  We investigate consequences of allowing the Hilbert space of a quantum system
to have a time-dependent metric. For a given possibly nonstationary quantum
system, we show that the requirement of having a unitary Schreodinger
time-evolution identifies the metric with a positive-definite (Ermakov-Lewis)
dynamical invariant of the system. Therefore the geometric phases are
determined by the metric. We construct a unitary map relating a given
time-independent Hilbert space to the time-dependent Hilbert space defined by a
positive-definite dynamical invariant. This map defines a transformation that
changes the metric of the Hilbert space but leaves the Hamiltonian of the
system invariant. We propose to identify this phenomenon with a quantum
mechanical analogue of the principle of general covariance of General
Relativity. We comment on the implications of this principle for geometrically
equivalent quantum systems and investigate the underlying symmetry group.
