Abstract
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 nonlinear interaction.
Highlights
Constructing interacting quantum field theory (QFT) models in 4 dimensions is one of the most important challenges facing modern theoretical physics
We have derived an explicit formula for the deformation quantization of a general class of infinite-dimensional Poisson manifolds
We have investigated the relation between our formula and the Kontsevich formula
Summary
Constructing interacting quantum field theory (QFT) models in 4 dimensions is one of the most important challenges facing modern theoretical physics. There, the theory was formulated in terms of formal power series in and the coupling constant λ In this (perturbative) approach, the starting point is an affine configuration space with an action functional that is split into a preferred quadratic (free) part and the remainder. What we achieve here is a construction of the interacting star product that depends only on the full action, the causal structure, and the affine structure of the configuration space This is a desirable result from the pAQFT perspective, since in many situations the split into free and interacting theory is unnatural and the physical results should not depend on this split. We discuss the relation to the formula of Kontsevich, provide some useful formulae for the quantum Møller operator, and we discuss the principle of perturbative agreement
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