The Wodzicki residue can be used to build Chern classes and Chern-Simons classes for infinite dimensional bundles with pseudodifferential operator-valued connections. This setup occurs naturally in the Riemannian geometry of loop spaces. The Chern-Simons classes give information on the topology of the diffeomorphism group of the line bundles in geometric quantization of symplectic manifolds. This has intriguing (i.e. vague) connections to the geometric quantization Fock space and to anomalies in abelian QFT.