Problems with Decoherence in Quantum Geometrodynamics

I illustrate some severe conceptual and technical problems that arise in attempts to apply quantum decoherence concepts to "quantum geometrodynamics" (i.e. Dirac constraint quantization of classical general relativity in the Hamiltonian formulation). In particular, I show that the Problem of Time, the Hilbert Space Problem, and the conceptual dependence of decoherence on time evolution and Hilbert space structure implies: 1) suppression of interference cannot take place between components of a superposition of quantum-gravitational (i.e. Wheeler-DeWitt) wave functionals; 2) approximate diagonalization of the reduced density matrix (associated with said superposition) in the 3-geometry basis cannot take place, and such a reduced density matrix is mathematically ill-defined; 3) time-dependent functional Schroedinger equations for the matter components of said superposition cannot be derived in a semiclassical approximation. I conclude, contrary to extant claims in the quantum gravity literature, that quantum decoherence concepts cannot be consistently applied to quantum geometrodynamics. I then suggest that quantum decoherence concepts can only be sensibly applied to quantum gravity theories that posit classical time parameters or matter-clock variables at a fundamental level.