INTERPRETATION OF TIME-REPARAMETRIZATION-INVARIANT QUANTUM-MECHANICS - AN EXACTLY SOLUBLE MODEL

The classical and quantum dynamics of simple time-reparametrization-in variant models containing two degrees of freedom are studied in detail . Elimination of one ''clock'' variable through the Hamiltonian constr aint leads to a description of time evolution for the remaining variab le which is essentially equivalent to the standard quantum mechanics o f an unconstrained system. In contrast with a similar proposal of Rove lli, evolution is with respect to a geometrical proper time, and the H eisenberg equation of motion is exact. The physical phase space contai ns no coordinate associated with the eliminated ''clock'' variable. Th erefore, 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, i n principle, by this observer. The possibility of a ''test clock,'' wh ich would reveal time evolution while contributing negligibly to the H amiltonian constraint, is examined, and found to be viable in the semi classical limit of large quantum numbers.