Restoring time dependence into quantum cosmology

Mini superspace cosmology treats the scale factor a(t), the lapse function n(t) and an optional dilation field (t) as canonical variables. While pre-fixing n(t) means losing the Hamiltonian constraint, pre-fixing a(t) is serendipitously harmless at this level. This suggests an alternative to the Hartle-Hawking approach, where the pre-fixed a(t) and its derivatives are treated as explicit functions of time, leaving n(t) and a now mandatory (t) to serve as canonical variables. The naive gauge pre-fix a(t) = const. is clearly forbidden, causing evolution to freeze altogether; so pre-fixing the scale factor, say a(t) = t, necessarily introduces explicit time dependence into the Lagrangian. Invoking Dirac's prescription for dealing with constraints, we construct the corresponding mini superspace time-dependent total Hamiltonian and calculate the Dirac brackets, characterized by {n}D= 0, which are promoted to commutation relations in the quantum theory.