  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.
