In Hawking and Page's paper "Operator ordering and flatness of the universe", there is a brief discussion about whether to interpret $J = \frac{i}{2} ( \Psi \nabla \bar{\Psi} - \bar{\Psi} \nabla \Psi)$ as a probability current for a wave function of the universe. They conclude that this interpretation is infeasible since the Hartle-Hawking wave function of the universe constructed via path integration over compact metrics will be real-valued and thus $J$ will vanish. However I am not sure why the question comes up in the first place. It seems to me that a wave function would need to satisfy a Schroedinger equation in order for $J$ to be interpretable as a probability current, so I would not have thought wave functions of the universe, which should satisfy a Wheeler-DeWitt equation instead, should give rise to a probability current in the same way.

Is there some other justification for interpreting $J$ as a probability current, without invoking a Schroedinger equation? Or is it implied that we are singling out some mode in superspace (e.g. conformal factor) as an internal clock (but this seems like too big of a leap to be left implicit)? Or...?

Thanks for any insight!!