Circle diffeomorphisms acting on fermionic and bosonic Fock space

Abstract

The group $Diff(S^1)$ of smooth orientation preserving diffeomorphisms of the circle has a natural action on the Hilbert spaces $L^2(S^1)$ and $H^{1/2}(S^1)$, preserving the canonical orthogonal, respectively symplectic, structures on these spaces. Applying a famous criterion of Shale and Stinespring, this yields a projective representation of $Diff(S^1)$ on the associated fermionic, respectively bosonic, Fock space. We prove this criterion in an abstract setting, treating the fermionic and bosonic cases analoguously. We investigate to which extent the smoothness condition on the circle diffeomorphisms can be relaxed.