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Left-Ideals, Dirac Fermions and SU(2)-Flavour

F. M. C. Witte
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Abstract

In this paper I reconsider the use of the left ideals of the even-grade subalgebra of spacetime algebra to describe fermionic excitations. When interpreted as rotors the general elements of an even-grade left-ideal describe massless particles in chiral flavour doublets. To study the application of these ideas to the standard Dirac formalism I construct a $2 \times 2$-matrix representation with bivector insertions for the Dirac algebra. This algebra has four ideals, and this approach clarifies how the identification of Dirac $\g_μ$-matrices with orthonormal basisvectors ${\bf e}_ν$ annihilates half of the ideals. For one possible choice of this mapping the remaining ideals the chiral left- and righthanded components of the fermion coincide with the even- and odd elements of spacetime algebra.

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F. M. C. Witte. 2007-07-17. Left-Ideals, Dirac Fermions and SU(2)-Flavour. https://arxiv.org/abs/math-ph/0612081

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