A wavelet-based set of constructive masses for the Minkowskian Feynman propagator

Abstract

We continue our pursuit of Minkowskian quantum field theory—specifically, a nonperturbative analysis of the characteristic functional generating the time-ordered Wightman distributions for the quartic scalar field interaction. As in a previous paper, our starting point is the Minkowskian Feynman propagator for the free scalar field theory analyzed by a system of expansion functions in space-time. Here, we choose a significantly different system of expansion functions. We start with an L2-orthonormal wavelet basis {Ψw:w∈W}, where each Ψw has the properties of a Daubechies wavelet in the temporal coordinate and the properties of a Meyer wavelet in the spatial coordinates. We consider the matrix elements Gww′(m) of the mass-m Feynman propagator based on the expansion functions vw=Lw−1Ψw, where Lw denotes the length scale of w∈W. For every finite Λ⊂W, we let GΛ(m) denote the Λ × Λ matrix of such elements. We define a constructive mass to be any value of m such that GΛ(m) is nonsingular for each finite Λ⊂W. We prove that the set of such masses is at least co-countable (although we expect every mass value to be constructive). For an arbitrary constructive mass m and an arbitrary finite Λ⊂W, we rigorously construct the interacting characteristic functional as an integral in amplitudes, where the Λ-cutoff on the quartic interaction is supplemented by a cutoff on the free-field action defined by GΛ(m)−1.