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arXiv:2512.23656

Ordering-Independent Wheeler-DeWitt Equation for Flat Minisuperspace Models

Victor Franken, Eftychios Kaimakkamis, Hervé Partouche, Nicolaos Toumbas

hep-thgr-qcquant-phSubmitted Dec 29, 2025• Announced Dec 30, 2025• Revised May 16, 2026 (v2)
v1 · Dec 29, 2025v2 · May 16, 2026
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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.

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1+28 pages

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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 ℏ\hbarℏ. 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.
The DeWitt boundary condition can be implemented in the Lorentzian path integral of minisuperspace quantum cosmology by summing only over non-singular paths. In this paper, we present a general definition of the resulting DeWitt kernels, along with specific rules for their construction, and evaluate them rigorously. The lapse integral over the positive half-line yields the Dirichlet Green function of the Wheeler--DeWitt operator, whereas the integral over the whole real line yields the Dirichlet group-averaging kernel; both vanish when either endpoint lies at the singularity. Since the minisuperspace actions are at most quadratic, all lapse integrals are performed in closed form without saddle-point approximations. We obtain the kernels for the closed universe without cosmological constant, for the flat de Sitter universe, and, in terms of Airy functions, for the closed and open de Sitter universes. Throughout, the DeWitt condition is imposed as the boundary condition at the singularity, which selects a self-adjoint constraint operator on the half-line and fixes the kernels uniquely, including their normalization.
We give an explicit, rigorous framework for calculating quantum probabilities in a model theory of quantum gravity. Specifically, we construct the decoherence functional for the Wheeler-DeWitt quantization of a flat Friedmann-Robertson-Walker cosmology with a free, massless, minimally coupled scalar field, thus providing a complete decoherent histories formulation for this quantum cosmological model. The decoherence functional is applied to study predictions concerning the model's Dirac (relational) observables; the behavior of semiclassical states and superpositions of such states; and to study the singular behavior of quantum Wheeler-DeWitt universes. Within this framework, rigorous formulas are given for calculating the corresponding probabilities from the wave function when those probabilities may be consistently defined, thus replacing earlier heuristics for interpreting the wave function of the universe with explicit constructions. It is shown according to a rigorously formulated standard, and in a quantum-mechanically consistent way, that in this quantization these models are generically singular. Independent of the choice of state we show that the probability for these Wheeler-DeWitt quantum universes to ever encounter a singularity is unity. In addition, the relation between histories formulations of quantum theory and relational Dirac observables is clarified.
One of the central difficulties in the quantization of the gravitational interactions is that they are described by a set of constraints. The standard strategy for dealing with the problem is the Dirac quantization procedure, which leads to the Wheeler-DeWitt equation. However, solutions to the equation are known only for specific symmetry-reduced systems, including models of quantum cosmology. Novel methods, which enable solving the equation for complex gravitational configurations, are therefore worth seeking. Here, we propose and investigate a new method of solving the Wheeler-DeWitt equation for minisuperspace models (with a finite number of classical degrees of freedom). The approach employs variational quantum algorithms and is possible to implement on quantum computers. For this purpose, the gravitational system is regularized, by performing spherical compactification of the phase space. This makes the system's Hilbert space finite dimensional and allows to use SU(2)SU(2)SU(2) variables, which are easy to handle in quantum computing. The validity of the method is examined in the case of the flat de Sitter universe. Both an emulator of a quantum computer and the IBM superconducting quantum computer have been used. The advantages and limitations of the approach are discussed.
We construct a Bohmian quantum cosmological model for a spatially flat Friedmann-Robertson-Walker universe filled with a single scalar field whose potential provides a unified description of cold dark matter and dark energy at the background level. Starting from the Einstein-Hilbert action supplemented by a scalar field, we derive the minisuperspace Lagrangian and the associated canonical Hamiltonian formulation. By means of a nontrivial canonical transformation, the minisuperspace dynamics is mapped into that of a two-dimensional hyperbolic oscillator with a fixed frequency ratio, rendering the Wheeler-DeWitt equation exactly solvable by separation of variables. The resulting Wheeler-DeWitt solutions are expressed in terms of parabolic cylinder functions and are parametrised by a continuous separation constant, reflecting the constrained nature of the theory and the absence of a standard Schrödinger time parameter. Adopting the de Broglie-Bohm formulation, we derive deterministic guidance equations in minisuperspace and construct a well-defined Bohmian Hubble parameter directly in terms of the pilot-wave phase. Finally, we present a Wheeler-DeWitt-derived toy wave function for which the Bohmian trajectories and the associated cosmological expansion history can be obtained analytically, reproducing the late-time ΛCDM behaviour while exhibiting quantum modifications at earlier epochs.
A variational framework for the quantization of gravitational fields is developed based on an extension of the stationary action principle. Within this framework, the Wheeler-DeWitt equation for the gravitational wave functional is recovered without assuming operator promotion of the canonical momentum, thus avoiding the ambiguity of operator ordering in canonical quantization. The derivation is based on three main ingredients. First, motivated by information-theoretic considerations, the classical stationary action principle is generalized by incorporating a correction term constructed from the relative entropy associated with field fluctuations. Second, an ensemble formulation on superspace is enhanced to incorporate this entropy correction. Third, the formalism is further refined to provide a unified treatment of quantization and constraints, thereby addressing the long-standing ambiguity concerning the ordering of quantization and constraint reduction. The framework is then applied to gravitational fields coupled to a massless scalar field. Using an emergent time parameter defined via the rate equation of the gravitational fields, a Schrodinger equation for the scalar-field wave functional is recovered, supplemented by an additional quantum correction term suppressed at order Gℏ2G\hbar^2Gℏ2. Finally, we comment on possible connections between the notion of relative entropy employed here and holographic dualities in quantum gravity.
This paper exploits an analogy with a time-independent Schrodinger equation to propose a concrete interpretation of a 'wavefunction of the universe' psi which solves the Wheeler-DeWitt equation for a minisuperspace quantum cosmology, and then uses this interpretation to define information theoretic notions of entropy. Given a wavefunction psi depending, in a configuration space representation, on some set of variables q 0 , . . ., q N , one is instructed to pick one of the q A , say q 0 identical to a, as a 'time' and then view psi (Q; a) as an unnormalised conditional probability amplitude for finding the values Q identical to (q 1 , . . ., q N ) at 'time' a. By identifying the 'natural' measure dQ on an a=constant hypersurface in minisuperspace, rho (Q, Q'; a)dQ varies as psi *(Q; a) psi (Q'; a)dQ is interpreted as representing a density matrix in the Q-representation; and, by viewing the Q as abstract operators for fixed a, rho is interpreted as an abstract density matrix. This interpretation admits no a priori notion of unitary evolution with respect to a 'translation' da, unitarity being viewed instead as a derived concept arising only in a semiclassical limit. The object then is to use rho to define reasonable measures of 'coarse-grained entropy' in accord with physical intuition which could serve as a diagnostic for an 'arrow of time'. Thus, e.g., by interpreting the diagonal components rho ( lambda i , lambda i ; a) in some O a -representation as conditional probabilities p i for measuring the eigenvalues lambda i of some operator O a , one is led to consider an observable dependent entropy S 0 (a)=- Sigma i p i log p i . And similarly, at least when the a=constant hypersurfaces are flat, one can map psi into a 'Wigner function' f w and then use that f w to define a 'Wigner function entropy' S w .