Note on Canonical Quantization and Unitary (in)-Equivalence in Field Theory

The problem of defining and constructing representations of the Canonical Commutation Relations can be systematically approached via the technique of Algebraic Quantization. In particular, when the phase space of the system is linear and finite dimensional, the "vertical polarization" provides an unambiguous quantization. For infinite dimensional field theory systems, where the Stone-von Neumann fails to be valid, even the simplest representation, the Schrodinger functional picture has some non-trivial subtleties. In this note we study the quantization of a real, free, scalar field --where the Fock quantization is well understood-- on an arbitrary background and show that the algebraic quantization approach needs some extra input in order to have full control of the resulting quantum theory.