Abstract
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.
Highlights
After the seminal paper by Lee and Yang [1], we have become accustomed to the idea that the physical processes associated with weak interactions are not invariant under space inversion
We have shown how such a system with ordinary and mirror particles coupled to the corresponding gauge fields can be described within a framework based on the algebraic spinors and their behavior under space inversion
The observation that space inversion P and time reversal T transform a spinor of one left ideal of Cl(1, 3) into a spinor of another left ideal brings a novel insight on the invariance of nature under P and T that, to my knowledge, has not yet been explicitly pointed out
Summary
After the seminal paper by Lee and Yang [1], we have become accustomed to the idea that the physical processes associated with weak interactions are not invariant under space inversion. In a more general theory, a spinor can be considered as being embedded within the Clifford algebra that operates on it This opens a Pandora’s box of possibilities that have been explored in the attempts to find a unified theory of fundamental particles and forces [11]–[13]. We construct a specific model starting from a generalized Dirac action that is a functional of 16 complex valued fields, components of a generalized algebraic spinor field Such a 16-component field is written as the 4 × 4 matrix ψαi, representing four Dirac spinors lying in four different left ideals. The ‘internal’ components of ψαi, denoted by the index i that runs over four different left ideals, form two irreducible representations of SU(2) that correspond to ordinary and mirror particles. We demonstrate the existence of such a parallel sector for the case of weak interactions only
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