Foundations for Relativistic Quantum Theory I: Feynman's Operator Calculus and the Dyson Conjectures

TL;DR

This paper develops a representation theory for Feynman's operator calculus, proving Dyson's conjectures in QED, and connects the theory to path integrals, providing insights into divergences and nonperturbative solutions.

Contribution

It introduces a representation framework for Feynman's operator calculus, proves Dyson's second conjecture on the asymptotic nature of the series, and relates the theory to path integrals.

Findings

Dyson's second conjecture proven: series is asymptotic.

Perturbative expansion can be exact to any finite order.

Divergences linked to violation of Heisenberg's uncertainty relations.

Abstract

In this paper, we provide a representation theory for the Feynman operator calculus. This allows us to solve the general initial-value problem and construct the Dyson series. We show that the series is asymptotic, thus proving Dyson's second conjecture for QED. In addition, we show that the expansion may be considered exact to any finite order by producing the remainder term. This implies that every nonperturbative solution has a perturbative expansion. Using a physical analysis of information from experiment versus that implied by our models, we reformulate our theory as a sum over paths. This allows us to relate our theory to Feynman's path integral, and to prove Dyson's first conjecture that the divergences are in part due to a violation of Heisenberg's uncertainly relations.