Observations on representations of the spatial diffeomorphism group and algebra in all dimensions

The canonical quantization 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 nonanomalous 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.