  By parametrizing the action integral for the standard Schrodinger equation we
present a derivation of the recently proposed method for quantizing a
parametrized theory. The reformulation suggests a natural extension from
conventional to nonlinear quantum mechanics. This generalization enables a
unitary description of the quantum evolution for a broad class of constrained
Hamiltonian systems with a nonlinear kinematic structure. In particular, the
new theory is applicable to the quantization of cosmological models where a
chosen gravitational degree of freedom acts as geometric time. This is
demonstrated explicitly using three cosmological models: the Friedmann universe
with a massless scalar field and Bianchi type I and IX models. Based on these
investigations, the prospect of further developing the proposed quantization
scheme in the context of quantum gravity is discussed.
