Composition of states and observables in Fock spaces

TL;DR

This paper investigates the composition of states and observables in Fock spaces across anti-Wick, Wick, and Weyl quantizations, providing convergent series and asymptotic expansions relevant to quantum field theory.

Contribution

It introduces new convergent series and asymptotic expansions for compositions of states and operators in Fock space quantizations, enhancing mathematical understanding of quantum observables.

Findings

Wick symbol composition yields an absolutely convergent series.

Weyl symbol asymptotic expansion remainder is absolutely convergent.

Results are applicable to quantum electrodynamics and related fields.

Abstract

This article is concerned with compositions in the context of three standard quantizations in the Fock space framework, namely, anti-Wick, Wick and Weyl quantizations. The first one is a composition of states and is closely related to the standard scattering identification operator encountered in Quantum Electrodynamics for time dynamics issues. Anti-Wick quantization and Segal Bargmann transforms are implied there for that purpose. The other compositions are for observables (operators in some specific classes) for the Wick and Weyl symbols. For the Wick symbol of the composition of two operators, we obtain an absolutely converging series, and for the Weyl symbol, the remainder term of the asymptotic expansion is absolutely converging, still in the Fock spaces framework.