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

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 (WDW) equations for the wavefunction ψ of the Universe. Similarly, ambiguity arises in the path-integral formulation of ψ , 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 WDW equation, including all corrections in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ℏ</mml:mi> </mml:mrow> </mml:math> . Additionally, we impose Hermiticity of the Hamiltonian to deduce the Hilbert-space measure µ , which appears in the definition of the Hermitian inner product. Remarkably, all quantum predictions can be expressed through the combination <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">Ψ</mml:mi> <mml:mo>=</mml:mo> <mml:msqrt> <mml:mi>μ</mml:mi> </mml:msqrt> <mml:mi>ψ</mml:mi> </mml:mrow> </mml:math> , which satisfies a universal WDW 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.