Wheeler-DeWitt Equation Research Articles

Spinorial Wheeler–DeWitt wave functions inside black hole horizons

We revisit the solutions of the Wheeler–DeWitt (WDW) equation inside the horizons of spherical black holes and planar topological black holes in arbitrary dimensions. For these systems, the solutions of the equations are found to have the same form. Therefore, Yeom’s Annihilation-to-nothing interpretation can be applied to each case. We have introduced the Dirac-type WDW equations into quantum cosmology in a recent paper, so we also apply our formulation to the quantum theory of the interior of the black hole in order to obtain the solution of the spinorial wave function. The shape of the wave packet of the spinorial WDW wave function indicates that the variation of Yeom’s interpretation holds in this scheme.

F(R) gravity for a FLRW universe in a deformed phase space

Modify gravity theories have received huge attention in the last decade. In this work, we find the Wheeler-DeWitt (WDW) equation in the Quantum Cosmology (QC) scenario for a Friedmann-Lemaitre-Robertson-Walker (FLRW) model using the deformed phase space in f(R) gravity. In addition to the already stated, we consider a function fΛ(R) that provides the Λ-Cold Dark Matter (ΛCDM) model as an immediate consideration.

Raychaudhuri equation from Lagrangian and Hamiltonian formulation: A quantum aspect

The paper deals with a suitable transformation related to the metric scalar of the hyper-surface so that the Raychaudhuri Equation (RE) can be written as a second order nonlinear differential equation. A first integral of this second order differential equation gives a possible analytic solution of the RE. Also, it is shown that construction of a Lagrangian (and hence a Hamiltonian) is possible, from which the RE can be derived. Wheeler-Dewitt equation has been formulated in canonical quantization scheme and norm of its solution (wave function of the universe) is shown to affect the singularity analysis in the quantum regime for any spatially homogeneous and isotropic cosmology. Finally Bohmian trajectories are formulated with causal interpretation and these quantum trajectories unlike classical geodesics obliterate the initial big-bang singularity when the quantum potential is included.

Quasi-Hermitian Formulation of Quantum Mechanics Using Two Conjugate Schrödinger Equations

To the existing list of alternative formulations of quantum mechanics, a new version of the non-Hermitian interaction picture is added. What is new is that, in contrast to the more conventional non-Hermitian model-building recipes, the primary information about the observable phenomena is provided not only by the Hamiltonian but also by an additional operator with a real spectrum (say, R(t)) representing another observable. In the language of physics, the information carried by R(t)≠R†(t) opens the possibility of reaching the exceptional-point degeneracy of the real eigenvalues, i.e., a specific quantum phase transition. In parallel, the unitarity of the system remains guaranteed, as usual, via a time-dependent inner-product metric Θ(t). From the point of view of mathematics, the control of evolution is provided by a pair of conjugate Schrödiner equations. This opens the possibility od an innovative dyadic representation of pure states, by which the direct use of Θ(t) is made redundant. The implementation of the formalism is illustrated via a schematic cosmological toy model in which the canonical quantization leads to the necessity of working with two conjugate Wheeler-DeWitt equations. From the point of view of physics, the “kinematical input” operator R(t) may represent either the radius of a homogeneous and isotropic expanding empty Universe or, if you wish, its Hubble radius, or the scale factor a(t) emerging in the popular Lemaitre-Friedmann-Robertson-Walker classical solutions, with the exceptional-point singularity of the spectrum of R(t) mimicking the birth of the Universe (“Big Bang”) at t=0.

Minisuperspace quantum cosmology in f(Q) gravity

f(Q) symmetric-teleparallel gravity is considered in view of quantum cosmology. Specifically, we derive cosmological equations for f(Q) models and then investigate the related energy conditions. In the Minisuperspace formalism, the point-like f(Q) Hamiltonian is taken into account. In this framework, we obtain and solve the Wheeler–De Witt equation, thus finding the wave function of the universe in different cases. We show that the Hartle criterion can be applied and classical observable universes occur.

On the decoherence of primordial gravitons

It is well-known that the primordial scalar curvature and tensor perturbations, ζ and γij, are conserved on super-horizon scales in minimal inflation models. However, their wave functional has a rapidly oscillating phase which is slow-roll unsuppressed, as can be seen either from boundary (total-derivative) terms of cosmological perturbations, or the WKB approximation of the Wheeler-DeWitt equation. Such an oscillatory phase involves gravitational non-linearity between scalar and tensor perturbations. By tracing out unobserved modes, the oscillatory phase causes faster decoherence of primordial gravitons compared to those by bulk interactions. Our results put a stronger lower bound of decoherence effect to the recent proposals probing squeezed primordial gravitons.

A Wheeler–DeWitt Quantum Approach to the Branch-Cut Gravitation with Ordering Parameters

In this contribution to the Festschrift for Prof. Remo Ruffini, we investigate a formulation of quantum gravity using the Hořava–Lifshitz theory of gravity, which is General Relativity augmented by counter-terms to render the theory regularized. We are then led to the Wheeler–DeWitt (WDW) equation combined with the classical concepts of the branch-cut gravitation, which contemplates as a new scenario for the origin of the Universe, a smooth transition region between the contraction and expansion phases. Through the introduction of an energy-dependent effective potential, which describes the space-time curvature associated with the embedding geometry and its coupling with the cosmological constant and matter fields, solutions of the WDW equation for the wave function of the Universe are obtained. The Lagrangian density is quantized through the standard procedure of raising the Hamiltonian, the helix-like complex scale factor of branched gravitation as well as the corresponding conjugate momentum to the category of quantum operators. Ambiguities in the ordering of the quantum operators are overcome with the introduction of a set of ordering factors α, whose values are restricted, to make contact with similar approaches, to the integers α=[0,1,2], allowing this way a broader class of solutions for the wave function of the Universe. In addition to a branched universe filled with underlying background vacuum energy, primordial matter and radiation, in order to connect with standard model calculations, we additionally supplement this formulation with baryon matter, dark matter and quintessence contributions. Finally, the boundary conditions for the wave function of the Universe are imposed by assuming the Bekenstein criterion. Our results indicate the consistency of a topological quantum leap, or alternatively a quantum tunneling, for the transition region of the early Universe in contrast to the classic branched cosmology view of a smooth transition.

Quintom fields from chiral anisotropic cosmology

In this paper we present an analysis of a chiral anisotropic cosmological scenario from the perspective of quintom fields. In this setup quintessence and phantom fields interact in a non-standard (chiral) way within an anisotropic Bianchi type I background. We present our examination from two fronts: classical and quantum approaches. In the classical program we find analytical solutions given by a particular choice of the emerged relevant parameters. Remarkably, we present an explanation of the “big-bang” singularity by means of a “big-bounce”. Moreover, isotropization is in fact reached as the time evolves. On the quantum counterpart the Wheeler–DeWitt equation is analytically solved for various instances given by the same parameter space from the classical study, and we also include the factor ordering $$\mathrm Q$$ . Having solutions in this scheme we compute the probability density, which is in effect damped as the volume function and the scalar fields evolve; and it also tends towards a flat FLRW framework when the factor ordering constant $$\textrm{Q}\ll 0$$ . This result might indicate that for a fixed set of parameters, the anisotropies quantum-mechanically vanish for very small values of the parameter $$\mathrm Q$$ . Finally, classical and quantum solutions reduce to their flat FLRW counterparts when the anisotropies vanish.

Quantum-gravitational corrections to the power spectrum for a closed universe

We study the quantum-gravitational corrections to the power spectrum of a gauge-invariant inflationary scalar perturbations in a closed model of a universe. We consider canonical quantum gravity as an approach to quantizing gravity. This leads to the Wheeler-DeWitt equation, which has been studied by applying a semiclassical Born–Oppenheimer type of approximation. At the corresponding orders of approximation, we recover both the uncorrected and quantum-gravitationally corrected Schrödinger equations for the perturbation modes from which we calculate the quantum-gravitational corrections to the power spectrum in the slow-roll regime. The results are compared to the power spectra for the flat model of the universe.

A local Wheeler-DeWitt measure for the string landscape

According to the ‘Cosmological Central Dogma’, de Sitter space can be viewed as a quantum mechanical system with a finite number of degrees of freedom, set by the horizon area. We use this assumption together with the Wheeler-DeWitt (WDW) equation to approach the measure problem of eternal inflation. Thus, our goal is to find a time-independent wave function of the universe on a total Hilbert space defined as the direct sum of a variety of subspaces: A finite-dimensional subspace for each de Sitter vacuum and an infinite-dimensional subspace for each terminal Minkowski or AdS vacuum. We argue that, to be consistent with semiclassical intuition, such a solution requires the presence of sources. These are implemented as an inhomogenous term in the WDW equation, induced by the Hartle-Hawking no-boundary or the Linde/Vilenkin tunneling proposal. Taken together, these steps unambiguously lead to what we would like to think of as a ‘Local WDW measure,’ where ‘local’ refers to the fact that the dS part of the resulting wave function describes a superposition of static patches. The global 3-sphere spatial section of the entire multiverse makes no appearance.

Spinor wave function of the Universe in non-minimally coupled varying constants cosmologies

In this paper, we introduce a non-minimally coupled varying speed of light and varying gravitational constant cosmological toy model. Using the Eisenhart–Duval lifting method, we extend the original minisuperspace of the model and depict the evolution of the system in the presence of the potential term as a geometrical flow associated with the lifted metric. We write the Dirac–Wheeler–DeWitt equation, which solution is a spinor wave function of the Universe. Then we find the solution of the Dirac–Wheeler–DeWitt equation, which describes the emergence of two early universe–antiuniverse pairs that differ with the conserved quantity, which is an analog of the spin.

A quantum cosmology approach to cosmic coincidence and inflation

This work studies the quantum cosmology of a closed, spatially homogeneous, and isotropic FLRW minisuperspace model with electromagnetic radiation as a matter content. We solve the associated Wheeler-DeWitt (WDW) equation using the holographic regularization method and show that the electromagnetic zero-point energy forms the vacuum energy and provides a unified resolution to the horizon, flatness, singularity, and cosmic coincidence problems. This quantum cosmology approach composes an alternative to the usual inflationary paradigm and can be extended to more general matter contents and emergent gravity schemes.

Quantum cosmology, eternal inflation, and swampland conjectures

In light of the recent swampland conjectures, we explore quantum cosmology and eternal inflation beyond the slow roll regime. We consider a model of a closed universe with a scalar field ϕ in the framework of tunneling approach to quantum cosmology. The scalar field potential is assumed to have a maximum at ϕ = 0 and can be approximated in its vicinity as V(ϕ)≈ 3H 2 - 1/2m 2 ϕ 2. Using the instanton method, we find that for m < 2H the dominant nucleation channel for the universe is tunneling to a homogeneous, spherical de Sitter space. For larger values of m/H, the most probable tunneling is to an inhomogeneous closed universe with a domain wall wrapped around its equator. We determine the quantum state of the field ϕ in the nucleated universe by solving the Wheeler-DeWitt equation with tunneling boundary conditions. Our results agree with earlier work which assumed a slow-roll regime m ≪ H. We finally show that spherical universes nucleating with m < 2H undergo stochastic eternal inflation with inflating regions forming a fractal of dimension d > 2. For larger values of m the field ϕ is unstable with respect to formation of domain walls and cannot be described by a perturbative stochastic approach.

QFT in curved spacetime from quantum gravity: Proper WKB decomposition of the gravitational component

Starting from a re-analysis of previous work, we construct the proper low energy quantum field theory (QFT) limit of a full quantum gravity theory in the Born-Oppenheimer approach. We separate the gravitational sector into a classical background, given by a vacuum diagonal Bianchi I cosmology, and its quantum perturbations represented by the two graviton degrees of freedom; we further include quantum matter in the form of a test scalar field. We then implement a Born-Oppenheimer separation, where the gravitons and matter play the role of "slow" and "fast" quantum components respectively, and perform a Wentzel-Kramers-Brillouin (WKB) expansion in a Planckian parameter. The functional Schr\"odinger evolution for matter is recovered after averaging over quantum gravitational effects, provided that a condition is imposed on the gravitons' wave functional. Such a condition fixes the graviton dynamics and is equivalent to the purely gravitational Wheeler-DeWitt constraint imposed in previous approaches. The main accomplishment of the present work is to clarify that QFT in curved spacetime can be recovered in the low energy limit of quantum gravity only after averaging over the graviton degrees of freedom, in the spirit of effective field theory. Furthermore, it justifies a posteriori the implementation of the gravitational Wheeler-DeWitt equation on the "slow" gravitons' wave functional rather than assuming its validity a priori.

Schrödinger from Wheeler–DeWitt: The issues of time and inner product in canonical quantum gravity

The wave-function in quantum gravity is supposed to obey the Wheeler–DeWitt (WDW) equation, however there is neither a satisfactory probability interpretation nor a successful solution to the problem of time in the WDW framework. To gain some insight on these issues we compare quantization of ordinary systems, first in the usual way having the Schrödinger equation and second by promoting them as parametrized theories by introducing embedding coordinate fields, which yields first class constraints and the WDW equation. We observe that the time evolution in the WDW framework can be described with respect to the embedding coordinates, where the WDW equation becomes Schrödinger like, i.e. it involves first order timelike functional derivatives. Moreover, the equivalence with the ordinary quantization procedure determines a suitable Hilbert space with a viable probability interpretation. We then apply the same construction to general relativity by adding embedding fields without any prior coordinate choice. The reparametrized general relativity has two different types of diffeomorphism invariance, which arises from world-volume and target-space reparametrizations. As in the case of ordinary systems, the time evolution can be described with respect to the embedding fields and the WDW equation becomes Schrödinger like; the construction is almost identical to an ordinary parametrized field theory in terms of time evolution and Hilbert space structure. However, this time, the constraint algebra enforces the wave-function to be in a subspace of states annihilated by an operator that can be identified as the Hamiltonian. The implications of these results for the canonical quantization program, and in particular for the minisuperspace quantum cosmology, are discussed.

On the Appearance of Time in the Classical Limit of Quantum Gravity

A possible solution of the problem of time in the Wheeler–DeWitt quantum geometrodynamics is that time appears within a semiclassical limit. Following this line of thinking, one can come to the Schrodinger equation for matter fields in curved spacetime with quantum-gravitational corrections. In the present paper, we study the semiclassical limit in the case of a closed isotropic model with a scalar field decomposed into modes. We analyse calculations made within frameworks of three approaches. The first approach was proposed by Kiefer and Singh. Since the Wheeler–DeWitt equation does not contain a time derivative, it is constructed by means of a special mathematical procedure, a time variable being a parameter along a classical trajectory of gravitational field. The second method was suggested in the paper of Maniccia and Montani, who introduced the Kuchař–Torre reference fluid as an origin of time. Furthermore, the third is the extended phase space approach to the quantisation of gravity. In this approach, the temporal Schrodinger equation is argued to be more fundamental than the Wheeler–DeWitt equation, and there is no problem of time. Time is introduced due to fixing a reference frame of a certain observer, who can register the macroscopic consequences of quantum gravitational phenomena in the Very Early Universe. To go to the semiclassical limit, the Born–Oppenheimer approximation for gravity is used. In each of the approaches, in the order of O(1/M), a temporal Schrödinger equation for matter fields in curved spacetime with quantum gravitational corrections is obtained. However, equations and corrections are different in various approaches, and the results depend on the additional assumptions made within the scopes of these approaches.

Spinor representation of the Hamiltonian constraint in 3D loop quantum gravity with a nonzero cosmological constant

We develop in a companion paper the kinematics of three-dimensional loop quantum gravity in Euclidean signature and with a negative cosmological constant, focusing in particular on the spinorial representation that is well known at zero cosmological constant. In this paper, we put this formalism to the test by quantizing the Hamiltonian constraint on the dual of a triangulation. The Hamiltonian constraints are obtained by projecting the flatness constraints onto spinors, as done in the flat case by the first author and Livine. Quantization then relies on $q$-deformed spinors. The quantum Hamiltonian constraint acts in the $q$-deformed spin network basis as difference equations on physical states, which are thus the Wheeler-DeWitt equations in this framework. Moreover, we study how physical states transform under Pachner moves of the canonical surface. We find that those transformations are in fact $q$ deformations of the transition amplitudes of the flat case as found by Noui and Perez. Our quantum Hamiltonian constraints, therefore, build a Turaev-Viro model at real $q$.

Wheeler-DeWitt states of the AdS-Schwarzschild interior

We solve the Wheeler-DeWitt equation for the planar AdS-Schwarzschild interior in a minisuperspace approximation involving the volume and spatial anisotropy of the interior. A Gaussian wavepacket is constructed that is peaked on the classical interior solution. Simple observables are computed using this wavepacket, demonstrating the freedom to a choose a relational notion of ‘clock’ in the interior and characterizing the approach to the spacelike singularity. The Wheeler-DeWitt equation may be extended out through the horizon, where it describes the holographic renormalization group flow of the black hole exterior. This amounts to the Hamilton-Jacobi evolution of the metric component gtt from positive interior values to negative exterior values. The interior Gaussian wavepacket is shown to evolve into the Lorentizan partition function of the boundary conformal field theory over a microcanonical energy window.

Quantization of a black-hole gravity: geometrodynamics and the quantum

Quantum gravity is effective in domains where both quantum effects and gravity are essential, such as in the vicinity of space-time singularities. This paper will investigate the quantization of a black-hole gravity, particularly the region surrounding the singularity at the origin of the coordinate system. Describing the system with a Hamiltonian formalism, we apply the covariant integral quantization method to find the Wheeler–DeWitt equation of the model. We find that the quantized system has a discrete energy spectrum in the region inside the event horizon. Through the Kantowski–Sachs metric, it is possible to correlate the entropic time, which gives the dynamics for this model, to the cosmic time in a non-trivial way. Different configurations for the phase space of a Schwarzschild black hole are obtained in a semi-classical analysis. For lower-energy states, the quantum corrections result in singularity removal and wormhole formation.

The paradigm of quantum cosmology through Dunkl fractional Laplacian operators and fractal dimensions

The semi-classical regime of fractional quantum cosmology has been discussed based on the Wheeler-DeWitt equation and ADM-Hamiltonian formalism. Fractionality has been introduced through two different approaches: the first one is based on fractional Dunkl Laplacian operator and the second one based on fractal calculus. Our models generalize the recent approaches of fractional quantum cosmology introduced in literature. In the first approach, it was revealed that power-law inflation and large number of e-folds may be obtained for various values of the fractional parameter. However, the second approach is plausible only if the cosmological constant decays with the scale factor of the universe. We have confronted our model with observations based on the Bayesian hierarchical model that describes the full distance ladder from nearby geometric anchors through Cepheids to Hubble-Flow SNe we have estimated the numerical values of the fractional order parameter. In both scenarios, the number of e-folds is too large.Pacs numbers: 05.45.Df, 45.10.Hj, 98.80.-k