  In this paper a new formulation of quantum dynamics of totally constrained
systems is developed, in which physical quantities representing time are
included as observables. In this formulation the hamiltonian constraints are
imposed on a relative probability amplitude functional $\Psi$ which determines
the relative probability for each state to be observed, instead of on the state
vectors as in the conventional Dirac quantization. This leads to a foliation of
the state space by linear manifolds on each of which $\Psi$ is constant, and
dynamics is described as linear mappings among acausal subspaces which are
transversal to these linear manifolds. This is a quantum analogue of the
classical statistical dynamics of totally constrained systems developed in the
previous paper. It is shown that if the von Neumann algebra $\C$ generated by
the constant of motion is of type I, $\Psi$ can be consistently normalizable on
the acausal subspaces on which a factor subalgebra of $\C$ is represented
irreducibly, and the mappings among these acausal subspaces are conformal. How
the formulation works is illustrated by simple totally constrained systems with
a single constraint such as the parametrized quantum mechanics, a relativistic
free particle in Minkowski and curved spacetimes, and a simple minisuperspace
model. It is pointed out that the inner product of the relative probability
amplitudes induced from the original Hilbert space picks up a special
decomposition of the wave functions to the positive and the negative frequency
modes.
