External Field QED on Cauchy Surfaces

TL;DR

This paper generalizes the conditions for implementing external field QED evolution from constant-time hyperplanes to arbitrary smooth space-like Cauchy surfaces, highlighting the role of tangential electromagnetic components.

Contribution

It extends the Shale-Stinespring and Ruijsenaar criteria to general Cauchy surfaces and characterizes admissible Fock spaces based on polarization classes influenced by tangential electromagnetic components.

Findings

Implementation is possible if and only if tangential components of A_mu are zero.

Polarization classes depend only on tangential components of A_mu.

Results are invariant under Lorentz and gauge transformations.

Abstract

The Shale-Stinespring Theorem (1965) together with Ruijsenaar's criterion (1977) provide a necessary and sufficient condition for the implementability of the evolution of external field quantum electrodynamics between constant-time hyperplanes on standard Fock space. The assertion states that an implementation is possible if and only if the spacial components of the external electromagnetic four-vector potential $A_\mu$ are zero. We generalize this result to smooth, space-like Cauchy surfaces and, for general $A_\mu$, show how the second-quantized Dirac evolution can always be implemented as a map between varying Fock spaces. Furthermore, we give equivalence classes of polarizations, including an explicit representative, that give rise to those admissible Fock spaces. We prove that the polarization classes only depend on the tangential components of $A_\mu$ w.r.t. the particular Cauchy surface, and show that they behave naturally under Lorentz and gauge transformations.