2021/11/11 by Oisin Kim, Kim, Oisin
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #math-ph #math.MP #math.QA
paper · pdf · doi:10.48550/arxiv.2111.06392
27 pages (including references), 7 figures
arxiv created 2021/11/11 · arxiv updated 2021/11/12
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 ℏ, so that the original pointwise product is recovered when ℏ=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 ℝn. 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.