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. Formal imposition of the resolved constraint as an operator restriction on the quantum states yields a functional Schrodinger 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 Schrodinger equation can be solved by separating the dust time from the geometric variables. Second. , disregarding the standard factor-ordering diKculties, 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 Schrodinger equation yields a formally conserved inner product that can be written in terms of either the instantaneous state functionals or the solutions of constraints. Gravitational observables admit a similar dual representation. Examples of observables are given, though neither the intrinsic metric nor the extrinsic curvature are observables. This comes as close as one can reasonably expect to a satisfactory phenomenological quantization scheme that is free of most of the problems of time. PACS number(s): 04.60.Ds, 04.20.Cv, 04.20.Fy I. INTKGDUCTIONThe Dirac constraint quantization of vacuum Einstein gravity yields the Wheeler-DeWitt equation for the quantum state of the intrinsic three geometry of space [1,2]. One can view this equation as a statement that only two out of three independent components of the intrinsic geometry are dynamical. The third component is an intrinsic time that specifies the location of space as a hypersurface in spacetime. The Wheeler-DeWitt equation is then interpreted as an evolution equation for the state in the intrinsic time.The Wheeler-DeWitt equation is a second-order variational differential equation. The space of its solutions carries no obvious Hilbert space structure [3,4]. This has prompted numerous attempts aimed at replacing the Wheeler-DeWitt equation by a first-order Schrodinger equation. In order to do that, one should identify the intrinsic time at the classical level, solve the Hamiltonian constraint for the momentum conjugate to time, and impose the resolved constraint as an operator restriction on the quantum states. Unfortunately, there is no natural candidate for the intrinsic time, and the procedure is beset by a number of conceptual and technical diKculties [51 Intrinsic clocks are strange contrap...

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