An Analysis of the First Order Form of Gauge Theories

TL;DR

This paper analyzes the structure and gauge invariance of first order formulations of gauge theories, including Maxwell, U(1), and non-Abelian cases, using the Dirac procedure and exploring their constraint structures.

Contribution

It provides a detailed analysis of the gauge transformation structure and constraints in first order gauge theories, including massive and non-Abelian cases, and compares first and second order forms.

Findings

Derived gauge transformations from constraints.

Rewrote 3D massive gauge theory with interacting vector fields.

Explored the relationship between first and second order Einstein-Hilbert actions.

Abstract

The first order form of a Maxwell theory and U(1) gauge theory in which a gauge invariant mass term appears is analyzed using the Dirac procedure. The form of the gauge transformation which leaves the action invariant is derived from the constraints present. A non-Abelian generalization is similarly analyzed. This first order three dimensional massive gauge theory is rewritten in terms of two interacting vector fields. The constraint structure when using light-cone coordinates is considered. The relationship between first and second order forms of the two-dimensional Einstein-Hilbert action is explored where a Lagrange multiplier is used to ensure their equivalence.