Resulting Wheeler-DeWitt Equation Research Articles

Bouncing Quantum Cosmology

The goal of this contribution is to present the properties of a class of quantum bouncing models in which the quantum bounce originates from the Dirac canonical quantization of a midi-superspace model composed of a homogeneous and isotropic background, together with small inhomogeneous perturbations. The resulting Wheeler-DeWitt equation is interpreted in the framework of the de Broglie-Bohm quantum theory, enormously simplifying the calculations, conceptually and technically. It is shown that the resulting models are stable and they never get to close to the Planck energy, where another more involved quantization scheme would have to be evoked, and they are compatible with present observations. Some physical effects around the bounce are discussed, like baryogenesis and magnetogenesis, and the crucial role of dark matter and dark energy is also studied.

Quantum cosmology and the inflationary spectra from a nonminimally coupled inflaton

We calculate the quantum gravitational corrections to the Mukhanov-Sasaki equation obtained by the canonical quantization of the inflaton-gravity system. Our approach, which is based on the Born-Oppenheimer decomposition of the resulting Wheeler-DeWitt equation, was previously applied to a minimally coupled inflaton. In this article we examine the case of a non minimally coupled inflaton and, in particular, the induced gravity case is also discussed. Finally, the equation governing the quantum evolution of the inflationary perturbations is derived on a de Sitter background. Moreover the problem of the introduction of time is addressed and a generalized method, with respect to that used for the minimal coupling case, is illustrated. Such a generalized method can be applied to the universe wave function when, through the Born-Oppenheimer factorization, we decompose it into a part which contains the minisuperspace degrees of freedom and another which describes the perturbations.

Classical and quantum Bianchi type III vacuum Hořava–Lifshitz cosmology

A diagonal Bianchi type III spacetime is treated, both at the classical and quantum level, in the context of Hořava–Lifshitz gravity. The system of the classical equations of motion is reduced to one independent Abel’s equation of the first kind. Closed form solutions are presented for various values of the coupling constants appearing in the action. Due to the method used, solutions of Euclidean, Lorentzian and neutral signatures are attained. The main general solution for λ=1 is seen to contain a curvature singularity at t=0, while in its Einsteinian limit it reproduces the known diagonal solution (with and/or without cosmological constant). The rest of the solutions correspond to somewhat isolated cases, as they emerge from either setting the coefficient of the lapse to zero or taking λ=1/3. They are singularity free, but they all have a singular limit as the other constants approach their Einsteinian values. At the quantum level, the resulting Wheeler–DeWitt equation is explicitly solved for λ=1,σ=0 and λ=13. The ensuing wave functions diverge in the Einsteinian limit.

A field theoretic generalization of Hajicek and Kuchar's quantization scheme in 3+1 canonical quantum gravity

The 3+1 (canonical) decomposition of all geometries admitting two-dimensional space-like surfaces is exhibited as a generalization of a previous work. A proposal, consisting of a specific re-normalization Assumption and an accompanying Requirement, which has been put forward in the 2+1 case is now generalized to 3+1 dimensions. This enables the canonical quantization of these geometries through a generalization of Kuchař's quantization scheme in the case of infinite degrees of freedom. The resulting Wheeler-deWitt equation is based on a re-normalized manifold parameterized by three smooth scalar functionals. The entire space of solutions to this equation is analytically given, a fact that is entirely new to the present case. This is made possible by exploiting the freedom left by the imposition of the Requirement and contained in the third functional.

Towards canonical quantum gravity for 3+1 geometries admitting maximally symmetric two-dimensional surfaces

The canonical decomposition of all 3+1 geometries admitting two-dimensional space-like surfaces is exhibited as a generalization of a previous work. A proposal, consisting of a specific renormalization Assumption and an accompanying Requirement, which has been put forward in the 2+1 case is now generalized to 3+1 dimensions. This enables the canonical quantization of these geometries through a generalization of Kuchař's quantization scheme in the case of infinite degrees of freedom. The resulting Wheeler–DeWitt equation is based on a renormalized manifold parameterized by three smooth scalar functionals. The entire space of solutions to this equation is analytically given, a fact that is entirely new to the present case. This is made possible through the exploitation of the residual freedom in the choice of the third functional, which is left by the imposition of the Requirement, and is proven to correspond to a general coordinate transformation in the renormalized manifold.

Generalization of Hajicek and Kuchař's canonical quantization scheme to the 3+1 geometries admitting maximally symmetric two‐dimensional surfaces

AbstractThe 3+1 (canonical) decomposition of all geometries admitting two‐dimensional space‐like surfaces is exhibited. A proposal consisting of a specific re‐normalization Assumption and an accompanying Requirement is put forward, which enables the canonical quantization of these geometries. The resulting Wheeler–deWitt equation is based on a re‐normalized manifold parameterized by three smooth scalar functionals and the entire space of solutions to this equation is analytically given.

Generalization of Hajicek and Kuchař's canonical quantization scheme to the 3+1 geometries admitting maximally symmetric two-dimensional surfaces

The 3+1 (canonical) decomposition of all geometries admitting two-dimensional space-like surfaces is exhibited. A proposal consisting of a specific re-normalization Assumption and an accompanying Requirement is put forward, which enables the canonical quantization of these geometries. The resulting Wheeler–deWitt equation is based on a re-normalized manifold parameterized by three smooth scalar functionals and the entire space of solutions to this equation is analytically given.

A cosmological viewpoint on the correspondence between deformed phase-space and canonical quantization

We employ the familiar canonical quantization procedure in a given cosmological setting to argue that it is equivalent to and results in the same physical picture if one considers the deformation of the phase-space instead. To show this we use a Probabilistic Evolutionary Process (PEP) to make the solutions of these different approaches comparable. Specific model theories are used to show that the independent solutions of the resulting Wheeler-DeWitt equation are equivalent to solutions of the deformation method with different signs for the deformation parameter. We also argued that since the Wheeler-DeWitt equation is a direct consequence of diffeomorphism invariance, this equivalence is only true provided that the deformation of phase-space does not break such an invariance.

Chaplygin gas quantum universe in the presence of the cosmological constant

We present a Chaplygin gas Friedmann-Robertson-Walker quantum cosmological model in the presence of the cosmological constant. We apply the Schutz’s variational formalism to recover the notion of time, and this gives rise to Wheeler-DeWitt equation for the scale factor. We study the early and late time universes and show that the presence of the Chaplygin gas leads to an effective positive cosmological constant for the late times. This suggests the possibility of changing the sign of the effective cosmological constant during the transition from the early times to the late times. For the case of an effective negative cosmological constant for both epoches, we solve the resulting Wheeler-DeWitt equation using the Spectral Method and find the eigenvalues and eigenfunctions for positive, zero, and negative constant spatial curvatures. Then, we use the eigenfunctions in order to construct wave packets for each case and obtain the time-dependent expectation value of the scale factors, which are found to oscillate between finite maximum and minimum values. Since the expectation value of the scale factors never tend to the singular point, we have an initial indication that this model may not have singularities at the quantum level.

Quantization of closed mini-superspace models as bound states

The Wheeler-DeWitt equation is applied to closedk>0 Friedmann-Robertson-Walker metric with various combination of cosmological constant and matter (e.g., radiation or pressureless gas). It is shown that if the universe ends in the matter dominated era (e.g., radiation or pressureless gas) with zero cosmological constant, then the resulting Wheeler-DeWitt equation describes a bound state problem. As solutions of a nondegenerate bound state system, the eigen-wave functions are real (Hartle-Hawking). Furthermore, as a bound state problem, there exists a quantization condition that relates the curvature of the three space with the various energy densities of the universe. If we assume that our universe is closed, then the quantum number of our universe isN∼(Gk)−1∼10122. The largeness of this quantum number is naturally explained by an early inflationary phase which resulted in a flat universe we observe today. It is also shown that if there is a cosmological constant Λ>0 in our universe that persists for all time, then the resulting Wheeler-DeWitt equation describes a non-bound state system, regardless of the magnitude of the cosmological constant. As a consequence, the wave functions are in general complex (Vilenkin).

Quantization of Robertson-Walker geometry coupled to a spin-(3/2 field.

A Robertson-Walker geometry coupled to a spin-(3/2 field is quantized applying Dirac's theory for constrained Hamiltonian systems. An ansatz for the matter field is made, dictated both by the iso- tropy and homogeneity of the geometry and the need to have no negative-norm states at the quantum level. This ansatz also avoids the difficulties of the derivative coupling naturally brought by the spin-(3/2 field. After the factor-ordering problem has been resolved the resulting Wheeler-DeWitt equation is solved. It is thus found that the probability density of finding the system with a scale factor R vanishes, as R approaches the classically singular value R=0.