Dirac-K\"Ahler Equation

TL;DR

This paper explores the mathematical structure of the Dirac-K"ahler equation using tensor, matrix, and quaternion formulations, revealing its symmetry group, conservation laws, and potential for gauge models.

Contribution

It introduces a simplified matrix formulation, identifies the symmetry group SO(4,2), and investigates supersymmetry and gauge interactions of Dirac-K"ahler fields.

Findings

Symmetry group of Dirac-K"ahler tensor fields is SO(4,2)

Constructed conservation currents for these fields

Showed the possibility of a gauge model with a noncompact gauge group

Abstract

Tensor, matrix and quaternion formulations of Dirac-K\"ahler equation for massive and massless fields are considered. The equation matrices obtained are simple linear combinations of matrix elements in the 16-dimensional space. The projection matrix-dyads defining all the 16 independent equation solutions are found. A method of computing the traces of 16-dimensional Petiau-Duffin-Kemmer matrix product is considered. We show that the symmetry group of the Dirac-K\"ahler tensor fields for charged particles is SO(4,2). The conservation currents corresponding this symmetry are constructed. We analyze transformations of the Lorentz group and quaternion fields. Supersymmetry of the Dirac-K\"ahler fields with tensor and spinor parameters is investigated. We show the possibility of constructing a gauge model of interacting Dirac-K\"ahler fields where the gauge group is the noncompact group under consideration.