Matter Time in Canonical Quantum Gravity

Summary

Abstract

The functional Schrodinger approach to canonical quantum gravity requires the construction of time and frame variables from the canonical data. I review the difficulties of basing such variables purely on the geometric data. I describe different ways of introducing matter variables for this purpose. I argue that the simplest phenomenological medium, an incoherent dust, does more than one can reasonably expect. The comoving coordinates of the dust particles and the proper time along their worldlines become canonical coordinates in the phase space of the coupled system. The Hamiltonian constraint can easily be solved for the momentum canonically conjugate to the dust time. The ensuing Hamiltonian density has an extraordinary feature not encountered in other systems: it depends only on the geometric variables, not on the dust frame and the dust time. This has three important consequences. Firstly, the functional Schrödinger equation can be solved by separating the dust time from geometric variables. Secondly, the Hamiltonian densities strongly commute and can therefore be simultaneously defined by spectral analysis. Thirdly, the Schrodinger equation can be solved independently of the supermomentum constraint, which is then satisfied by parametrizing this solution.

Canonical quantum gravity

One can never pay off old debts; the most one can do is to acknowledge them. Not that I lacked opportunity to acknowledge those which I owe Charlie Misner. He helped to set the entire framework of canonical gravity, and there is hardly a paper which I wrote since coming to the United States in which I did not have the chance – and the pleasure – to refer to his contributions.