Dirac-K\"ahler Equation (Review)

TL;DR

This review explores the tensor and matrix formulations of the Dirac-K"ahler equation, analyzing its solutions, symmetries, and potential gauge models, highlighting its mathematical structure and physical implications.

Contribution

It provides a comprehensive analysis of the Dirac-K"ahler equation's tensor and matrix forms, including solutions, symmetry groups, and gauge model possibilities.

Findings

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

Explicit projection matrices for solutions are derived.

A potential gauge model with a noncompact gauge group is proposed.

Abstract

Tensor and matrix 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. It is shown that the symmetry group of the Dirac-K\"ahler tensor fields is SO(4,2). The conservation currents corresponding this symmetry are constructed. Supersymmetry of the Dirac-K\"ahler fields with tensor and spinor parameters is analyzed. 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.