Subgroups of Clifford algebras

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.

arXiv commentIncluded more detailed calculations, and a new notation to simplify these calculations. Now 16 pages. To appear in Advances in Applied Clifford Algebras. 20 pages

References in corpus (12)

• Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra
• The Higgs sector of gravitational gauge theories
• Octonions, E6, and Particle Physics
• Generations: Three Prints, in Colour
• The Standard Model Algebra - Leptons, Quarks, and Gauge from the Complex Clifford Algebra Cl6
• Classification of Lie algebras of specific type in complexified Clifford algebras
• On some Lie groups containing spin group in Clifford algebra
• The Algebra of Grand Unified Theories
• Symplectic, orthogonal and linear Lie groups in Clifford algebra
• Superselection of the weak hypercharge and the algebra of the Standard Model
• A group-theorist's perspective on symmetry groups in physics
• Lectures on twistor theory

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• A group-theorist's perspective on symmetry groups in physics
• Finite symmetry groups in physics
• Potential applications of modular representation theory to quantum mechanics
• Options for a finite group model of quantum mechanics
• Remarks on the group-theoretical foundations of particle physics
• Possible uses of the binary icosahedral group in grand unified theories