Second quantized quantum field theory based on invariance properties of locally conformally flat space-times

Abstract

Well defined quantum field theory (QFT) for the electroweak force including quantum electrodynamics (QED) and the weak force is obtained by considering natural unitary representations of a group $K\subset U(2,2)$, where $K$ is locally isomorphic to $SL(2,{\bf C})\times U(1)$, on a state space of Schwartz spinors, a Fock space ${\mathcal F}$ of multiparticle states and a space ${\mathcal H}$ of fermionic multiparticle states which forms a Grassmann algebra. These algebras are defined constructively and emerge from the requirement of covariance associated with the geometry of space-time. (Here $K$ is the structure group of a certain principal bundle associated with a given M\"{o}bius structure modeling space-time.) Scattering processes are associated with intertwining operators between various algebras, which are encoded in an associated bundle of kernel algebras. Supersymmetry emerges naturally from the algebraic structure of the theory. Kernels can be generated using $K$ covariant matrix valued measures given a suitable definition of covariance. It is shown how Feynman propagators, fermion loops and the electron self energy can be given well defined interpretations as measures covariant in this sense. An example of the methods described in the paper is given in which the first order Feynman amplitude of electro-electron scattering ($ee\rightarrow ee$) is derived from a simple order (2,2) kernel. A second example is given explaining muon decay which is a manifestation of the weak force.