Geometric Spinors, Generalized Dirac Equation and Mirror Particles

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.