# The deformation quantization mapping of Poisson- to associative structures in field theory

## TL;DR

This paper constructs a deformation quantization map for Poisson structures in field theory, transforming the commutative algebra of local functionals into a noncommutative star-product algebra using geometric techniques.

## Contribution

It introduces a well-defined deformation quantization procedure for variational Poisson brackets in field theories, extending the geometric approach to iterated variations.

## Key findings

- Defined a star-product in the algebra of local functionals
- Established a deformation quantization map for Poisson structures
- Provided a geometric framework for quantization in field theory

## Abstract

Let $\{{\cdot},{\cdot}\}_{\boldsymbol{\mathcal{P}}}$ be a variational Poisson bracket in a field model on an affine bundle $\pi$ over an affine base manifold $M^m$. Denote by $\times$ the commutative associative multiplication in the Poisson algebra $\boldsymbol{\mathcal{A}}$ of local functionals $\Gamma(\pi)\to\Bbbk$ that take field configurations to numbers. By applying the techniques from geometry of iterated variations, we make well defined the deformation quantization map ${\times}\mapsto{\star}={\times}+\hbar\,\{{\cdot},{\cdot}\}_{\boldsymbol{\mathcal{P}}}+\bar{o}(\hbar)$ that produces a noncommutative $\Bbbk[[\hbar]]$-linear star-product $\star$ in $\boldsymbol{\mathcal{A}}$.