External Field QED on Cauchy Surfaces for Varying Electromagnetic Fields

Abstract

The Shale–Stinespring Theorem (J Math Mech 14:315–322, 1965) together with Ruijsenaar’s criterion (J Math Phys 18(4):720–737, 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 spatial 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.