A construction of full qed using finite dimensional Hilbert space

Abstract

While causal perturbation theory and lattice regularisation allow treatment of the ultraviolet divergences in qed, they do not resolve the issues of constructive field theory, or show the validity of qed except as a perturbation theory. I present a rigorous construction of quantum and classical electrodynamics from fundamental principles of quantum theory. Hilbert space of dimension N is justified from statements about measurements with finite range and resolution. Using linear combinations of basis kets, a continuum of kets, |x> for x in R^3, is constructed such that the inner product can be expressed as a finite sum or as an integral. Vectors are smooth wave functions such that differential operators are defined and the choice of basis has no affect on underlying physics. Quantum field operators, phi(x) for x in R^4, are constructed from creation and annihilation operators on Fock space, obey quantum covariance and locality, and are suitable for a description of particle interactions under the Feynman-Stückelberg interpretation. It is shown that the formulation is consistent and that any dependency on a lattice arises from measurement, not from underlying physics. In consequence, and because the continuum is constructed from linear combinations of basis kets, it is not required to take the limit N-->oo. Quantum fields are defined on a continuum, and are operator valued functions, not distributions. The interacting Dirac equation, Maxwell's equations and the Lorentz force law are derived, showing that qed is a complete theory of the electromagnetic interaction, not just a perturbation theory and that bare mass and charge are the physical values. Up to the accuracy of measurement, predictions of perturbation theory are identical to those of standard qed with all loop divergences removed. The standard perturbation expansion is asymptotic to the finite expansion given here.