A distributional approach to Feynman's operational calculus (Q2512547)

\textit{R. P. Feynman} [Phys. Rev., II. Ser. 84, 108--128 (1951; Zbl 0044.23304)] was the first to formulate what is nowadays called operational calculus. Feynman's original paper concerned itself with the formation of functions of non-commuting operators. Clearly, with a particular problem in mind, one has to decide how to define a given function of two or more operators on some Hilbert space. In quite a number of papers, Jefferies and Johnson chose a special approach which is followed in the present paper. In this setting, measures on some time interval are used to determine when a given operator will act in products. Naturally, the question arises how these measures are chosen und used to determine the order of operators in a product. The answer is: attach time indices to the operators and apply time ordering. The present author constructs an operator-valued distribution that extends the calculus of Feynman in the sense of Jefferies and Johnson. He finds a way to extend the operational calculus to the Schwartz space of functions on \(\mathbb R^n\). This streamlined presentation will help mathematically oriented readers. Physical applications are not proposed.