Across the mass hyperboloid

Abstract

This paper is another step in our pursuit of nonperturbative Minkowskian quantum field theory. Instead of studying the Schwinger functions arising from the Euclidean approach, we study the Wightman distributions directly. We use the non-rigorous Feynman integral for a given quantum field theory as a guide for the rigorous definition of the characteristic functional corresponding to the Lagrangian, but this approach limits us to time-ordered Wightman distributions. We use the idea that if a classical action is perturbed by an interaction involving only N excitations, then one should be able to express the resulting characteristic functional rigorously in terms of the previous characteristic functional. Implementation requires the modification of the classical action with imaginary terms involving the excitation amplitudes. If the previous characteristic functional has a measure-theoretic representation of some kind, then one can subsequently remove these auxiliary terms with a limiting argument. We continue to consider the scalar Minkowskian quantum field theory with the free, massive case as our starting point. The characteristic functional can be defined in terms of the Minkowskian Feynman propagator, and previously, we constructed a probabilistic representation of the functional. In the appendix of that previous paper, we introduced the scalar quartic field interaction with only N space–time excitations. Under the condition that their Fourier transforms be all supported on one side of the mass hyperboloid, we constructed the resulting characteristic functional. In this paper, we construct the characteristic functional in the case where both regions of energy–momentum space contribute to the finite collection of excitations.