Geometric Spinors, Generalized Dirac Equation and Mirror Particles

TL;DR

This paper explores how geometric spinors, as elements of Clifford algebras, transform under Lorentz transformations in Minkowski spacetime, with implications for understanding parity non-conservation.

Contribution

It demonstrates the transformation properties of geometric spinors within Clifford algebras and their implications for Dirac equations and mirror particles.

Findings

Clifford spinors transform via multiplication by Clifford numbers from both sides.

Transformations can convert vectors into 3-vectors and spinors into other ideals.

Implications for non-conservation of parity in physics.

Abstract

It is shown that since the geometric spinors are elements of Clifford algebras, they must have the same transformation properties as any other Clifford number. In general, a Clifford number $\Phi$ transforms into a new Clifford number $\Phi'$ according to $\Phi \to \Phi ' = {\rm{R}}\,\Phi \,{\rm{S}}$, i.e., by the multiplication from the left and from the right by two Clifford numbers ${\rm R}$ and ${\rm S}$. We study the case of $Cl(1,3)$, which is the Clifford algebra of the Minkowski spacetime. Depending on choice of ${\rm R}$ and ${\rm S}$, there are various possibilities, including the transformations of vectors into 3-vectors, and the transformations of the spinors of one minimal left ideal of $Cl(1,3)$ into another minimal left ideal. This, among others, has implications for understanding the observed non-conservation of parity.