Correlation Functions Involving Dirac Fields from Homotopy Algebras I: The Free Theory

TL;DR

This paper extends a homotopy algebra framework for scalar field correlation functions to include Dirac fields, demonstrating that the formula satisfies the Schwinger-Dyson equations in the free theory and relating it to factorization algebra approaches.

Contribution

It introduces a unified formula for correlation functions involving Dirac fields within the quantum $A_{ abla}$ algebra framework, matching the scalar case and satisfying key equations.

Findings

Correlation functions satisfy Schwinger-Dyson equations in the free theory.

The formula applies equally to scalar and Dirac fields.

Connection established with factorization algebra approach.

Abstract

We extend the formula for correlation functions of scalar field theories in terms of quantum $A_{\infty}$ algebras, presented in arXiv:2203.05366, to incorporate Dirac fields. We use a description that is analogous to string field theory, and the formula for correlation functions takes the same form for both scalar fields and Dirac fields. We prove that correlation functions from our formula satisfy the Schwinger-Dyson equations in the free theory. The proof for interacting theories is presented in the companion paper arXiv:2305.13103 by one of the authors. We also explain the relation of our formula to the definition of correlation functions in the approach by Costello and Gwilliam based on factorization algebras.