Lorentz Invariance of the Multidimensional Dirac-Hestenes Equation

TL;DR

This work investigates the Lorentz invariance of the multidimensional Dirac--Hestenes equation within the geometric algebra framework. It contrasts tensor and spinor approaches: the tensor route transforms the algebra generators under ${ m O}(1,n)$, while the spinor route keeps the generators fixed and transforms the wavefunction via the spin group ${ m Spin}(1,n)$ (or ${ m Pin}(1,n)$ with caveats). The authors extend the 4D theory to general $(1,n)$, classify solutions (spinor, semi-spinor, double spinor) via Hermitian idempotents in left ideals, and prove invariance for both formalisms, with a careful treatment of even dimensions by restricting to ${ m Spin}(1,2d)$. Overall, the spinor approach provides a natural and invariant formulation of the multidimensional Dirac--Hestenes equation, while the tensor approach requires explicit invariants, highlighting the deep connection between pseudo-orthogonal and spin groups in geometric algebra.

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.

Theorems & Definitions (23)

• Lemma 2.1
• Remark 2.2
• Example
• Remark 3.1
• Theorem 3.2
• Remark 3.3
• proof
• Theorem 3.4
• proof
• Remark 3.5