A Schwinger term in q-deformed su(2) algebra

TL;DR

This paper identifies an additional Schwinger-like term in the q-deformed su(2) algebra arising from the positive norm requirement, impacting the algebra's structure and its application to the Bloch electron problem.

Contribution

It introduces a new Schwinger term in the q-deformed su(2) algebra resulting from combining specific algebraic constructions with positive norm constraints.

Findings

Extra term appears in q-deformed su(2) algebra for q=exp(2πiθ)

The term is analogous to the Schwinger term in current algebra

Implications discussed for the Bloch electron problem

Abstract

An extra term generally appears in the q-deformed $su(2)$ algebra for the deformation parameter $q = \exp{ 2 \pi i\theta}$, if one combines the Biedenharn-Macfarlane construction of q-deformed $su(2)$, which is a generalization of Schwinger's construction of conventional $su(2)$, with the representation of the q-deformed oscillator algebra which is manifestly free of negative norm. This extra term introduced by the requirement of positive norm is analogous to the Schwinger term in current algebra. Implications of this extra term on the Bloch electron problem analyzed by Wiegmann and Zabrodin are briefly discussed.