can be identified with the set of closed subspaces of H, denoted L(H). With set inclusion as implication and complementation A f-----7 A..L, L(H) forms a O"-complete orthomodularposet. Analogous to probability measures on o--algebras, probability valuations of each· assertion of the system are given by states, that is, functions s : L(H) -. [O, 1] such that for any collection {An: n EN} of pairwise orthogonal elements of L(H)