A distributional approach to Feynman’s operational calculus

In this paper we will construct an operator-valued distribution that will extend Feynman’s operational calculus in the setting of Jefferies and Johnson, 2001–2003, and Johnson–Lapidus–Nielsen, 2014, from the disentangling of holomorphic functions of several variables to the disentangling of Schwartz functions on R. It will be shown that the disentangled operator corresponding to a Schwartz function (i.e., the disentangling of a Schwartz function) can be realized as the limit of a sequence of operator-valued distributions of compact support in a ball of a certain radius centered at 0 ∈ R. In this way, we can extend the operational calculi to the Schwartz space.