  Covariant phase space quantization attempts to quantize the full space of
classical solutions, leading to a quantum theory in which the usual time
coordinate is missing. In this paper we explore how the time evolution of the
quantum states of this system is restored. We compare covariant phase space
methods for a nonrelativistic system (the harmonic oscillator) and three
relativistic quantum cosmologies. We find that for the nonrelativistic case the
preferred time coordinate emerges in a phase factor when one transforms among
certain representations, giving exactly the phase that appears when the time
evolution operator acts on the stationary states in the usual quantization
method. For the relativistic cases, on the other hand, this phase factor is no
longer present, and covariance appears to be restored. These results are
consistent with other quantization methods for these models.
