Abstract

The Kontsevich deformation quantization combines Poisson dynamics, noncommutative geometry, number theory, and calculus of oriented graphs. To manage the algebra and differential calculus of series of weighted graphs, we present software modules: these allow generating the Kontsevich graphs, expanding the noncommutative ⋆-product by using a priori undetermined coefficients, and deriving linear relations between the weights of graphs. Throughout this text we illustrate the assembly of the Kontsevich ⋆-product up to order 4 in the deformation parameter $\hslash .$ Already at this stage, the ⋆-product involves hundreds of graphs; expressing all their coefficients via 149 weights of basic graphs (of which 67 weights are now known exactly), we express the remaining 82 weights in terms of only 10 parameters (more specifically, in terms of only 6 parameters modulo gauge-equivalence). Finally, we outline a scheme for computer-assisted proof of the associativity, modulo $\overline{o}({\hslash }^4),$ for the newly built ⋆-product expansion.

摘要

Kontsevich 变形量化结合了泊松动力学、非交换几何、数论和有向图的微积分。为了管理一系列加权图的代数和微积分，我们提出了软件模块：这些模块允许生成 Kontsevich 图，通过使用先验未定系数来扩展非交换 ⋆ 积，并推导图权重之间的线性关系。在整个文本中，我们说明了 Kontsevich ⋆ 乘积在变形参数中高达 4 阶的组装$\hslash .$在这个阶段，⋆-product 已经包含了数百个图；通过基本图的 149 个权重（其中 67 个权重现在确切知道）来表达它们的所有系数，我们仅用 10 个参数（更具体地说，仅用 6 个模规范等效参数）来表达剩余的 82 个权重。最后，我们概述了关联性的计算机辅助证明方案，模$\overline{○}({\hslash }^4),$用于新建 ⋆ 产品扩展。