  The canonical quantisation of General Relativity including matter on a
spacetime manifold in the globally hyperbolic setting involves in particular
the representation theory of the spatial diffeomorphism group (SDG), and/or its
Lie algebra (SDA), of the underlying spatial submanifold. There are well known
Fock representations of the SDA in one spatial dimension and non-Fock
representations of the SDG in all dimensions. The latter are not strongly
continuous and do not descend to representations of the SDA.
  In this work we report some partial results on non anomalous representations
of the SDA for both geometry and matter: 1. Background independent Fock
representations of the SDA by operators exist in all dimensions. 2. Infinitely
many unitary equivalence classes of background dependent Fock representations
of the SDA by operators exist in one dimension but these do not extend to
higher dimensions. 3. Infinitely many unitary equivalence classes of background
dependent Fock representations of the SDA of volume preserving diffeomorphisms
by operators exist in all dimensions. 4. Infinitely many unitary equivalence
classes of background dependent Fock representations of the SDA by quadratic
forms exist in all dimensions.
  Except for 1. these representations do not descend from an invariant state of
the Weyl algebra and 4. points to a new strategy for solving the quantum
constraints.
