Dirac Traces and the Tutte Polynomial

TL;DR

This paper establishes a link between Dirac matrix traces in quantum field theory and the Tutte polynomial of graphs, enabling improved algorithms for calculating these traces.

Contribution

It introduces a novel connection between Dirac traces and the Tutte polynomial, leading to more efficient computational methods.

Findings

Proven connection between Dirac traces and the Tutte polynomial.

Analysis of the computational complexity of Dirac trace calculations.

Proposed algorithmic improvements for computing Dirac traces.

Abstract

Perturbative calculations involving fermion loops in quantum field theories require tracing over Dirac matrices. A simple way to regulate the divergences that generically appear in these calculations is dimensional regularisation, which has the consequence of replacing 4-dimensional Dirac matrices with d-dimensional counterparts for arbitrary complex values of d. In this work, a connection between traces of d-dimensional Dirac matrices and computations of the Tutte polynomial of associated graphs is proven. The time complexity of computing Dirac traces is analysed by this connection, and improvements to algorithms for computing Dirac traces are proposed.