  As time is not an observable, we use Fisher information (FI) to address the
problem of time. We show that the Hamiltonian constraint operator cannot be
used to analyze any quantum process for quantum geometries that are associated
with time-reparametrization invariant classical geometries. This is because the
Hamiltonian constraint does not contain FI about time. We demonstrate that
although the Hamiltonian operator is the generator of time, the Hamiltonian
constraint operator can not observe the change that arises through the passage
of time. This means that the problem of time is inescapably problematic in the
associated quantum gravitational theories. Although we explicitly derive these
results on the world-sheet of bosonic strings, we argue that it holds in
general. We also identify an operator on the world-sheet which contains FI
about time in a string theoretical processes. Motivated by this observation, we
propose that a criteria for a meaningful operator of any quantum gravitational
process, is that it should contain non-vanishing FI about time.
