Application of Geometric Algebra to the Electroweak Sector of the Standard Model of Particle Physics

Abstract

Geometric algebra (GA) offers an intriguing approach to understanding the fields of the standard model (SM) of elementary particle physics. This paper examines a geometric view of electron and neutrino fields in the electroweak sector of the SM. These fields are related by the transformations of the \({SU(2)}\) Lie group, with generators customarily represented by the \({2\times 2}\) complex Pauli matrices. In \({\mathcal{G}_3}\), the GA of three-dimensional Euclidean space, the three unit basis vectors may be used to provide a more geometrically oriented representation of \({SU(2)}\). In fact, \({\mathcal{G}_3}\) is sometimes referred to as the Pauli algebra. However, a more general representation of the special unitary group \({SU(n)}\) in GA is in terms of generators that are compound (non-blade) bivectors in \({\mathcal{G}_{2n}}\), the GA of \({2n}\)-dimensional Euclidean space. Therefore, a natural approach to electroweak theory mathematically is to work with \({SU(2)}\) generators as compound bivectors in \({\mathcal{G}_4}\). This approach leads one to consider electroweak fields as multivector fields in \({\mathcal{G}_4}\) that are solutions of the Dirac equation in four spatial dimensions and one time dimension. This paper examines such multivector fields and offers a new point of view on chiral projection of \({\mathcal{G}_3}\) fields. It is shown that \({SU(2)}\) representation in \({\mathcal{G}_4}\) leads naturally to the singlet/doublet structure of the chiral electroweak fields.