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We reviewed the canonical quantisation of the geometry of the spacetime in the cases of a simply and a non-simply connected manifold. In the former, we analysed the information contained in the solutions of the Wheeler--DeWitt equation and showed their interpretation in terms of the customary boundary conditions that are typically imposed on the semiclassical wave functions. In particular, we reviewed three different paradigms for the quantum creation of a homogeneous and isotropic universe. For the quantisation of a non-simply connected manifold, the best framework is the third quantisation formalism, in which the wave function of the universe is seen as a field that propagates in the space of Riemannian 3-geometries, which turns out to be isomorphic to a (part of a) 1 + 5 Minkowski spacetime. Thus, the quantisation of the wave function follows the customary formalism of a quantum field theory. A general review of the formalism is given, and the creation of the universes is analysed, including their initial expansion and the appearance of matter after inflation. These features are presented in more detail in the case of a homogeneous and isotropic universe. The main conclusion in both cases is that the most natural way in which the universes should be created is in entangled universe--antiuniverse pairs. [ABSTRACT FROM AUTHOR]

Given two sets S1, S2 and unital C*-algebras A1, A2 of functions thereon, we show that a map σ:S1⟶S2 can be lifted to a continuous map σ¯:specA1⟶specA2 iff σ*A2:={σ*f|f∈A2}⊆A1. Moreover, σ¯ is unique if existing, and injective iff σ*A2 is dense. Then, we apply these results to loop quantum gravity and loop quantum cosmology. For all usual technical conventions, we decide whether the cosmological quantum configuration space is embedded into the gravitational one; indeed, both are spectra of some C*-algebras, say, Acosm and Agrav, respectively. Typically, there is no embedding, but one can always get an embedding by the defining Acosm:=C*(σ*Agrav), where σ denotes the embedding between the classical configuration spaces. Finally, we explicitly determine C*(σ*Agrav) in the homogeneous isotropic case for Agrav generated by the matrix functions of parallel transports along analytic paths. The cosmological quantum configuration space so equals the disjoint union of R and the Bohr compactification of R, appropriately glued together. [ABSTRACT FROM AUTHOR]

The present work shows that the mathematical equivalence of the Jordan frame and its conformally transformed version, the Einstein frame, so as far as Brans–Dicke theory is concerned, survives a quantization of cosmological models, arising as solutions to the Brans–Dicke theory. We work with the Wheeler–deWitt quantization scheme and take up quite a few anisotropic cosmological models as examples. We effectively show that the transformation from the Jordan to the Einstein frame is a canonical one and hence two frames furnish equivalent description of same physical scenario. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the classical geometric flow as a dynamical system. We obtain an action for this system, such that its equation of motion is the Raychaudhuri equation. This action will be used to quantize this system. As the Raychaudhuri equation is the basis for deriving the singularity theorems, we will be able to understand the effects and such a quantization will have on the classical singularity theorems. Thus, quantizing the geometric flow, we can demonstrate that a quantum space-time is complete (nonsingular). This is because the existence of a conjugate point is a necessary condition for the occurrence of singularities, and we will be able to demonstrate that such conjugate points cannot occur due to such quantum effects. [ABSTRACT FROM AUTHOR]

We address the quantisation of a model that induces the Little Sibling of the Big Rip (LSBR) abrupt event, where the dark energy content is described by means of a phantom-like fluid or a phantom scalar field. The quantisation is done in the framework of the Wheeler-DeWitt (WDW) equation and imposing the DeWitt boundary condition; i.e., the wave function vanishes close to the abrupt event. We analyse the WDW equation within two descriptions: First, when the dark energy content is described with a perfect fluid. This leaves the problem with the scale factor as the single degree of freedom. Second, when the dark energy content is described with a phantom scalar field in such a way that an additional degree of freedom is incorporated. Here, we have applied the Born-Oppenheimer (BO) approximation in order to simplify the WDW equation. In all cases, the obtained wave function vanishes when the LSBR takes place, thus fulfilling the DeWitt boundary condition. [ABSTRACT FROM AUTHOR]

Within the framework of the third quantization, we consider the possibility that an initially recollapsing baby universe can enter a stage of near de Sitter inflation by tunnelling through a Euclidean wormhole that connects the recollapsing and inflationary geometries. We present the solutions for the evolution of the scale factor in the Lorentzian and Euclidean regions as well as the probability that the baby universe indeed crosses the wormhole when it reaches its maximum size. [ABSTRACT FROM AUTHOR]

In the canonical approach to quantization of gravity, one often uses relational clock variables and an interpretation in terms of conditional probabilities to overcome the problem of time. In this essay, we show that these suffer from serious conceptual issues. [ABSTRACT FROM AUTHOR]

In this paper we study the quantum cosmology of homogeneous and isotropic cosmology, via the Weyl-Wigner-Groenewold-Moyal formalism of phase space quantization, with perfect fluid as a matter source. The corresponding quantum cosmology is described by the Moyal-Wheeler-DeWitt equation which has exact solutions in Moyal phase space, resulting in Wigner quasiprobability distribution functions peaking around the classical paths for large values of scale factor. We show that the Wigner functions of these models are peaked around the non-singular universes with quantum modified density parameter of radiation. [ABSTRACT FROM AUTHOR]

We consider the loop quantization of the (diagonal) Bianchi type IX cosmological model. We explore different quantization prescriptions that extend the work of Wilson-Ewing and Singh. In particular, we study two different ways of implementing the so-called inverse triad corrections. We construct the corresponding Hamiltonian constraint operators and show that the singularity is formally resolved. We find the effective equations associated with the different quantization prescriptions, and study the relation with the isotropic model that, classically, is contained within the Bianchi IX model. Somewhat surprisingly, we find the most natural quantization does not reduce to the model. We use geometrically defined scalar observables to explore the physical implications of each of these theories. This is the first part in a series of papers analyzing different aspects of the Bianchi IX model, with inverse corrections, within loop quantum cosmology (LQC). [ABSTRACT FROM AUTHOR]

In the framework of Loop Quantum Cosmology (LQC), inhomogeneous models are usually quantized by means of a hybrid approach that combines loop quantization techniques with standard quantum field theory methods. This approach is based on a splitting of the phase space in a homogeneous sector, formed by global, zero-modes and an inhomogeneous sector, formed by the remaining, infinite number of modes, that describe the local degrees of freedom. Then, the hybrid quantization is attained by adopting a loop representation for the homogeneous gravitational sector, while a Fock representation is used for the inhomogeneities. The zero-mode of the Hamiltonian constraint operator couples the homogeneous and inhomogeneous sectors. The hybrid approach, therefore, is expected to provide a suitable quantum theory in regimes where the main quantum effects of the geometry are those affecting the zero-modes, while the inhomogeneities, still being quantum, can be treated in a more conventional way. This hybrid strategy was first proposed for the simplest cosmological midisuperspaces: the Gowdy models, and it has been later applied to the case of cosmological perturbations. This paper reviews the construction and main applications of hybrid LQC. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the quantum aspects of the big-brake singularity. This is a singularity where the expansion of the universe stops abruptly, with infinity deceleration, caused by the divergence of the pressure of the fluid which describes the matter content of the model. In order to obtain our results, we interpret the quantum solutions of the Wheeler-DeWitt equation obtained from the canonical quantization of the classical model using the de Broglie-Bohm (dBB) quantum theory. Analyzing the Bohmian trajectories, we show that when one approaches the big-brake singularity, the universe still stops expanding, but now with finite deceleration, and initiates a smooth contracting phase. The pressure and the curvature never diverge. [ABSTRACT FROM AUTHOR]

Quantum gravity is sometimes considered as a kind of metaphysical speculation. In this review, we show that, although still extremely difficult to reach, observational signatures can in fact be expected. The early universe is an invaluable laboratory to probe ‘Planck scale physics’. Focusing on loop quantum gravity as one of the best candidate for a non-perturbative and background-independent quantization of gravity, we detail some expected features. [ABSTRACT FROM AUTHOR]

We consider the k = 1 Friedman–Lemaitre–Robertson–Walker model within loop quantum cosmology, from the perspective of the two available quantization prescriptions. We focus our attention on the existence of the so-called ‘inverse corrections’ in the quantization process. We derive the corresponding quantum constraint operators, and study in detail the issue of the self-adjointness of the quantum Hamiltonian constraint for two different quantizations based on open holonomies. Furthermore, we analyse in detail the resulting effective theories, paying special attention to issues such as the boundedness of different scalar observables such as energy density and expansion. We show that the inclusion of the inverse corrections solves one of the main undesirable features of the previous formalism, namely the unboundedness of the energy density and expansion near the bounce on the full (effective) phase space. We briefly comment on possible consequences of this new feature. Communicated by P Singh [ABSTRACT FROM AUTHOR]

We study quantization ambiguities in loop quantum cosmology that arise for space-times with non-zero spatial curvature and anisotropies. Motivated by lessons from different possible loop quantizations of the closed Friedmann–Lemaître–Robertson–Walker cosmology, we find that using open holonomies of the extrinsic curvature, which due to gauge-fixing can be treated as a connection, leads to the same quantum geometry effects that are found in spatially flat cosmologies. More specifically, in contrast to the quantization based on open holonomies of the Ashtekar–Barbero connection, the expansion and shear scalars in the effective theories of the Bianchi type II and Bianchi type IX models have upper bounds, and these are in exact agreement with the bounds found in the effective theories of the Friedmann–Lemaître–Robertson–Walker and Bianchi type I models in loop quantum cosmology. We also comment on some ambiguities present in the definition of inverse triad operators and their role. [ABSTRACT FROM AUTHOR]

Simple Summary: We clarify the question of clock transformations and trajectories in quantum cosmology in a vacuum Bianchi I minisuperspace. We consider a simple cosmological model consisting of an empty Bianchi I Universe, whose Hamiltonian we deparametrise to provide a natural clock variable. The model thus effectively describes an isotropic universe with an induced clock given by the shear. By quantising this model, we obtain various different possible bouncing trajectories (semiquantum expectation values on coherent states or obtained by the de Broglie–Bohm formulation) and explicit their clock dependence, specifically emphasising the question of symmetry across the bounce. [ABSTRACT FROM AUTHOR]

We offer a new, physically transparent argument for the existence of the critical, universal maximum matter density in loop quantum cosmology for the case of a flat Friedmann-Lemâıtre-Robertson-Walker cosmology with scalar matter. The argument is based on the existence of a sharp exponential ultraviolet cutoff in momentum space on the eigenfunctions of the quantum cosmological dynamical evolution operator (the gravitational part of the Hamiltonian constraint), attributable to the fundamental discreteness of spatial volume in loop quantum cosmology. The existence of the cutoff is proved directly from recently found exact solutions for the eigenfunctions for this model. As a consequence, the operators corresponding to the momentum of the scalar field and the spatial volume approximately commute. The ultraviolet cutoff then implies that the scalar momentum, though not a bounded operator, is in effect bounded on subspaces of constant volume, leading to the upper bound on the expectation value of the matter density. The maximum matter density is universal (i.e. independent of the quantum state) because of the linear scaling of the cutoff with volume. These heuristic arguments are supplemented by a new proof in the volume representation of the existence of the maximum matter density. The techniques employed to demonstrate the existence of the cutoff also allow us to extract the large-volume limit of the exact eigenfunctions, confirming earlier numerical and analytical work showing that the eigenfunctions approach superpositions of the eigenfunctions of the Wheeler-DeWitt quantization of the same model. We argue that generic (not just semiclassical) quantum states approach symmetric superpositions of expanding and contracting universes. [ABSTRACT FROM AUTHOR]

The present paper deals with quantization of perfect fluid anisotropic cosmological models. Bianchi type V and IX models are discussed following Schutz's method of expressing fluid velocities in terms of six potentials. The wave functions are found for several examples of equations of state. In one case a complete wave packet could be formed analytically. The initial singularity of a zero proper volume can be avoided in this case, but it is plagued by the usual problem of non-unitarity of anisotropic quantum cosmological models. It is seen that a particular operator ordering alleviates this problem. [ABSTRACT FROM AUTHOR]

A brief review of various numerical techniques used in loop quantum cosmology and results is presented. These include the way extensive numerical simulations provided insights into the resolution of classical singularities, resulting in the key prediction of the bounce at the Planck scale in different models, and the numerical methods used to analyze the properties of the quantum difference operator and the von Neumann stability issues. Using the quantization of a massless scalar field in an isotropic spacetime as a template, an attempt is made to highlight the complementarity of different methods to gain understanding of the new physics emerging from the quantum theory. Open directions which need to be explored with more refined numerical methods are discussed. [ABSTRACT FROM AUTHOR]

In this paper, a generalized quantization principle for the gravitational field in canonical quantum gravity, especially with respect to quantum geometrodynamics is considered. This assumption can be interpreted as a transfer from the generalized uncertainty principle in quantum mechanics, which is postulated as generalization of the Heisenberg algebra to introduce a minimal length, to a corresponding quantization principle concerning the quantities of quantum gravity. According to this presupposition there have to be given generalized representations of the operators referring to the observables in the canonical approach of a quantum description of general relativity. This also leads to generalized constraints for the states and thus to a generalized Wheeler-DeWitt equation determining a new dynamical behavior. As a special manifestation of this modified canonical theory of quantum gravity, quantum cosmology is explored. The generalized cosmological Wheeler-DeWitt equation corresponding to the application of the generalized quantization principle to the cosmological degree of freedom is solved by using Sommerfelds polynomial method. [ABSTRACT FROM AUTHOR]

We present the method of describing an evolution in quantum cosmology in the framework of the reduced phase space quantization of loop cosmology. We apply our method to the flat Friedmann-Robertson-Walker model coupled to a massless scalar field. We identify the physical quantum Hamiltonian that is positive-definite and generates globally a unitary evolution of the considered quantum system. We examine the properties of expectation values of physical observables in the process of the quantum big bounce transition. The dispersion of evolved observables is studied for the Gaussian state. Calculated relative fluctuations enable an examination of the semi-classicality conditions and possible occurrence of the cosmic forgetfulness. Preliminary estimations based on the cosmological data suggest that there was no cosmic amnesia. Presented results are analytical, and numerical computations are only used for the visualization purposes. Our method may be generalized to sophisticated cosmological models including the Bianchi-type universes. [ABSTRACT FROM AUTHOR]

The problem of time evolution in quantum cosmology is studied in the context of a dust-filled, spatially flat Friedmann-Robertson-Walker universe. In this model, two versions of the commonly-adopted notion of internal time can be implemented in the same quantization, and are found to yield contradictory views of the same quantum state: with one choice, the big-bang singularity appears to be resolved, but with another choice it does not. This and other considerations lead to the conclusion that the notion of internal time as it is usually implemented has no satisfactory physical interpretation. A recently proposed variant of the relational-time construction, using a test clock that is regarded as internal to a specific observer, appears to provide an improved account of time evolution relative to the proper time that elapses along the observer's worldline. This construction permits the derivation of consistent joint probability densities for observable quantities, which can be viewed either as evolving with proper time or as describing correlations in a timeless manner. It turns out that the observer whose sense of time originates in this test clock will find herself to be living in a bouncing universe. [ABSTRACT FROM AUTHOR]

The deformation quantization formalism is applied to the linearized gravitational field. Standard aspects of this formalism are worked out before the ground state Wigner functional is obtained. Finally, the propagator for the graviton is also found within the context of this formalism. [ABSTRACT FROM AUTHOR]