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

TL;DR

<p>Canonical quantum gravity with incoherent dust provides a physically well-motivated time reference by coupling gravity to a dust fluid, yielding a functional Schrödinger equation in which the Hamiltonian density depends only on geometry. The dust enables separation of dust time from geometric degrees of freedom, and transforms the Dirac constraint algebra into an Abelian set, with a subsequent reformulation that yields a true Lie algebra for vacuum gravity when dust is present. This framework allows a conserved inner product, a unitary action of dust-space diffeomorphisms, and a spectrum analysis leading to a distinguished, positive-definite Hilbert space ${oldsymbol{ m H}}^{+}$ for gravity observables, albeit with caveats about operator ordering and the status of fundamental gravitational variables as observables. Overall, the dust clock offers a phenomenologically appealing route to addressing the problem of time in canonical quantum gravity and provides concrete tools for constructing and interpreting gravitational observables in a quantum setting.</p>

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 Schrödinger 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 Schrödinger 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 Schrödinger 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.