  The condition for the unitarity of a quantum field is investigated in
semiclassical gravity from the Wheeler-DeWitt equation. It is found that the
quantum field preserves unitarity asymptotically in the Lorentzian universe,
but does not preserve unitarity completely in the Euclidean universe. In
particular we obtain a very simple matter field equation in the basis of the
generalized invariant of the matter field Hamiltonian whose asymptotic solution
is found explicitly.
