# Obtaining non-Abelian field theories via Faddeev-Jackiw symplectic formalism

## TL;DR

This paper demonstrates a systematic method to derive non-Abelian gauge theories, such as SU(2) and SU(2)xU(1), from Abelian theories using the Faddeev-Jackiw symplectic formalism, highlighting a novel approach to gauge theory construction.

## Contribution

It introduces a new systematic procedure to construct non-Abelian field theories from Abelian ones via the Faddeev-Jackiw formalism, including the implementation of gauge symmetries.

## Key findings

- Successfully constructed SU(2) and SU(2)xU(1) Yang-Mills theories from U(1) Maxwell theory.
- Showed how to incorporate non-Abelian gauge symmetries using the Faddeev-Jackiw method.
- Provided a step-by-step framework for deriving non-Abelian theories systematically.

## Abstract

In this work we have shown that it is possible to construct non-Abelian field theories employing, in a systematic way, the Faddeev-Jackiw symplectic formalism. This approach follows two steps. In the first step, the original Abelian fields are modified in order to introduce the non-Abelian algebra. After that, the Faddeev-Jackiw method is implemented and the gauge symmetry relative to some non-Abelian symmetry group, is introduced through the zero-mode of the symplectic matrix. We construct the SU(2) and SU(2)xU(1) Yang-Mills theories having as starting point the U(1) Maxwell electromagnetic theory.