Ward - Takahashi identities and Noether's theorem in quantum field theory

TL;DR

This paper introduces a rigorous variational calculus for operator fields in quantum field theory, demonstrating that classical results like Noether's theorem and Ward-Takahashi identities remain valid after quantization.

Contribution

It provides a mathematically rigorous framework for variational calculus in quantum field theory, clarifying the logical consistency of key symmetries and identities.

Findings

Naive classical results are confirmed to be correct in quantum context

Mathematically rigorous variational calculus for operator fields established

Noether's theorem and Ward-Takahashi identities hold in quantum field theory

Abstract

A gap in the mathematical logic in derivations in quantum field theory arises as consequence of variation before quantization. To close this gap the present paper introduces a mathematically rigorous variational calculus for operator fields. Using quantization before variation it is demonstrated that the so-called naive results are correct; in particular both Noether's theorem and the Ward-Takahashi identities retain full validity in quantum field theory.