Feynman’s Operational Calculi: Decomposing Disentanglings

Let X be a Banach space and suppose that A ₁,…,A _n 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 μ _i be a continuous probability measure on ℬ(0,1]), the Borel class of 0,1]. Each such n-tuple of operator-measure pairs (A _i ,μ _i ), i=1,…,n, determines an operational calculus or disentangling map T_m1,...,mn{\mathcal{T}}_{\mu_{1},\dots,\mu_{n}} from a commutative Banach algebra \mathbbD(A₁,...,A_n){\mathbb{D}}(A_{1},\dots,A_{n}) 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.