Houseman, DK; (2012) A Function-Analytic Development of Field Theory. Doctoral thesis , UCL (University College London).

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Abstract

This thesis presents a system of coupled differential equations as a simple model of quantum electrodynamics (QED). A key feature of the model is the Riemann- Silberstein (RS) representation of the photon. The RS representation leads to a natural configuration-space description for a system of multiple, non-interacting electrons and photons. Relativistic covariance is shown by extending the dynamics to a representation of the Poincar´e group on the space of configuration-space amplitudes. Because the differential system forms a well-posed initial-value problem, this model features a natural concept of time evolution, and concretely parametrises the system even at intermediate times during scattering processes. If QED could be formulated in a framework of this type, both the analysis and rigorous formulation of quantum field theory may benefit from a useful new perspective. Towards this aim, I consider deformation of the free theory, preserving the initial-value nature while incorporating interactions between particles. The deformation takes the form of a coupling between states of different particle content. I present some simple criteria to show whether the deformation is compatible with relativity. For a specific choice of the deformation, I perform a perturbative expansion on this system. I demonstrate agreement between some of the leading terms and QED. Although further extensions are required, these appear to be compatible with the existing framework, and these results are an encouraging first step towards a complete configuration-space/differential representation of QED.

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