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

Abstract

Let $\{\cdot ,\cdot {\}}_{\mathcal{P}}$ 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 $\mathcal{A}$ of local functionals $\Gamma (\pi )\to \mathbb{k}$ that take field configurations to numbers. By applying the techniques from geometry of iterated variations, we make well defined the deformation quantization map $\times \mapsto \star =\times +\hslash \{\cdot ,\cdot {\}}_{\mathcal{P}}+\bar{o}(\hslash )$ that produces a noncommutative $\mathbb{k}[[\hslash ]]$-linear star-product $\star$ in $\mathcal{A}$.

Proc. 50th Sophus Lie Seminar (26-30 September 2016, Bedlewo, Poland), 8 figures, 24 pages

References in corpus (2)

• The Kontsevich tetrahedral flow revisited
• A geometric derivation of KdV-type hierarchies from root systems

Cited by in corpus (2)

• The Kontsevich tetrahedral flow revisited
• The expansion $\star$ mod $\bar{o}({\hslash }^4)$ and computer-assisted proof schemes in the Kontsevich deformation quantization