The Feynman integral and Feynman's operational calculus : a heuristic and mathematical introduction

Abstract

We provide a short heuristic introduction to Feynman's operational calculus for noncommuting operators, as well as discuss briefly a mathematical approach to this subject developed by Gerald W. Johnson and the author. We also evoke some of the connections between this topic and the ``Feynman path integrals'' from quantum physics. In particular, analytic (operator-valued) Feynman path integrals, along with (suitable) noncommutative operations on certain algebras of Wiener functionals (called ``disentangling algebras''), play a prominent role in this context. The interested reader can find a much more thorough discussion of these and related developments in Chapters 14 through 19 of the book by {\it G. W. Johnson} and the author entitled ``The Feynman Integral and Feynman's Operational Calculus'', to be published by Oxford University Press.