Subgroups of Clifford algebras

TL;DR

This paper explores various subgroups of Clifford algebras, highlighting their potential broader applications in understanding fundamental properties of matter beyond their traditional use in quantum mechanics.

Contribution

It provides an algebraist's perspective on Clifford subgroups and suggests possible physical interpretations related to fermion generations and symmetry-breaking.

Findings

Extension of Dirac algebra from complex numbers to quaternions suggests natural emergence of fermion generations.

Clifford algebra subgroups may have broader applications in describing matter properties.

Potential links between algebraic structures and fundamental physical phenomena.

Abstract

Clifford algebras are used for constructing spin groups, and are therefore of particular importance in the theory of quantum mechanics. But the spin group is not the only subgroup of the Clifford algebra. An algebraist's perspective on these groups and algebras may suggest ways in which they might be applied more widely to describe the fundamental properties of matter. I do not claim to build a physical theory on top of the fundamental algebra, and my suggestions for possible physical interpretations are indicative only, and may not work. Nevertheless, both the existence of three generations of fermions and the symmetry-breaking of the weak interaction seem to emerge naturally from an extension of the Dirac algebra from complex numbers to quaternions.