Gauge Transformations, BRST Cohomology and Wigner's Little Group

TL;DR

This paper explores the relationship between gauge transformations, BRST cohomology, and Wigner's little group in various gauge theories, revealing connections and properties of gauge states and symmetries in different dimensions.

Contribution

It demonstrates the connection between gauge transformations and Wigner's little group through BRST cohomology in 2D and 4D Abelian gauge theories, highlighting their symmetry structures.

Findings

Gauge transformations relate to Wigner's little group parameters.

BRST cohomology classifies gauge states as exact or physical.

The 4D 2-form gauge theory exhibits quasi-topological features.

Abstract

We discuss the (dual-)gauge transformations and BRST cohomology for the two (1 + 1)-dimensional (2D) free Abelian one-form and four (3 + 1)-dimensional (4D) free Abelian 2-form gauge theories by exploiting the (co-)BRST symmetries (and their corresponding generators) for the Lagrangian densities of these theories. For the 4D free 2-form gauge theory, we show that the changes on the antisymmetric polarization tensor e^{\mu\nu} (k) due to (i) the (dual-)gauge transformations corresponding to the internal symmetry group, and (ii) the translation subgroup T(2) of the Wigner's little group, are connected with each-other for the specific relationships among the parameters of these transformation groups. In the language of BRST cohomology defined w.r.t. the conserved and nilpotent (co-)BRST charges, the (dual-)gauge transformed states turn out to be the sum of the original state and the (co-)BRST exact states. We comment on (i) the quasi-topological nature of the 4D free 2-form gauge theory from the degrees of freedom count on e^{\mu\nu} (k), and (ii) the Wigner's little group and the BRST cohomology for the 2D one-form gauge theory {\it vis-{\`a}-vis} our analysis for the 4D 2-form gauge theory.