Extrinsic time in quantum cosmology

Extrinsic time is identified in most isotropic and homogeneous cosmological models by matching it with the ideal clock—a parametrized system whose only ‘‘degree of freedom’’ is time. Once this matching is established, the cosmological models are quantized in the same way as the ideal clock. The space of solutions of the Wheeler-DeWitt equation is turned into a Hilbert space by inserting a time-dependent operator in the inner product, yielding a unitary theory equivalent to the phase-space-reduced theory.