Commutativity and Generalized Transition Probability in Quantum Logic

Abstract

In this paper we discuss the relation between commutativity and transition probability in quantum mechanics based on Hilbert space. To depict this relation, we shall use the framework of axiomatic quantum mechanics, leading to an orthomudular lattice corresponding to the lattice of closed subspaces of a Hilbert space and called the logic of quantum mechanics.^{1, 6} In Hilbert space quantum mechanics, if ϕ and ψ are unit vectors representing pure states, then (ϕ, ψ) is interpreted as the probability of transition between the states ψ and ϕ. On the other hand, if P _ϕ and P _ψ are two projection operators onto the subspaces generated by the vectors ϕ and ψ, respectively, then the commutativity or non-commutativity of P _ϕ and P _ψ is clearly connected with the transition probability between the states ϕ and ψ, which are eigenstates of P _ϕ and P _ψ , respectively. In general, two observables represented by self-adjoint operators A and B commute if they have common eigenvectors (for simplicity, we speak only of observables with pure point spectrum). If they do not commute, there is at least one pair ϕ, ψ of two eigenvectors, ϕ being an eigenstate of A and not B, and ψ being an eigenstate of B but not A, and during interchanging measurements of A and B there is a positive probability of transition between ϕ and ψ. Hence we see that if we wish to introduce the measure of non-commutativity, this measure should be related to transition probability.