Lorentz Invariance of the Multidimensional Dirac-Hestenes Equation

TL;DR

This paper explores whether the multidimensional Dirac-Hestenes equation remains Lorentz invariant under coordinate transformations, comparing tensor and spinor approaches, and generalizing results to higher dimensions.

Contribution

It provides a detailed comparison of tensor and spinor formulations for Lorentz invariance of the Dirac-Hestenes equation, extending analysis to multidimensional spaces.

Findings

Tensor approach requires explicit invariants

Spinor formulation maintains natural Lorentz covariance

Results generalized to (1,n) signature spaces

Abstract

This paper investigates the Lorentz invariance of the multidimensional Dirac-Hestenes equation, that is, whether the equation remains form-invariant under pseudo-orthogonal transformations of the coordinates. We examine two distinct approaches: the tensor formulation and the spinor formulation. We first present a detailed examination of the four-dimensional Dirac-Hestenes equation, comparing both transformation approaches. These results are subsequently generalized to the multidimensional case with (1,n) signature. The tensor approach requires explicit invariants, while the spinor formulation naturally maintains Lorentz covariance through spin group action.