Time Evolution in the external field problem of Quantum Electrodynamics

TL;DR

This paper investigates the mathematical challenges of defining time evolution in quantum electrodynamics with external fields, comparing different approaches and establishing conditions for consistent second quantization.

Contribution

It provides rigorous analysis and proofs of various methods for second quantization in external field QED, including phase fixing and gauge invariance.

Findings

Second quantization via parallel transport preserves causality.

Phase of the second quantized S-matrix is not solely determined by causality.

Gauge-invariant second quantization can be achieved through suitable renormalizations.

Abstract

A general problem of quantum field theories is the fact that the free vacuum and the vacuum for an interacting theory belong to different, non-equivalent representations of the canonical (anti-)commutation relations. In the external field problem of QED, we encounter this problem in the form that the Dirac time evolution for an external field with non-vanishing magnetic components will not satisfy the Shale-Stinespring condition, known to be necessary and sufficient for the existence of an implementation on the fermionic Fock space. Therefore, a second quantization of the time evolution in the usual way is impossible. In this thesis, we present several rigorous approaches to QED with time-dependent, external fields and analyze in what sense a time evolution can exist in the second quantized theory. We study different constructions of the fermionic Fock space and prove their equivalence. We study and compare the results of Deckert et. al. (2010), where the time evolution is realized as unitary transformations between time-varying Fock spaces, with those of Langmann and Mickelsson (1996), who construct a "renormalization" for the time evolution and present a method to fix the phase of the second quantized scattering operator by parallel transport in a principle fibre bundle over the restricted, general linear group acting on the fermionic Fock space. We provide rigorous proof for the fact that the second quantization by parallel transport preserves causality. These findings seem to refute claims made in Scharf (1995) that the phase of the second quantized S-matrix is essentially determined by the requirement of causality. We propose a simple solution to the problem of gauge anomalies in the procedure of Langmann and Mickelsson, showing that the second quantization of the scattering operator can be made gauge-invariant by using a suitable class of renormalizations.