  For some time the York time parameter has been identified as a candidate for
a physically meaningful time in cosmology. An associated Hamiltonian may be
found by solving the Hamiltonian constraint for the momentum conjugate to the
York time variable, although an explicit solution can only be found in highly
symmetric cases. The Poisson structure of the remaining variables is not
canonical. Here we quantise this dynamics in an anisotropic minisuperspace
model via a natural extension of canonical quantisation. The resulting quantum
theory has no momentum representation. Instead the position basis takes a
fundamental role. We illustrate how the quantum theory and the modified
representation of its momentum operators lead to a consistent theory in the
presence of the constraints that arose during the Hamiltonian reduction. We are
able to solve for the eigenspectrum of the Hamiltonian. Finally we discuss how
far the results of this model extend to the general non-homogeneous case, in
particular perturbation theory with York time.
