The Star Product in Interacting Quantum Field Theory

We propose a new formula for the star product in deformation quantization of Poisson structures related in a specific way to a variational problem for a function $S$, interpreted as the action functional. Our approach is motivated by perturbative Algebraic Quantum Field Theory (pAQFT). We provide a direct combinatorial formula for the star product and we show that it can be applied to a certain class of infinite dimensional manifolds (e.g., regular observables in pAQFT). This is the first step towards understanding how pAQFT can be formulated such that the only formal parameter is $\hbar$, while the coupling constant can be treated as a number. In the introductory part of the paper, apart from reviewing the framework, we make precise several statements present in the pAQFT literature and recast these in the language of (formal) deformation quantization. Finally, we use our formalism to streamline the proof of perturbative agreement provided by Drago, Hack, and Pinamonti and to generalize some of the results obtained in that work to the case of a non-linear interaction.