Interpretation of time-reparametrization-invariant quantum mechanics: An exactly soluble model

The classical and quantum dynamics of simple time-reparametrization-invariant models containing two degrees of freedom are studied in detail. Elimination of one ''clock'' variable through the Hamiltonian constraint leads to a description of time evolution for the remaining variable which is essentially equivalent to the standard quantum mechanics of an unconstrained system. In contrast with a similar proposal of Rovelli, evolution is with respect to a geometrical proper time, and the Heisenberg equation of motion is exact. The physical phase space contains no coordinate associated with the eliminated ''clock'' variable. Therefore, time evolution is not with respect to the observable readings of a physical clock. Rather, the eliminated variable can be construed as intrinsic to a particular observer, and, hence, as unobservable, in principle, by this observer. The possibility of a ''test clock,'' which would reveal time evolution while contributing negligibly to the Hamiltonian constraint, is examined, and found to be viable in the semiclassical limit of large quantum numbers.