Coherent representation of fields and deformation quantization

TL;DR

This paper develops a holomorphic representation for quantum fields within deformation quantization, linking phase space distributions like Husimi and Wigner functionals to simplify quantum observable calculations.

Contribution

It introduces a holomorphic formalism for quantum fields in deformation quantization, connecting star-products and phase space distributions explicitly.

Findings

Established a relation between Husimi distribution and symmetric operator symbols.

Derived a $c$-equivalence between Moyal and normal star-products.

Presented a series representation of the Wigner functional for quantum fields.

Abstract

Motivated by some well-known results in the phase space description of quantum optics and quantum information theory, we aim to describe the formalism of quantum field theory by explicitly considering the holomorphic representation for a scalar field within the deformation quantization program. Notably, the symbol of a symmetric ordered operator in terms of holomorphic variables may be straightforwardly obtained by the quantum field analogue of the Husimi distribution associated with a normal ordered operator. This relation also allows establishing a $c$-equivalence between the Moyal and the normal star-products. In addition, by writing the density operator in terms of coherent states we are able to directly introduce a series representation of the Wigner functional distribution which may be convenient in order to calculate probability distributions of quantum field observables without performing formal phase space integrals at all.