Representations of the Renormalization Group as Matrix Lie Algebra

TL;DR

This paper presents a novel Lie algebra framework for renormalization using infinite triangular matrices, linking it to the Connes-Kreimer algebra and demonstrating its application in a three-loop scalar field theory example.

Contribution

It introduces a matrix Lie algebra approach to renormalization, connecting it to existing algebraic structures and exploring potential applications in physics and mathematics.

Findings

Matrices generate counterterms for Feynman diagrams with subdivergences.

The matrix Lie algebra relates to the Connes-Kreimer Lie algebra and Ihara brackets.

Validated in a three-loop scalar field theory example.

Abstract

Renormalization is cast in the form of a Lie algebra of infinite triangular matrices. By exponentiation, these matrices generate counterterms for Feynman diagrams with subdivergences. As representations of an insertion operator, the matrices are related to the Connes-Kreimer Lie algebra. In fact, the right-symmetric nonassociative algebra of the Connes-Kreimer insertion product is equivalent to an "Ihara bracket" in the matrix Lie algebra. We check our results in a three-loop example in scalar field theory. Apart from possible applications in high-precision phenomenology, we give a few ideas about possible applications in noncommutative geometry and functional integration.