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.