Quantum Noether Method

TL;DR

This paper introduces a quantum Noether method for constructing consistent quantum field theories with global symmetries, using a causal approach that avoids regularization and handles massless theories effectively.

Contribution

It develops a quantum version of Noether's method within the causal perturbation framework, enabling the systematic construction of symmetric quantum theories without anomalies.

Findings

Constructed pure Yang-Mills theory with BRST symmetry.

Built the N=1 supersymmetric Wess-Zumino model.

Ensured symmetry and unitarity without anomalies in massless theories.

Abstract

We present a general method for constructing consistent quantum field theories with global symmetries. We start from a free non-interacting quantum field theory with given global symmetries and we determine all consistent perturbative quantum deformations assuming the construction is not obstructed by anomalies. The method is established within the causal Bogoliubov-Shirkov-Epstein-Glaser approach to perturbative quantum field theory (which leads directly to a finite perturbative series and does not rely on an intermediary regularization). Our construction can be regarded as a direct implementation of Noether's method at the quantum level. We illustrate the method by constructing the pure Yang-Mills theory (where the relevant global symmetry is BRST symmetry), and the N=1 supersymmetric model of Wess and Zumino. The whole construction is done before the so-called adiabatic limit is taken. Thus, all considerations regarding symmetry, unitarity and anomalies are well-defined even for massless theories.