Path integrals, complex probabilities and the discrete Weyl representation

TL;DR

This paper develops a discrete real-time path integral framework using complex probabilities derived from Weyl algebra, enabling exact unitarity and applications to scattering and quantum field theory.

Contribution

It introduces a novel discrete formulation of path integrals based on complex probabilities and Weyl algebra, maintaining unitarity and enabling practical approximations.

Findings

Complex probabilities factor into products of conditional probabilities.

Exact unitarity is preserved at each approximation level.

Applications demonstrated in scattering theory and quantum field theory.

Abstract

A discrete formulation of the real-time path integral as the expectation value of a functional of paths with respect to a complex probability on a sample space of discrete valued paths is explored. The formulation in terms of complex probabilities is motivated by a recent reinterpretation of the real-time path integral as the expectation value of a potential functional with respect to a complex probability distribution on cylinder sets of paths. The discrete formulation in this work is based on a discrete version of Weyl algebra that can be applied to any observable with a finite number of outcomes. The origin of the complex probability in this work is the completeness relation. In the discrete formulation the complex probability exactly factors into products of conditional probabilities and exact unitarity is maintained at each level of approximation. The approximation of infinite dimensional quantum systems by discrete systems is discussed. Applications to scattering theory and quantum field theory are illustrated.