Ordering-Independent Wheeler-DeWitt Equation for Flat Minisuperspace Models

TL;DR

The paper resolves operator-ordering ambiguities in the Wheeler–DeWitt equation for flat minisuperspace models with quadratic kinetic terms by showing that each admissible path-integral measure—equivalently, each Jacobian from field redefinitions—defines a specific ordering that leads to a universal dressed-WDW equation for the wavefunction. By constructing a Hermitian inner product with $\mu=J^2$ and introducing the dressed wavefunction $\Psi=J\psi$, the authors demonstrate that all consistent quantum theories are physically equivalent and share identical observables to all orders in $\hbar$, despite different choices of path-integral measure. The results are illustrated with explicit applications to Starobinsky inflation and de Sitter JT gravity, where the explicit WDW equations and positive inner products are derived for the flat minisuperspace. A key limitation is that the construction relies on flat target space; extending to curved ${\cal T}$ remains an open challenge, with potential implications for a broader class of quantum-gravity models.

Abstract

We consider minisuperspace models with quadratic kinetic terms, assuming a flat target space and a closed Universe. We show that, upon canonical quantization of the Hamiltonian, only a restricted subset of operator orderings is consistent with the path-integral viewpoint. Remarkably, all consistent orderings are physically equivalent to all orders in $\hbar$. Specifically, each choice of path-integral measure in the definition of the wavefunction path integral uniquely determines an operator ordering and hence a corresponding Wheeler-DeWitt equation. The consistent orderings are in one-to-one correspondence with the Jacobians associated with all field redefinitions of a set of canonical degrees of freedom. For each admissible operator ordering--or equivalently, each path-integral measure--we identify a definite, positive Hilbert-space inner product. All such prescriptions define the same quantum theory, in the sense that they lead to identical physical observables. We illustrate our formalism by applying it to de Sitter Jackiw-Teitelboim gravity and to the Starobinsky model.