Quantum electrodynamics in external fields from the spin representation

Abstract

Systematic use of the infinite-dimensional spin representation simplifies and rigorizes several questions in quantum field theory. This representation permutes "Gaussian" elements in the fermion Fock space, and is necessarily projective: we compute its cocycle at the group level, and obtain Schwinger terms and anomalies from infinitesimal versions of this cocycle. Quantization, in this framework, depends on the choice of the "right" complex structure on the space of solutions of the Dirac equation. We show how the spin representation allows one to compute exactly the S-matrix for fermions in an external field; the cocycle yields a causality condition needed to determine the phase.