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Motivated by some well-known results in the phase space description ofquantum optics and quantum information theory, we aim to describe the formalismof quantum field theory by explicitly considering the holomorphicrepresentation for a scalar field within the deformation quantization program.Notably, the symbol of a symmetric ordered operator in terms of holomorphicvariables may be straightforwardly obtained by the quantum field analogue ofthe Husimi distribution associated with a normal ordered operator. Thisrelation also allows establishing a $c$-equivalence between the Moyal and thenormal star-products. In addition, by writing the density operator in terms ofcoherent states we are able to directly introduce a series representation ofthe Wigner functional distribution which may be convenient in order tocalculate probability distributions of quantum field observables withoutperforming formal phase space integrals at all.

Coherent representation of fields and deformation quantization