Wheeler-DeWitt Equation Research Articles

Quantum cosmology in teleparallel gravity with a boundary term

We quantize a homogeneous and isotropic universe for two models of modified teleparallel gravity: one wherein an arbitrary function of the boundary term, namely B, is present in the action, and in the other model, a scalar field that is non-minimally coupled to both the torsion and boundary term. In this regard, we study exact solutions of both the classical and quantum frameworks by utilizing the corresponding Wheeler–DeWitt (WDW) equations of the models. To correspond to the comprehensive classical and quantum levels, in the second model, we propose an appropriate initial condition for the wave packets and observe that they closely adhere to the classical trajectories and reach their peak. We quantify this correspondence using the de Broglie–Bohm interpretation of quantum mechanics. According to this proposal, the classical and Bohmian trajectories coincide when the quantum potential vanishes along the Bohmian paths. Furthermore, we apply the de-parameterization technique to our model in the realm of the problem of time in quantum cosmological models based on the WDW equation, utilizing the global internal time denoted as χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi $$\\end{document}, which represents a scalar field.

Novel Phenomena of the Hartle–Hawking Wave Function

We find a novel phenomenon in the solution to the Wheeler–DeWitt equation by solving numerically the equation assuming O(4)-symmetry and imposing the Hartle–Hawking wave function as a boundary condition. In the slow-roll limit, as expected, the numerical solution gives the most dominant steepest-descent that describes the probability distribution for the initial condition of a universe. The probability is consistent with the Euclidean computations, and the overall shape of the wave function is compatible with analytical approximations, although there exist novel differences in the detailed probability computation. Our approach gives an alternative point of view for the no-boundary wave function from the wave function point of view. Possible interpretations and conceptual issues of this wave function are discussed.

Classical versus quantum features of the Berthelot cosmological model

In this paper, we compare and contrast the classical versus quantum dynamics of a cosmological model based on the literature (Modified) Berthelot equation of state for the description of the dark sector of the universe. At the classical background level we identify a Minkowski-like and a de Sitter-like equilibrium epochs, with the latter occurring only beyond a certain threshold for a parameter in the equation of state; at the classical perturbed level we find that this same parameter realizes a duality in the adiabatic speed of sound between the two equilibrium epochs. The quantum evolution of this model is studied in the context of quantum geometrodynamics by solving analytically the Wheeler–DeWitt equation in the Born–Oppenheimer approximation for the scalar field potentials about the two equilibrium epochs. We identify the phenomenon of quantum decoherence to arise at the same threshold which constitutes the bifurcation between the two equilibrium epochs at the classical level. We comment on the quantum modified power spectrum focusing on some consequences dealing with the formation of astrophysical structures within the Press–Schechter framework. Our paper is intended to scrutinize which classical features of a certain cosmological model are preserved at its quantum level, and under which assumptions.

Summing over non-singular paths in quantum cosmology

In this paper we provide the DeWitt propagator and its wave function in quantum cosmology using the path integral formulation of quantum gravity. The DeWitt boundary condition is introduced as a way to avoid the Big Bang singularity by positing that the wave function of the Universe vanishes near the Big Bang. However, there is currently no clear definition of the DeWitt boundary condition in the path integral formulation. To address this issue, we use the image method, which eliminates singular paths in the forbidden region of the infinite potential, and apply this method to quantum cosmology based on the Batalin–Fradkin–Vilkovisky formulation of the path integral. We investigate the validity of the image method, and in particular, find that this method is only appropriate when the potential has symmetry with respect to the boundary. We then show that the DeWitt propagator and the DeWitt wave function derived from the image method are consistent with solutions of the Wheeler–DeWitt equation for certain models of quantum cosmology.

Classical and quantum cosmology for two scalar field Brans–Dicke type theory: a Noether symmetry approach

The paper deals with a cosmological model containing two scalar fields which can be considered as an extension of the Brans–Dicke scalar field model. Due to highly coupled non linear field equations, Noether symmetry analysis has been imposed and as a result the Lagrangian as well as the field equations become much simple in form to have the classical solutions. The relevant cosmological parameters are analyzed graphically. Finally, quantum cosmology has been studied by constructing the Wheeler–DeWitt equation and the solution of this second order partial differential equation has been done using this symmetry analysis.

The Hilbert space of de Sitter quantum gravity

We obtain solutions of the Wheeler-DeWitt equation with positive cosmological constant for a closed universe in the large-volume limit. We argue that this space of solutions provides a complete basis for the Hilbert space of quantum gravity in an asymptotically de Sitter spacetime. Our solutions take the form of a universal phase factor multiplied by distinct diffeomorphism invariant functionals, with simple Weyl transformation properties, that obey the same Ward identities as a CFT partition function. The Euclidean vacuum corresponds to a specific choice of such a functional but other choices are equally valid. Each functional can be thought of as specifying a “theory” and, in this sense, the space of solutions is like “theory space”. We describe another basis for the Hilbert space where all states are represented as excitations of the vacuum that have a specific constrained structure. This gives the finite GN generalization of the basis proposed by Higuchi in terms of group averaging, which we recover in the nongravitational limit.

Inflationary quantum spectrum of the quasi-isotropic Universe

We investigate the quantum dynamics of the quasi-isotropic inflationary solution. This is achieved by deriving the Lagrangian and Hamiltonian for both the FLRW background and the inhomogeneous correction, via an expansion of the Einstein–Hilbert action up to second order in the perturbation amplitudes. Then we implement a semiclassical WKB scenario for which the inhomogeneous component of the Universe is treated as a “small” quantum subsystem, evolving on the classical isotropic background. Starting from the Wheeler–DeWitt equation, we recover a Schrödinger dynamics for the perturbations, in which the time dependence of the wave function emerges thanks to the classicality of the background, and we solve it for an inflationary phase. The main result of this paper is to show that, while the scalar component of the power spectrum has the standard scale invariant profile, the tensor one results to be not constrained by the inflationary expansion, apart from an overall normalization factor which guarantees a small tensor-to-scalar ratio. This means that the spatial distribution of the quasi-isotropic correction to the metric remains preserved, with the consequence that some information about the pre-inflationary Universe survives to the de Sitter expansion.

Phase space deformations in SUSY cosmology

In this paper we propose a SUSY generalization for deformed phase-space cosmology. In particular, scalar field and phantom cosmology are studied. We construct the supercharges of the models and the SUSY Wheeler-DeWitt equation. We also construct and derive the classical dynamics using the WKB approximation.

Gravitational collapse via Wheeler–DeWitt equation

We analyze the Wheeler–DeWitt (WDW) equation in the context of a gravitational collapse. The physics of an expanding/collapsing universe and many details of a collapsing star can classically be described by the Robertson–Walker metric in which the WDW equation takes the form of a times-less Schrödinger equation. We set up the corresponding WDW potential for the collapse and study the solutions of the wave function. The results show that the central singularity appearing in classical general relativity is avoided, the density is quantized in terms of the Planck density and the expectation value of the scale factor exhibits a discrete behavior.

Holography of information in de Sitter space

We study the natural norm on the space of solutions to the Wheeler-DeWitt equation in an asymptotically de Sitter spacetime. We propose that the norm is obtained by integrating the squared wavefunctional over field configurations and dividing by the volume of the diff-and-Weyl group. We impose appropriate gauge conditions to fix the diff- and-Weyl redundancy and obtain a finite expression for the norm using the Faddeev-Popov procedure. This leads to a ghost action that has zero modes corresponding to a residual conformal subgroup of the diff-and-Weyl group. By keeping track of these zero modes, we show that Higuchi’s norm for group-averaged states emerges from our prescription in the nongravitational limit. We apply our formalism to cosmological correlators and propose that they should be understood as gauge-fixed observables. We identify the symmetries of these observables. In a nongravitational theory, it is necessary to specify such correlators everywhere on a Cauchy slice to identify a state in the Hilbert space. In a theory of quantum gravity, we demonstrate a version of the principle of holography of information: cosmological correlators in an arbitrarily small region suffice to completely specify the state.

Noncommutative branch‐cut quantum gravity with a self‐coupling inflaton scalar field: The wave function of the Universe

AbstractThis article focuses on the implications of a noncommutative formulation of branch‐cut quantum gravity. Based on a mini‐superspace structure that obeys the noncommutative Poisson algebra, combined with the Wheeler–DeWitt equation and Hořava–Lifshitz quantum gravity, we explore the impact of a scalar field of the inflaton‐type in the evolution of the Universe's wave function. Taking as a starting point the Hořava–Lifshitz action, which depends on the scalar curvature of the branched Universe and its derivatives, the corresponding wave equations are derived and solved. The noncommutative quantum gravity approach adopted preserves the diffeomorphism property of General Relativity, maintaining compatibility with the Arnowitt–Deser–Misner Formalism. In this work we delve deeper into a mini‐superspace of noncommutative variables, incorporating scalar inflaton fields and exploring inflationary models, particularly chaotic and nonchaotic scenarios. We obtained solutions to the wave equations without resorting to numerical approximations. The results indicate that the noncommutative algebraic space captures low and high spacetime scales, driving the exponential acceleration of the Universe.

The branch‐cut quantum gravity with a self‐coupling inflation scalar field: Dynamical equations

AbstractThis article focuses on the implications of the recently developed commutative formulation based on branch‐cutting cosmology, the Wheeler–DeWitt equation, and Hořava–Lifshitz quantum gravity. Assuming a mini‐superspace of variables, we explore the impact of an inflaton‐type scalar field on the dynamical equations that describe the trajectories evolution of the scale factor of the Universe, characterized by the dimensionless helix‐like function . This scale factor characterizes a Riemannian foliated spacetime that topologically overcomes the big bang and big crunch singularities. Taking the Hořava–Lifshitz action as our starting point, which depends on the scalar curvature of the branched Universe and its derivatives, with running coupling constants denoted as , the commutative quantum gravity approach preserves the diffeomorphism property of General Relativity, maintaining compatibility with the Arnowitt–Deser–Misner formalism. We investigate both chaotic and nonchaotic inflationary scenarios, demonstrating the sensitivity of the branch‐cut Universe's dynamics to initial conditions and parameterizations of primordial matter content. The results suggest a continuous connection of Riemann surfaces, overcoming primordial singularities and exhibiting diverse evolutionary behaviors, from big crunch to moderate acceleration.

The branch‐cut quantum gravity with a self‐coupling inflation scalar field: The wave function of the Universe

AbstractThis paper focuses on the implications of a commutative formulation that integrates branch‐cutting cosmology, the Wheeler–DeWitt equation, and Hořava–Lifshitz quantum gravity. Building on a mini‐superspace structure, we explore the impact of an inflaton‐type scalar field on the wave function of the Universe. Specifically analyzing the dynamical solutions of branch‐cut gravity within a mini‐superspace framework, we emphasize the scalar field's influence on the evolution of the evolution of the wave function of the Universe. Our research unveils a helix‐like function that characterizes a topologically foliated spacetime structure. The starting point is the Hořava–Lifshitz action, which depends on the scalar curvature of the branched Universe and its derivatives, with running coupling constants denoted as . The corresponding wave equations are derived and are resolved. The commutative quantum gravity approach preserves the diffeomorphism property of General Relativity, maintaining compatibility with the Arnowitt–Deser–Misner formalism. Additionally, we delve into a mini‐superspace of variables, incorporating scalar‐inflaton fields and exploring inflationary models, particularly chaotic and nonchaotic scenarios. We obtained solutions for the wave equations without recurring to numerical approximations.

Cosmological quantum states of de Sitter-Schwarzschild are static patch partition functions

We solve the Wheeler-DeWitt equation in the ‘cosmological interior’ (the past causal diamond of future infinity) of four dimensional dS-Schwarzschild spacetimes. Within minisuperspace there is a basis of solutions labelled by a constant c, conjugate to the mass of the black hole. We propose that these solutions are in correspondence with partition functions of a dual quantum mechanical theory where c plays the role of time. The quantum mechanical theory lives on worldtubes in the ‘static patch’ of dS-Schwarzschild, and the partition function is obtained by evolving the corresponding Wheeler-DeWitt wavefunction through the cosmological horizon, where a metric component gtt changes sign. We establish that the dual theory admits a symmetry algebra given by a central extension of the Poincaré algebra e\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{e} $$\\end{document}(1, 1) and that the entropy of the dS black hole is encoded as an averaging of the dual partition function over the background gtt.

Just some simple (but nontrivial) analytical solutions for de Broglie–Bohm quantum cosmology

In this work, we provide some simple analytical solutions to the Wheeler–DeWitt equation for the minisuperspace applied to de Broglie–Bohm quantum cosmology for particular potentials of a scalar matter field ϕ. One solution describes a constantly rolling scalar field with a quadratic potential plus a negative cosmological constant. The scale factor is given by a Gaussian function. The other solution describes a flat Minkowski universe with the potential of a hilltop form.

Big-bang is a boundary condition

There is a common expectation that the big-bang singularity must be resolved in quantum gravity but it is not clear how this can be achieved. A major obstacle here is the difficulty of interpreting wave-functions in quantum gravity. The standard quantum mechanical framework requires a notion of time evolution and a proper definition of an invariant inner product having a probability interpretation, both of which are seemingly problematic in quantum gravity. We show that these two issues can actually be solved by introducing the embedding coordinates as dynamical variables à la Isham and Kuchar. The extended theory is identical to general relativity but has a larger group of gauge symmetries. The Wheeler-DeWitt equations describe the change of the wave-function from one arbitrary spacelike slice to another, however the constraint algebra makes this evolution purely kinematical and furthermore enforces the wave-function to be constrained in the subspace of zero-energy states. An inner product can also be introduced having all the necessary requirements. In this formalism big-bang appears as a finite field space boundary on which certain boundary conditions must be imposed for mathematical consistency. We explicitly illustrate this point both in the full theory and in the minisuperspace approximation.

Cosmogenesis as symmetry transformation

We consider the quantized bi-scalar gravity, which may serve as a locally Lorentz invariant cosmological model with varying speed of light and varying gravitational constant. The equation governing the quantum regime for the case of homogeneous and isotropic cosmological setup is a Dirac-like equation which replaces the standard Wheeler–DeWitt equation. We show that particular cosmogenesis may occur as a result of the action of the symmetry transformation which due to Wigner’s theorem can either be unitary or antiunitary. We demonstrate that the transition from the pre-big-bang contraction to the post-big-bang expansion – a scenario that also occurs in string quantum cosmologies – can be attributed to the action of charge conjugation, which belongs to the class of antiunitary transformations. We also demonstrate that the emergence of the two classical expanding post-big-bang universe–antiuniverse pairs, each with opposite spin projections, can be understood as being triggered by the action of a unitary transformation resembling the Hadamard gate.

Quantum Big Bounce of the Isotropic Universe Using Relational Time

We analyze the canonical quantum dynamics of the isotropic Universe with a metric approach by adopting a self-interacting scalar field as relational time. When the potential term is absent, we are able to associate the expanding and collapsing dynamics of the Universe with the positive- and negative-frequency modes that emerge in the Wheeler–DeWitt equation. On the other side, when the potential term is present, a non-zero transition amplitude from positive- to negative-frequency states arises, as in standard relativistic scattering theory below the particle creation threshold. In particular, we are able to compute the transition probability for an expanding Universe that emerges from a collapsing regime both in the standard quantization procedure and in the polymer formulation. The probability distribution results similar in the two cases, and its maximum takes place when the mean values of the momentum essentially coincide in the in-going and out-going wave packets, as it would take place in a semiclassical Big Bounce dynamics.

A description of classical and quantum cosmology for a single scalar field torsion gravity

In the background of homogeneous and isotropic flat FLRW spacetime, both classical and quantum cosmologies have been studied for teleparallel dark energy (DE) model. Using Noether symmetry analysis, not only the symmetry vector but also the coupling function in the Lagrangian and the potential of the scalar field have been determined. Also symmetry analysis identifies a cyclic variable in the Lagrangian along the symmetry vector and as a result the Lagrangian simplifies to a great extent so that classical solution is obtained. Subsequently, in quantum cosmology Wheeler–DeWitt (WD) equation has been constructed and the quantum version of the conserved momenta corresponding to Noether symmetry identifies the periodic part of the wave function of the universe and as a result the WD equation becomes solvable. Finally, quantum description shows finite nonzero probability at the classical big bang singularity.

Wheeler-DeWitt equation beyond the cosmological horizon: Annihilation to nothing, infinity avoidance, and loss of quantum coherence

Wheeler-DeWitt equation beyond the cosmological horizon: Annihilation to nothing, infinity avoidance, and loss of quantum coherence