Renormalization of current algebra

TL;DR

This paper discusses the renormalization process of gauge currents in quantum field theories, highlighting differences in current algebra anomalies between 1+1 and 3+1 dimensions, and proposing a new renormalization scheme for interactions.

Contribution

It introduces a method for renormalizing gauge currents that accounts for anomalies and modifies local commutation relations, extending to spatial components and interaction Hamiltonians.

Findings

In 1+1 dimensions, the Schwinger term is a central extension independent of background fields.

In 3+1 dimensions, anomalies depend linearly on the background potential.

A new renormalization scheme for fermionic interactions is proposed.

Abstract

In this talk I want to explain the operator substractions needed to renormalize gauge currents in a second quantized theory. The case of space-time dimensions $3+1$ is considered in detail. In presence of chiral fermions the renormalization effects a modification of the local commutation relations of the currents by local Schwinger terms. In $1+1$ dimensions on gets the usual central extension (Schwinger term does not depend on background gauge field) whereas in $3+1$ dimensions one gets an anomaly linear in the background potential. We extend our method to the spatial components of currents. Since the bose-fermi interaction hamiltonian is of the form $j^k A_k$ (in the temporal gauge) we get a new renormalization scheme for the interaction. The idea is to define a field dependent conjugation for the fermi hamiltonian in the one-particle space such that after the conjugation the hamiltonian can be quantized just by normal ordering prescription.