Schwinger terms in 1+1 dimensions

TL;DR

This paper compares two methods for calculating Schwinger terms in 1+1 dimensional current algebras, examines the Jacobi identity's validity, and links Schwinger terms to anomalies, highlighting dimensional differences.

Contribution

It provides a detailed comparison of Källen's and Brandt's methods for Schwinger term calculations and explores the role of regularization and anomalies in 1+1 dimensions.

Findings

Jacobi identity holds in 1+1 dimensions with certain regularizations

Schwinger terms are connected to anomalies in the Schwinger model

Regularization issues differ between 1+1 and 3+1 dimensions

Abstract

The two different approaches - K\"{a}llen's and Brandt's methods - for the calculations of the Schwinger terms in the 1+1 dimensional Abelian and non-Abelian free current algebras are discussed. These methods are applied to calculation of the single and double commutators. The validity of the Jacobi identities is examined in 1+1 and 3+1 dimensions and in this way is given natural restriction on the regularization. It is shown that the Jacobi identity cannot be broken in 1+1 dimensions even using the regularization which fails in the 3+1 dimensional case. A connection between the Schwinger term and anomaly is shown in the simplest case of the Schwinger model.