A Construction of Full QED Using Finite Dimensional Hilbert Space

摘要

Context. While causal perturbation theory and lattice regularization allow treatment of theultraviolet divergences in qed, they do not resolve the mathematical issues of constructive field theory,or show the validity of qed except as a perturbation theory.Aims. To present a mathematically rigorous construction of quantum and classical electrodynamics fromfundamental principles of quantum theory.Methods. Hilbert space of dimension N is justified from statements about measurements with finiterange and resolution. Using linear combinations of basis kets, a continuum of kets, |x for x ∈ R3,is constructed such that the inner product can be expressed either as a finite sum or as an integral.Vectors are smooth wave functions such that differential operators are defined and a form of covarianceis obeyed (the choice of basis has no affect on underlying physics). Quantum field operators, φ(x)for x ∈ R4, are constructed from creation and annihilation operators on Fock space, obey quantumcovariance and locality, and are suitable for a description of particle interactions under the FeynmanStückelberg interpretation.Results. It is shown that the formulation is consistent and that any dependency on a lattice arises frommeasurement, not from underlying physics. In consequence, and because the continuum is constructedfrom linear combinations of basis kets, it is not required to take the limit N → ∞. Quantum fieldsare defined on a continuum, and are operator valued functions, not distributions. The interacting Diracequation, Maxwell’s equations and the Lorentz force law are derived, showing that qed is a completetheory of the electromagnetic interaction, not just a perturbation theory, and that bare mass and charge arethe physical values. Up to the accuracy of measurement, predictions of perturbation theory are identicalto those of standard qed with all loop divergences removed. The standard perturbation expansion isasymptotic to the finite expansion given here.