Should Metric Signature Matter in Clifford Algebra Formulations of Physical Theories?

TL;DR

This paper investigates the significance of metric signature in Clifford algebra formulations of physical theories, proposing a fully signature-invariant wave equation to address the ambiguity in current models.

Contribution

It introduces a generalized multivector wave equation that is manifestly tilt covariant, incorporating all algebra components and interaction possibilities for signature independence.

Findings

The proposed wave equation is fully signature invariant in form.

Mapping between different signatures is possible via tilt transformations.

The approach may unify Clifford algebra formulations regardless of metric signature.

Abstract

Standard formulation is unable to distinguish between the (+++-) and (---+) spacetime metric signatures. However, the Clifford algebras associated with each are inequivalent, R(4) in the first case (real 4 by 4 matrices), H(2) in the latter (quaternionic 2 by 2). Multivector reformulations of Dirac theory by various authors look quite inequivalent pending the algebra assumed. It is not clear if this is mere artifact, or if there is a right/wrong choice as to which one describes reality. However, recently it has been shown that one can map from one signature to the other using a "tilt transformation" [see P. Lounesto, "Clifford Algebras and Hestenes Spinors", Found. Phys. 23, 1203-1237 (1993)]. The broader question is that if the universe is signature blind, then perhaps a complete theory should be manifestly tilt covariant. A generalized multivector wave equation is proposed which is fully signature invariant in form, because it includes all the components of the algebra in the wavefunction (instead of restricting it to half) as well as all the possibilities for interaction terms.