Topological Graph Polynomials and Quantum Field Theory, Part I: Heat Kernel Theories

TL;DR

This paper explores the connection between graph polynomials and quantum field theory, revealing how certain polynomials encode Feynman amplitudes and their extensions in noncommutative theories.

Contribution

It demonstrates that Symanzik polynomials are special cases of Tutte polynomials and introduces how noncommutative quantum field theory polynomials relate to Bollobás-Riordan polynomials.

Findings

Symanzik polynomials are multivariate Tutte polynomials.

Noncommutative QFT polynomials relate to Bollobás-Riordan polynomials.

Provides a unified topological framework for Feynman amplitude polynomials.

Abstract

We investigate the relationship between the universal topological polynomials for graphs in mathematics and the parametric representation of Feynman amplitudes in quantum field theory. In this first paper we consider translation invariant theories with the usual heat-kernel-based propagator. We show how the Symanzik polynomials of quantum field theory are particular multivariate versions of the Tutte polynomial, and how the new polynomials of noncommutative quantum field theory are particular versions of the Bollob\'as-Riordan polynomials.