Clifford residues and charge quantization

Abstract

We derive the quantization of action, particle number, andelectric charge in a Lagrangian spin bundle over${\mathbb{M}} \equiv {\mathbb{M}}_\# \backslash \cup D_J$ Penrose’s conformal compactification of Minkowsky space, with the world tubes of massive particles removed.

Our Lagrangian density,${\mathcal{L}}_g$, is the spinor factorization of the Maurer-Cartan 4-form Ω⁴; it’s action,S _g , measures the covering number of the 4internal u (1)×su (2) phases over external spacetime${\mathbb{M}}$. UnderPTC symmetry,${\mathcal{L}}_g$ reduces to the second Chern formTrK _L ⋏K _R for a left ⊕ right chirality spin bundle. We prove aresidue theorem forgl (2, ℂ)-valued forms, which says that, when we “sew-in” singular lociD _J over which theu (1)×su (2) phases of the matter fields have some extra twists compared to the8 vacuum modes, the additional contributions to the action, electric charge, lepton and baryon numbers are alltopologically quantized. Because left and right chirality 2-forms arechiral dual, forms are quantized over theirdual cycles. Thus it is the interactionc ₂ (E), with a globally nontrivialmagnetic field, that forceselectric fields to be topologically quantized overspatial 2 cycles,$\int_{{\mathbb{S}}^2 } { K_{or} } e^\theta \wedge e^\varphi = 4\pi {\rm N}$.