Canonically quantized gravity: Disentangling the super-Hamiltonian and supermomentum constraints

This paper discusses some aspects of canonically quantized gravity as viewed in the Wheeler-DeWitt formalism. (2+1)-dimensional quantum gravity is described in great detail, with particular emphasis on the role played by the moduli space of Riemann surfaces. Wheeler's superspace in this case decomposes into a product of conformal deformations and moduli. A general treatment of the conformal mode of the (spatial) $d$-metric relevant to ($d+1\mathrm{}$)-dimensional quantum gravity is exhibited. The super-Hamiltonian constraint for 2+1 gravity is disentangled into independent constraints on the conformal mode and on the moduli. Finally, some suggestions are made on how these ideas might be applied to give some information about the physically interesting (3+1)-dimensional case.