Kontsevich's deformation quantization: from Dirac to multiple zeta values

TL;DR

This paper explains Kontsevich's deformation quantization for Poisson manifolds, highlighting its connections to multiple zeta values and recent mathematical developments, with an emphasis on the underlying structures and implications.

Contribution

It provides an accessible explanation of Kontsevich's deformation quantization and explores its novel links to multiple zeta values and contemporary research.

Findings

Kontsevich's formula for deformation quantization of Poisson structures.

Connections between deformation quantization and multiple zeta values.

Original calculations illustrating the mathematical structures involved.

Abstract

One way of reconciling classical and quantum mechanics is deformation quantization, which involves deforming the commutative algebra of functions on a Poisson manifold to a non-commutative, associative algebra, reminiscent of the space of quantum observables. This depends on a formal parameter $\hbar$, so that the original pointwise product is recovered when $\hbar=0$. In 1997 Kontsevich showed that a deformation quantization exists for every Poisson manifold. He furthermore gave a simple, combinatorial formula for producing a quantization of any Poisson structure on $\mathbb{R}^n$. The primary aim of this essay, largely drawn from the author's MMath dissertation at Oxford, is to present and explain Kontsevich's results. Starting with the motivation, we discuss how the problem is solved by situating it in a richer mathematical structure, performing a few original calculations along the way. We hope to communicate a sense of the strange links between this subject and seemingly distant areas of mathematics, and also to describe some of the contemporary research in the field. To these ends, we consider a recent paper, arXiv:1812.11649 [math.QA], which connects deformation quantization to multiple zeta values.