Noether's theorem and Ward-Takahashi identities from homotopy algebras

TL;DR

This paper establishes a new identity in homotopy algebras that corresponds to Schwinger-Dyson equations, deriving Ward-Takahashi identities and addressing anomalies through regularization techniques.

Contribution

It introduces a novel identity linking homotopy algebras to quantum field theory equations, providing a new framework for understanding gauge identities and anomalies.

Findings

Derived Ward-Takahashi identities from homotopy algebra identities.

Demonstrated the identities in several quantum field theory examples.

Addressed divergence issues using string field theory stubs and regularization.

Abstract

We derive the new identity in homotopy algebras which directly corresponds to the Schwinger-Dyson equations in quantum field theory. As an application, we derive the Ward-Takahashi identities. We demonstrate that the Ward-Takahashi identities are reproduced in several examples. In general, our formula contains divergence. We mediate this problem by introducing stubs known in the context of string field theory. With the regularization, we can calculate the anomaly such as axial U(1) anomaly in vector-like U(1) gauge theory.