Generalized Fock space and contextuality

Abstract

This paper is devoted to linear space representations of contextual probabilities -in generalized Fock space. This gives the possibility to use the calculus of creation and annihilation operators to express probabilistic dynamics in the Fock space (in particular, the wide class of classical kinetic equations). In this way we reproduce the Doi-Peliti formalism. The contextdependence of probabilities can be quantified with the aid of the generalized formula of total probabilityby the magnitude of the interference term. Keywords: Fock space; Doi-Peliti formalism; quantum probability; contextuality; interference of probabilities; formula of total probability; calculi of creation-annihilation operators, classical versus quantum dynamics, kinetic equation We recall this notion. Let a, b, c be three quantum observables represented by Hermitian operators A, B,C. Let a is compatible with b and with c, i.e., [A, B]=0 and [A, C]=0. If the outputs of the a-measurements depend on whether it is measured jointly with b or with c, this phenomenon is called contextuality.The contextuality notion explored in our paper goes back to the Copenhagen interpretation and Bohr's complementarity principle. By this principle all components of the experimental arrangement should be taken into account: outputs of measurements depend on the complete experimental context. It is impossible to split contributions of a system under measurement and the experimental context. 1 We can call this sort of contextuality "Bohr-like contextuality". This notion is more general than Bell-like contextuality. The latter is reduced to the class of contexts determined by observables compatible with an observable a. Rising-lowering (creation-annihilation) operators and Fock spaceConsider the abstract Fock space, which is given by the basis | ⟩, where are integers. The dual space to it is given by the basis ⟨ |. By definition, the inner product of orts satisfies the condition