Ordering-Independent Wheeler-DeWitt Equation for Flat Minisuperspace Models

Abstract

We consider minisuperspace models with two-derivative kinetic terms, assuming a flat target space and a closed Universe. We show that, upon canonical quantization of the Hamiltonian, only a restricted class of operator orderings is compatible with the path-integral formulation. Remarkably, these orderings are physically equivalent to all orders in $\hbar$. More precisely, 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. These orderings are in one-to-one correspondence with the Jacobians arising from field redefinitions of a set of canonical fields. For each operator ordering consistent with a path-integral measure, we identify a positive definite Hilbert-space inner product. All such prescriptions define the same quantum theory, in the sense that they yield identical physical observables. We illustrate our formalism by applying it to de Sitter Jackiw-Teitelboim gravity and to the Starobinsky model.