Triality and Dual Equivalence Between Dirac Field and Topologically Massive Gauge Field

TL;DR

This paper demonstrates a novel triality and duality between Dirac spinor fields and topologically massive gauge fields, extending Cartan's triality concept to Dirac spinors and establishing a dual Lagrangian relationship.

Contribution

It introduces a vector representation of Dirac spinors equivalent to biquaternions and generalizes triality to Dirac fields, linking Dirac and gauge theories via dual Lagrangians.

Findings

Vector representation of Dirac spinors as biquaternions.

Dual equivalence between Dirac Lagrangian and gauge field Lagrangian.

Identification of a self-dual gauge field with combined Lie and Jordan structures.

Abstract

It is proved that there exist a vector representation of Dirac's spinor field and in one sense it is equivalent to biquaternion (i.e. complexified quaternion) representation. This can be considered as a generalization of Cartan's idea of triality to Dirac's spinors. In the vector representation the first order Dirac Lagrangian is dual equivalent to the two order Lagrangian of topologically massive gauge field. The potential field which corresponds to the Dirac field is obtained by using master (or parent) action approach. The novel gauge field is self-dual and contains both anti-symmetric Lie and symmetric Jordan structure.