  In this paper, we survey recent progress on the constructive theory of the
Feynman operator calculus. (The theory is constructive in that, operators
acting at different times, actually commute.) We first develop an operator
version of the Henstock-Kurzweil integral, and a new Hilbert space that allows
us to construct the elementary path integral in the manner originally
envisioned by Feynman. After developing our time-ordered operator theory we
extend a few of the important theorems of semigroup theory, including the
Hille-Yosida theorem. As an application, we unify and extend the theory of
time-dependent parabolic and hyperbolic evolution equations. We then develop a
general perturbation theory and use it to prove that all theories generated by
semigroups are asympotic in the operator-valued sense of Poincare. This allows
us to provide a general theory for the interaction representation of
relativistic quantum theory. We then show that our theory can be reformulated
as a physically motivated sum over paths, and use this version to extend the
Feynman path integral to include more general interactions. Our approach is
independent of the space of continuous functions and thus makes the question of
the existence of a measure more of a natural expectation than a death blow to
the foundations for the Feynman integral.
