Proposal for solving the ‘‘problem of time’’ in canonical quantum gravity

The ``problem of time'' in canonical quantum gravity refers to the difficulties involved in defining a Hilbert space structure on states, and local observables on this Hilbert space, for a theory in which the spacetime metric is treated as a quantum field, so no classical metrical or causal structure is present on spacetime. We describe an approach, much in the spirit of ideas proposed by Misner, Kucha {r} {r}{}, and others, to defining states and local observables in quantum gravity which exploits the analogy between the Hamiltonian formulation of general relativity and that of a relativistic particle. In the case of minisuperspace models, a concrete theory is obtained which appears to be mathematically and physically viable, although it contains some radical features with regard to the presence of an ``arrow of time.'' The viability of this approach in the case of infinitely many degrees of freedom rests on a number of fairly well-defined issues, which, however, remain unresolved. As a by-product of our analysis, the theory of a relativistic particle in curved spacetime is developed.