[Paper Review] 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 Physics1 citations

TL;DR

This paper presents a highly efficient algorithm for computing the trace of the symmetrized product of n Dirac gamma matrices in 2n-dimensional spacetime, a critical step in constructing Chern-Simons gravity theories. By expressing the trace as a sum over integer partitions of n with computable coefficients, and using a recurrence relation, the method computes results in under one minute for n ≤ 25—dramatically outperforming existing tools.

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.

Motivation & Objective

• To address the computational bottleneck in calculating the trace of symmetrized products of Dirac gamma matrices for large n.
• To overcome the limitations of existing computer algebra systems that fail or become impractically slow for n > 10.
• To develop a scalable algorithm that computes the trace efficiently for applications in gauge-theoretic gravity models.
• To derive a recurrence relation between coefficients to accelerate computation of the trace formula.

Proposed method

• The trace is expressed as a sum over integer partitions s of n, with each term weighted by a numerical coefficient αs.
• The coefficients αs are computed by solving a linear system of equations derived from evaluating the trace formula with random tensor inputs B^{ab}.
• A recurrence relation between the coefficients αs is derived and exploited to reduce computational complexity.
• The algorithm is implemented in a minimal form that maintains high performance across all n ≤ 25.
• The method avoids symbolic explosion by relying on numerical evaluation and linear algebra rather than symbolic simplification.
• The approach is validated by testing the algorithm on random tensor configurations and confirming consistency with known results.

Experimental results

Research questions

• RQ1 How can the trace of the symmetrized product of n Dirac gamma matrices be computed efficiently for large n?
• RQ2 What is the structure of the coefficients αs in the partition-based expansion of the trace formula?
• RQ3 Can a recurrence relation between αs coefficients be derived to accelerate computation?
• RQ4 How does the proposed algorithm compare in performance to existing computer algebra systems for n > 10?
• RQ5 What is the maximum n for which the trace can be computed reliably and quickly using this method?

Key findings

• The algorithm computes the trace of the symmetrized product of n Dirac gamma matrices in under one minute for all n ≤ 25 on a standard desktop computer.
• The coefficients αs are determined by solving a linear system generated from random tensor inputs B^{ab}, ensuring numerical stability and correctness.
• A recurrence relation between αs coefficients is derived and used to significantly reduce computation time.
• The method avoids memory overflow and performance degradation seen in traditional computer algebra systems for large n.
• The approach is robust and scalable, covering all foreseeable applications in Chern-Simons gravity theories.
• The algorithm’s efficiency is demonstrated through runtime benchmarks, confirming practical usability for theoretical physics applications.