Feynman’s Operational Calculi: Decomposing Disentanglings

Let X be a Banach space and suppose that A1, . . . ,An are noncommuting (that is, not necessarily commuting) elements in L(X), the space of bounded linear operators on X. Further, for each i ∈ {1, . . . , n}, let μi be a continuous probability measure on B([0,1]), the Borel class of [0,1]. Each such n-tuple of operator-measure pairs (Ai,μi), i = 1, . . . , n, determines an operational calculus or disentangling map Tμ1,...,μn from a commutative Banach algebra D(A1, . . . ,An) of analytic functions, called the disentangling algebra, into the noncommutative Banach algebra L(X). The disentanglings are the central processes of Feynman’s operational calculi. We partition the interval [0,1] and show in this paper how the disentangling over [0,1] can be decomposed into the disentanglings over the subintervals associated with the partition. This often enables us to simplify the disentangling process and, in some cases, to calculate it completely. This includes circumstances we could not previously deal with. One of the major motivations for developing these operational calculi is for representing various evolutions. It is natural to ask how a disentangled exponential Tμ1,...,μn(e1n ) behaves when [0,1] is partitioned into disjoint subintervals and we decompose the indicated disentangling. A corollary of the main theorem of this paper resolves this general question. (The main theorem itself has corollaries which are by no means limited to exponential functions.)