# Non-abelian gauge symmetry for fields in phase space: a realization of the Seiberg-Witten non-abelian gauge theory

## TL;DR

This paper extends the Seiberg-Witten non-abelian gauge theory to phase space, formulating an SU(2) gauge field for the Dirac equation and exploring its application to nucleons.

## Contribution

It introduces a non-abelian gauge formalism in phase space for the Dirac field, generalizing the Seiberg-Witten approach to SU(2) symmetry.

## Key findings

- Derived a non-abelian field strength in phase space.
- Formulated an SU(2) gauge theory for the Dirac field in phase space.
- Discussed potential applications to nucleon physics.

## Abstract

The Seiberg-Witten formalism has been realized as an electrodynamics in phase space (associated to the Dirac equation written in phase space) and this fact is explored here with non-abelian gauge group. First, a physically heuristic presentation of the Seiberg-Witten approach is carried out for non-abelian gauge in order to guide the calculation procedures. These results are realized by starting with the Lagrangian density for the free Dirac field in phase space. Then a field strength is derived, where the non-abelian gauge group is the SU(2), corresponding to an isospin (non-abelian) field theory in phase space. An application to nucleon is then discussed.