  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-St\"uckelberg 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.
