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## Banach Center Publications

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## The deformation quantisation mappingof Poisson to associative structuresin field theory

### Volume 113 / 2017

Banach Center Publications 113 (2017), 219-242 MSC: Primary 53D55, 58E30, 81S10;Secondary 53D17, 58Z05, 70S20. DOI: 10.4064/bc113-0-12

#### 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 $\boldsymbol{\mathcal{A}}$.

#### Authors

• Arthemy V. KiselevJohann Bernoulli Institute for Mathematics and Computer Science
University of Groningen
P.O. Box 407
9700 AK Groningen, The Netherlands
e-mail

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