Dust as a Standard of Space and Time in Canonical Quantum Gravity

Abstract

The coupling of the metric to an incoherent dust introduces into spacetime a privileged dynamical reference frame and time foliation. The comoving coordinates of the dust particles and the proper time along the dust worldlines become canonical coordinates in the phase space of the system. The Hamiltonian constraint can be resolved with respect to the momentum that is canonically conjugate to the dust time. Imposition of the resolved constraint as an operator restriction on the quantum states yields a functional Schr\"{o}dinger equation. The ensuing Hamiltonian density has an extraordinary feature: it depends only on the geometric variables, not on the dust coordinates or time. This has three important consequences. First, the functional Schr\"{o}dinger equation can be solved by separating the dust time from the geometric variables. Second, the Hamiltonian densities strongly commute and therefore can be simultaneously defined by spectral analysis. Third, the standard constraint system of vacuum gravity is cast into a form in which it generates a true Lie algebra. The particles of dust introduce into space a privileged system of coordinates that allows the supermomentum constraint to be solved explicitly. The Schr\"{o}dinger equation yields a conserved inner product that can be written in terms of either the instantaneous state functionals or the solutions of constraints. Examples of gravitational observables are given, though neither the intrinsic metric nor the extrinsic curvature are observables. Disregarding factor--ordering difficulties, the introduction of dust provides a satisfactory phenomenological approach to the problem of time in canonical quantum gravity.