Classical and quantum cosmology with York time
  We consider a solution to the problem of time in quantum gravity by
deparameterisation of the ADM action in terms of York time, a parameter
proportional to the extrinsic curvature of a spatial hypersurface. We study a
minisuperspace model together with a homogeneous scalar field, for which we can
solve the Hamiltonian constraint exactly and arrive at an explicit expression
for the physical (non-vanishing) Hamiltonian. The scale factor and associated
momentum cease to be dynamical variables, leaving the scalar field as the only
physical degree of freedom. We investigate the resulting classical theory,
showing how the dynamics of the scale factor can be recovered via an
appropriate interpretation of the Hamiltonian as a volume. We then quantise the
system in the Schr\"odinger picture. In the quantum theory we recover the
dynamics of the scale factor by interpreting the spectrum and expectation value
of the Hamiltonian as being associated with volume rather than energy. If
trajectories in the sense of de~Broglie-Bohm are introduced in the quantum
theory, these are found to match those of the classical theory. We suggest that
these trajectories may provide the basis for a perturbation theory in which
both background and perturbations are quantised.
