A Treatment of Quantum Electrodynamics as a Model of Interactions between Sizeless Particles in Relational Quantum Gravity

A Relational Quantum Theory Incorporating Gravity developed the concept of quantum covariance and argued that this is the correct expression of the fundamental physical principle that the behaviour of matter is everywhere the same in the quantum domain, as well as being the required condition for the unification of general relativity with quantum mechanics for non-interacting particles. This paper considers the interactions of elementary particles. Quantum covariance describes families of finite dimensional Hilbert spaces with an inbuilt cut-off in energy-momentum and using flat space metric (quantum coordinates) between initial and final states. It is shown by direct construction that it is possible to construct a quantum field theory of operators on members of these families, obeying locality, suitable for a description of particle interactions, and leading to a general formulation of particle theoretic field theory incorporating qed. The construction is consistent and effectively identical in the continuum limit to classical quantum electrodynamics with all loop divergences removed by the method of Epstein and Glaser. The model avoids the Landau pole. The model does not start from a classical Lagrangian and it is shown that the interaction leads to Maxwell's equations in the classical limit and to Feynman rules as in the standard theory after renormalisation