Time Evolution of the External Field Problem in QED

TL;DR

This paper investigates the time evolution of the Dirac equation in QED under time-dependent electromagnetic fields, clarifying conditions for Fock space implementation and establishing the uniqueness of transition probabilities.

Contribution

It provides a detailed analysis of the conditions under which the quantum evolution can be implemented across varying Fock spaces, extending previous results on the Shale-Stinespring condition.

Findings

Implementation of time evolution is phase-independent and unique.

Transition probabilities between different Fock spaces are well-defined.

External magnetic fields prevent lifting the evolution to Fock space.

Abstract

We construct the time-evolution for the second quantized Dirac equation subject to a smooth, compactly supported, time dependent electromagnetic potential and identify the degrees of freedom involved. Earlier works on this (e.g. Ruijsenaars) observed the Shale-Stinespring condition and showed that the one-particle time-evolution can be lifted to Fock space if and only if the external field had zero magnetic components. We scrutinize the idea, observed earlier by Fierz and Scharf, that the time-evolution can be implemented between time varying Fock spaces. In order to define these Fock spaces we are led to consider classes of reference vacua and polarizations. We show that this implementation is up to a phase independent of the chosen reference vacuum or polarization and that all induced transition probabilities are well-defined and unique.