The exact Wheeler-DeWitt equation for the scale-factor minisuperspace model

TL;DR

This paper derives the exact Wheeler-DeWitt equation for a closed, homogeneous, isotropic universe with a positive cosmological constant, showing that different quantization choices lead to equivalent quantum descriptions.

Contribution

It provides a method to determine the exact form of the Wheeler-DeWitt equation for each path-integral measure and shows the fundamental equivalence of different quantum theories for the model.

Findings

Derived the exact Wheeler-DeWitt equation including all quantum corrections.

Established the universal form of the wavefunction satisfying a measure-independent equation.

Demonstrated the equivalence of quantum theories from different quantization schemes.

Abstract

We consider the classical minisuperspace model describing a closed, homogeneous and isotropic Universe, with a positive cosmological constant. Upon canonical quantization, the infinite number of possible operator orderings in the quantum Hamiltonian leads to distinct Wheeler-DeWitt equations for the wavefunction $\psi$ of the Universe. Similarly, ambiguity arises in the path-integral formulation of $\psi$, due to the infinite choices of path-integral measures for the single degree of freedom of the model. For each choice of path-integral measure, we determine the correct operator ordering of the quantum Hamiltonian and derive the exact form of the Wheeler-DeWitt equation, including all corrections in $\hbar$. Additionally, we impose Hermiticity of the Hamiltonian to deduce the Hilbert-space measure $\mu$, which appears in the definition of the Hermitian inner product. Remarkably, all quantum predictions can be expressed through the combination $\Psi=\sqrt{\mu}\, \psi$, which satisfies a universal Wheeler-DeWitt equation, independent of the initial path-integral prescription. This result demonstrates that all seemingly distinct quantum theories arising from different quantization schemes of the same classical model are fundamentally equivalent.