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arXiv:2512.18818 · Physics Letters B

Bohmian quantum cosmology from the Wheeler-DeWitt equation

Spyros Basilakos, Gerasimos Kouniatalis, Emmanuel N. Saridakis, Charalampos Tzerefos

Published Mar 7, 2026• 2 citations

Abstract

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.

DOI: 10.1016/j.physletb.2026.140339

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The quantized Friedmann-Lema\^ıtre-Robertson-Walker (FLRW) model minimally coupled to a free massless scalar field is studied and interpreted in the Bohm-de Broglie framework. We analyze the quantum bohmian trajectories corresponding to a certain class of gaussian packets, solutions of the Wheeler-DeWitt equation. We show that these bohmian trajectories undergo an accelerated expansion in the middle of its evolution due to the presence of quantum cosmological effects in this period. It is shown that the luminosity-redshift relation in the quantum cosmological model can be made close to the corresponding relation coming from the classical model suplemented by a cosmological constant, for

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. In this way we have the posibility of interpreting the present observations of high redshift supernovae as the consequence of a quantum cosmological effect.

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.

Friedmann equations play a central role in cosmology for describing the evolution of the universe. Initially, the Friedmann equations were derived by Alexander Friedmann in 1922. He derived the equations from Einstein’s field equations for a universe which is spatially homogeneous and isotropic. The equations were again derived from Newtonian mechanics by Milne. However in this paper, we derive the spatially flat Friedmann equation from Wheeler-DeWitt equation. We apply the Bohm-de Broglie interpretation to the wave function of universe. In addition, we also set a condition to the wave function, where a large scale factor of the universe is needed. Key words: Canonical quantum cosmology, Friedmann equations, Wheeler-DeWitt equation.

We summarize a fully worked-out consistent histories formulation of a quantum cosmological model, and illustrate with an example calculation of the probability that the quantum universe becomes singular. Specifically, we take a complete Wheeler-DeWitt canonical quantization of a flat Friedmann-Robertson- Walker cosmological model sourced with a free massless scalar field, and construct the decoherence functional which determines which quantum histories may be consistently assigned probabilities. As an example, we exhibit the probability that the quantum universe is singular in the sense that it assumes zero volume. The classical solutions of this model are all singular; we show that the quantum solutions are as well.We comment on the crucial role played by quantum decoherence in arriving at this conclusion. © 2010 American Institute of Physics.

We consider a Bohmian approach to the Wheeler-DeWitt quantization of the Friedmann-Lema\^{\i}tre-Robertson-Walker model and investigate the question of whether or not there are singularities, in the sense that the Universe reaches zero volume. We find that for generic wave functions (i.e., nonclassical wave functions), there is a nonzero probability for a trajectory to be nonsingular. This should be contrasted to the consistent histories approach for which it was recently shown by Craig and Singh that there is always a singularity. This result illustrates that the question of singularities depends much on which version of quantum theory one adopts. This was already pointed out by Pinto-Neto et al., albeit with a different Bohmian approach. Our current Bohmian approach agrees with the consistent histories approach by Craig and Singh for single-time histories, unlike the one studied earlier by Pinto-Neto et al. Although the trajectories are usually different in the two Bohmian approaches, their qualitative behavior is the same for generic wave functions.