Space inversion of spinors revisited: A possible explanation of chiral behavior in weak interactions

We investigate a model in which spinors are considered as being embedded within the Clifford algebra that operates on them. In Minkowski space M1,3, we have four independent 4-component spinors, each living in a different minimal left ideal of Cl(1,3). We show that under space inversion, a spinor of one left ideal transforms into a spinor of another left ideal. This brings novel insight to the role of chirality in weak interactions. We demonstrate the latter role by considering an action for a generalized spinor field ψαi that has not only a spinor index α but also an extra index i running over four ideals. The covariant derivative of ψαi contains the generalized spin connection, the extra components of which are interpreted as the SU(2) gauge fields of weak interactions and their generalization. We thus arrive at a system that is left-right symmetric due to the presence of a “parallel sector”, postulated a long time ago, that contains mirror particles coupled to mirror SU(2) gauge fields.