Software modules and computer-assisted proof schemes in the Kontsevich deformation quantization
Software modules and computer-assisted proof schemes in the Kontsevich deformation quantization
复制标题
Kontsevich变形量化中的软件模块和计算机辅助证明方案
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
A. Kiselev
中科院分区：
文献类型：
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作者：
R. Buring;A. Kiselev
Kontsevich 变形量化结合了泊松动力学、非交换几何、数论和定向图微积分。为了管理一系列加权图的代数和微分演算，我们提出了软件模块：这些模块允许生成 Kontsevich 图，通过使用先验未确定的系数来扩展非交换 $star$-产品，并导出图权重之间的线性关系。在本文中，我们说明了变形参数 $hbar$ 中 Kontsevich $star$ 乘积高达 4 阶的组装。在此阶段，$star$ 产品已涉及数百个图表；通过基本图的 149 个权重（其中 67 个权重现在已准确知道）来表达它们的所有系数，我们仅用 10 个参数来表示其余 82 个权重（更具体地说，仅用模规范等价的 6 个参数来表示）。最后，我们为新构建的 $star$ 乘积展开概述了计算机辅助关联性证明的方案，模 $ ar{o}(hbar^4)$。
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 $star$-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 $star$-product up to order 4 in the deformation parameter $hbar$. Already at this stage, the $star$-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 $ar{o}(hbar^4)$, for the newly built $star$-product expansion.