The metaplectic representation and boson fields

Abstract

We construct explicitly the infinite-dimensional metaplectic representation and show how its use simplifies and rigorizes several questions in bosonic Quantum Field Theory. The representation permutes Gaussian elements in the boson Fock space, and is necessarily projective. We compute its cocycle at the group level, and obtain Schwinger terms and anomalies from different versions of the cocycle; for instance, the Virasoro anomalous terms are obtained in this manner. We show how the choice of a complex structure on the space of solutions of a wave equation is related to the covariant Feynman propagator methods. We then show how the metaplectic representation allows one to compute exactly the S-matrix for bosons in an external field from the classical scattering operator.