The deformation quantisation mappingof Poisson to associative structuresin field theory
Tom 113 / 2017
Banach Center Publications 113 (2017), 219-242
MSC: Primary 53D55, 58E30, 81S10;Secondary 53D17, 58Z05, 70S20.
DOI: 10.4064/bc113-0-12
Streszczenie
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 $\boldsymbol{\mathcal{A}}$.
