On relationship of gauge transformation with Wigner's little group

TL;DR

This paper explores how the Wigner little group for massless particles relates to gauge transformations, extending from abelian to non-abelian cases by including unphysical modes and analyzing their role in gauge symmetry realization.

Contribution

It generalizes the connection between Wigner's little group and gauge transformations from abelian to non-abelian cases, incorporating unphysical modes and physical state conditions.

Findings

Non-abelian gauge transformations can be derived from the little group translations.

Unphysical modes are essential for realizing non-abelian gauge symmetries.

Results are applicable in any spacetime dimension.

Abstract

Wigner's little group of a massless particle is ISO(2) which contains rotation and two translations. As well-known, eigenvalues of the rotation are helicity. On the other hand, by S. Weinberg et al., it has been shown that two translations generate abelian gauge transformation by acting on polarization vectors. In this paper, we include unphysical modes and show abelian case result can be generalized to the case of non-abelian gauge transformation. By including the unphysical modes, we obtain Nakanishi-Lautrup physical state condition from the requirement of unitarity of the transformation. As a result, non-abelian gauge transformation is realized as the translation of the little group which acts on gauge group. We also obtain similar results for any spacetime dimensions.