Using Raising and Lowering Operators from Geometric Algebra for Electroweak Theory in Particle Physics

This paper has two objectives. The first is to explore the form and action of raising and lowering operators expressed in geometric algebra (GA). The second is to show how increasing the number of dimensions of Euclidean space from three to four opens a new avenue for understanding the chiral asymmetry of electroweak interactions. These explorations are guided by isomorphisms among groups represented in complex Clifford algebra, matrix algebra, and GA. With these isomorphisms, expressions for raising and lowering operators for electron and neutrino states in complex Clifford algebra are translated into GA and elaborated to include positrons and antineutrinos. This paper addresses such operators in the context of the electroweak sector of the SM utilizing (1) the GA G(3) for the Hestenes-Dirac equation in a Euclidean lab frame, (2) G(4) to introduce chiral asymmetry, and (3) G(4,1) to express the electroweak fermion states of the first generation of the SM and demonstrate their SU(2) relationships.