Parametric Representation of Noncommutative Field Theory

TL;DR

This paper explores a new parametric representation for noncommutative quantum field theory, introducing hyperbolic polynomials that encode richer topological information than traditional polynomials.

Contribution

It develops a Schwinger parametric representation for noncommutative $ield{phi}^4_4$ theory using hyperbolic polynomials, advancing the mathematical tools for noncommutative quantum field theories.

Findings

Introduces hyperbolic polynomials as noncommutative analogs of Symanzik polynomials.

Provides a new representation that captures topological features of noncommutative Feynman amplitudes.

Enhances understanding of renormalizable noncommutative quantum field theories.

Abstract

In this paper we investigate the Schwinger parametric representation for the Feynman amplitudes of the recently discovered renormalizable $\phi^4_4$ quantum field theory on the Moyal non commutative ${\mathbb R^4}$ space. This representation involves new {\it hyperbolic} polynomials which are the non-commutative analogs of the usual "Kirchoff" or "Symanzik" polynomials of commutative field theory, but contain richer topological information.