Sign of the Feynman Propagator and Irreversibility

TL;DR

This paper investigates the sign properties of the interacting Feynman propagator in scalar electrodynamics, linking them to time reversibility and exploring how approximations affect these properties.

Contribution

It demonstrates the dependence of the propagator's sign on time reversibility and analyzes how common approximations influence this property.

Findings

Re( ext{i}\Delta_{F,int}) \\geq 0$ hinges on reversibility.

Under weak coupling, the positive semidefinite sign is generally lost.

Rotating wave approximation can restore the sign under certain conditions.

Abstract

For the interacting Feynman propagator $ \Delta_{F,int}(x,y) $ of scalar electrodynamics, we show that the sign property, $ \operatorname{Re} i\Delta_{F,int} \geq 0 $, hinges on the reversibility of time evolution. In contrast, $ \operatorname{Im} i\Delta_{F,int} $ is indeterminate. When we switch to reduced dynamics under the weak coupling approximation, the positive semidefinite sign of $ \operatorname{Re} i\Delta_{F,int} $ is generally lost, unless we impose severe restrictions on the Kraus operators that govern time evolution. With another approximation, the rotating wave approximation, we may recover the sign by restricting the test functions to exponentials under certain conditions.