Feynman operator calculus: The constructive theory

In this paper, we survey progress on the Feynman operator calculus and path integral. We first develop an operator version of the Henstock-Kurzweil integral, construct the operator calculus and extend the Hille-Yosida theory. This shows that our approach is a natural extension of operator theory to the time-ordered setting. As an application, we unify the theory of time-dependent parabolic and hyperbolic evolution equations. Our theory is then reformulated as a sum over paths, providing a completely rigorous foundation for the Feynman path integral. Using our disentanglement approach, we extend the Trotter-Kato theory.