[论文解读] An efficient algorithm for the computation of the trace of the symmetrized product of an arbitrary number of Dirac matrices with two indices

Fernando Izaurieta, R. Ramirez-Gomez+1 more|arXiv (Cornell University)|Jun 8, 2011

Black Holes and Theoretical Physics被引用 1

一句话总结

本文提出了一种高度高效的算法，用于计算2n维时空中n个狄拉克伽马矩阵对称乘积的迹，这是构建陈-西蒙斯引力理论的关键步骤。通过将迹表示为n的整数分拆之和，其中各项具有可计算的系数，并利用递推关系，该方法在n ≤ 25时可在一分钟内完成计算——显著优于现有工具。

ABSTRACT

A genuine gauge theory for the Poincare, de Sitter or anti-de Sitter algebras can be constructed in (2n-1)-dimensional spacetime by means of the Chern-Simons form, yielding a gravitational theory that differs from General Relativity but shares many of its properties, such as second order field equations for the metric. The particular form of the Lagrangian is determined by a rank n, symmetric tensor invariant under the relevant algebra. In practice, the calculation of this invariant tensor can be reduced to the computation of the trace of the symmetrized product of n Dirac Gamma matrices \Gamma_{ab} in 2n-dimensional spacetime. While straightforward in principle, this calculation can become extremely cumbersome in practice. For large enough n, existing computer algebra packages take an inordinate long time to produce the answer or plainly fail having used up all available memory. In this talk we show that the general formula for the trace of the symmetrized product of 2n Gamma matrices \Gamma_{ab} can be written as a certain sum over the integer partitions s of n, with every term being multiplied by a numerical coefficient \alpha_{s}. We then give a general algorithm that computes the \alpha-coefficients as the solution of a linear system of equations generated by evaluating the general formula for different sets of tensors B^{ab} with random numerical entries. A recurrence relation between different coefficients is shown to hold and is used in a second, algorithm to greatly speed up the computations. Runtime of the minimal algorithm stays below 1 min on a typical desktop computer for up to n=25, which easily covers all foreseeable applications of the trace formula.

研究动机与目标

• 解决在大n下计算狄拉克伽马矩阵对称乘积迹时的计算瓶颈问题。
• 克服现有计算机代数系统在n > 10时失效或变得极不切实际的性能问题。
• 开发一种可扩展的算法，高效计算规范理论引力模型中的迹。
• 推导系数之间的递推关系，以加速迹公式的计算。

提出的方法

• 迹被表示为n的整数分拆s的和，每一项乘以一个数值系数αs。
• 系数αs通过求解从随机张量输入B^{ab}代入迹公式所得的线性方程组来计算。
• 推导并利用系数αs之间的递推关系，以降低计算复杂度。
• 该算法以极简形式实现，确保在所有n ≤ 25下保持高性能。
• 通过依赖数值计算和线性代数而非符号简化，避免了符号爆炸问题。
• 通过在随机张量配置上测试算法，并与已知结果保持一致，验证了该方法的正确性。

实验结果

研究问题

• RQ1 如何高效计算大n下n个狄拉克伽马矩阵对称乘积的迹？
• RQ2 迹公式基于分拆展开时，系数αs的结构是什么？
• RQ3 能否推导出αs系数之间的递推关系以加速计算？
• RQ4 该算法在n > 10时与现有计算机代数系统相比性能如何？
• RQ5 该方法可可靠且快速计算的最大n值是多少？

主要发现

• 该算法在标准台式计算机上，对所有n ≤ 25，计算狄拉克伽马矩阵对称乘积的迹时间少于一分钟。
• 系数αs通过从随机张量输入B^{ab}生成的线性系统求解得到，确保了数值稳定性和正确性。
• 推导并使用了αs系数之间的递推关系，显著减少了计算时间。
• 该方法避免了传统计算机代数系统在大n时出现的内存溢出和性能下降问题。
• 该方法具有鲁棒性和可扩展性，覆盖了陈-西蒙斯引力理论中所有可预见的应用。
• 通过运行时间基准测试，验证了算法的高效性，证实其在理论物理应用中的实际可用性。