Perturbative Quantum Field Theory via Vertex Algebras

TL;DR

This paper formulates perturbative quantum field theory using vertex algebras derived from the operator product expansion, providing a new algebraic perspective and practical computational methods for interactions.

Contribution

It introduces a vertex algebra framework for perturbative QFT based on OPE coefficients and develops a deformation theory approach for calculating perturbations.

Findings

Repackaging OPE coefficients into vertex operators

Derivation of graphical rules for vertex operators

Application to interactions like b1 b4^4"

Abstract

In this paper, we explain how perturbative quantum field theory can be formulated in terms of (a version of) vertex algebras. Our starting point is the Wilson-Zimmermann operator product expansion (OPE). Following ideas of a previous paper [arXiv:0802.2198], we consider a consistency (essentially associativity) condition satisfied by the coefficients in this expansion. We observe that the information in the OPE coefficients can be repackaged straightforwardly into "vertex operators" and that the consistency condition then has essentially the same form as the key condition in the theory of vertex algebras. We develop a general theory of perturbations of the algebras that we encounter, similar in nature to the Hochschild cohomology describing the deformation theory of ordinary algebras. The main part of the paper is devoted to the question how one can calculate the perturbations corresponding to a given interaction Lagrangian (such as $\lambda \varphi^4$) in practice, using the consistency condition and the corresponding non-linear field equation. We derive graphical rules, which display the vertex operators (i.e., OPE coefficients) in terms of certain multiple series of hypergeometric type.