  We investigate the origin of the arrow of time in quantum mechanics in the
context of quantum cosmology. The ``Copenhagen'' quantum mechanics of measured
subsystems incorporates a fundamental arrow of time. Extending discussions of
Aharonov, Bergmann and Lebovitz, Griffiths, and others we investigate a
generalized quantum mechanics for cosmology that utilizes both an initial and a
final density matrix to give a time-neutral formulation without a fundamental
arrow of time. Time asymmetries can arise for particular universes from
differences between their initial and final conditions. Theories for both would
be a goal of quantum cosmology. A special initial condition and a final
condition of indifference would be sufficient to explain the observed time
asymmetries of the universe. In this essay we ask under what circumstances a
completely time symmetric universe, with T-symmetric initial and final
condition, could be consistent with the time asymmetries of the limited domain
of our experience. We discuss the approach to equilibrium, the electromagnetic
system, and T-violation in the weak interactions in such universes.
  The decoherence of alternatives that is necessary for the prediction of
probabilities in the quantum mechanics of closed systems is the origin of a
fundamental limitation on quantum cosmologies with initial and final
conditions. The initial and final density matrices cannot both be pure; if they
are there is no decoherence and no prediction.
