A formalism for the ambiguities of the Wheeler-DeWitt equation

We study ambiguities in the precise formulation of the Wheeler-DeWitt equation for the wavefunction of the Universe that arise due to different operator orderings in the quantum Hamiltonian. We first examine the simpler case of the 1-dimensional minisuperspace model and derive the inner product measure that renders the Hamiltonian hermitian. Based on this, we establish that the Hilbert space inner products and quantum probabilities are universal, free of any ambiguities, at the semiclassical level. Recasting the Wheeler-DeWitt equation in a form invariant under field redefinitions of the minisuperspace variable, we show that all ambiguity functions are contained in a higher order scalar function, which can be used to define classes of models with universal predictions to all orders in $\hbar$. We then generalize to minisuperspace models of arbitrary dimension, upon the inclusion of an arbitrary number of scalar matter fields. We show that the hermiticity of the Hamiltonian in these cases provides a set of constraints, which can be used to cast the WDW equation in a covariant form and establish the universality of the inner products at the semiclassical level. In these cases as well, all ambiguity functions appear in a higher order scalar function of the minisuperspace manifold, which leads to distinct universality classes.