A new description of macroscopic Kruskal black holes that incorporates the quantum geometry corrections of loop quantum gravity is presented. It encompasses both the 'interior' region that contains classical singularities and the 'exterior' asymptotic region. Singularities are naturally resolved by the quantum geometry effects of loop quantum gravity. The resulting quantum extension of space-time has the following features: (i) It admits an infinite number of trapped, anti-trapped and asymptotic regions; (ii) All curvature scalars have uniform (i.e., mass independent) upper bounds; (iii) In the large mass limit, all asymptotic regions of the extension have the same ADM mass; (iv) In the low curvature region (e.g., near horizons) quantum effects are negligible, as one would physically expect; and (v) Final results are insensitive to the fiducial structures that have to be introduced to construct the classical phase space description (as they must be). Previous effective theories [1-10] shared some but not all of these features. We compare and contrast our results with those of these effective theories and also with expectations based on the AdS/CFT conjecture [11]. We conclude with a discussion of limitations of our framework, especially for the analysis of evaporating black holes.1 In the cosmological models, effective equations were first introduced by examining the form of the quantum Hamiltonian constraint, then writing down an effective Hamiltonian constraint on the classical phase space that includes key quantum corrections due to quantum geometry effects of LQG, and calculating its dynamical flow, again on the classical phase space. Later these equations were shown to follow from the full quantum dynamics of sharply peaked states [14,29,30]. In the Schwarzschild case this last step has not been carried out in any of the approaches, including ours. 2 The same turns out to be true for the δ b , δ c prescriptions studied in Refs. [23,24] in the context of Kantowski-Sachs cosmologies, when they are used to model the Schwarzschild interior.

We present a new effective description of macroscopic Kruskal black holes that incorporates corrections due to quantum geometry effects of loop quantum gravity. It encompasses both the 'interior' region that contains classical singularities and the 'exterior' asymptotic region. Singularities are naturally resolved by the quantum geometry effects of loop quantum gravity, and the resulting quantum extension of the full Kruskal space-time is free of all the known limitations of previous investigations [1][2][3][4][5][6][7][8][9][10][11] of the Schwarzschild interior. We compare and contrast our results with these investigations and also with the expectations based on the AdS/CFT duality [12].

The flat, homogeneous, and isotropic universe with a massless scalar field is a paradigmatic model in Loop Quantum Cosmology. In spite of the prominent role that the model has played in the development of this branch of physics, there still remain some aspects of its quantization which deserve a more detailed discussion. These aspects include the kinematical resolution of the cosmological singularity, the precise relation between the solutions of the densitized and non-densitized versions of the quantum Hamiltonian constraint, the possibility of identifying superselection sectors which are as simple as possible, and a clear comprehension of the Wheeler-DeWitt (WDW) limit associated with the theory in those sectors. We propose an alternative operator to represent the Hamiltonian constraint which is specially suitable to deal with all these issues in a detailed and satisfactory way. In particular, with our constraint operator, the singularity decouples in the kinematical Hilbert space and can be removed already at this level. Thanks to this fact, we can densitize the quantum Hamiltonian constraint in a well-controlled manner. Besides, together with the physical observables, this constraint superselects simple sectors for the universe volume, with a discrete support contained in a single semiaxis of the real line and for which the basic functions that encode the information about the geometry possess optimal physical properties. Namely, they provide a no-boundary description around the cosmological singularity and admit a well-defined WDW limit in terms of standing waves. Both properties explain the presence of a generic quantum bounce replacing the classical singularity at a fundamental level, in contrast with previous studies where the bounce was proved in concrete regimes -focusing on states with a marked semiclassical behavior-or for a simplified model.

We quantize to completion an inflationary universe with small inhomogeneities in the framework of loop quantum cosmology. The homogeneous setting consists of a massive scalar field propagating in a closed, homogeneous scenario. We provide a complete quantum description of the system employing loop quantization techniques. After introducing small inhomogeneities as scalar perturbations, we identify the true physical degrees of freedom by means of a partial gauge fixing, removing all the local degrees of freedom except the matter perturbations. We finally combine a Fock description for the inhomogeneities with the polymeric quantization of the homogeneous background, providing the quantum Hamiltonian constraint of the composed system. Its solutions are then completely characterized, owing to the suitable choice of quantum constraint, and the physical Hilbert space is constructed. Finally, we consider the analog description for an alternate gauge and, moreover, in terms of gauge-invariant quantities. In the deparametrized model, all these descriptions are unitarily equivalent at the quantum level.

We present a complete quantization of an approximately homogeneous and isotropic universe with small scalar perturbations. We consider the case in which the matter content is a minimally coupled scalar field and the spatial sections are flat and compact, with the topology of a three-torus. The quantization is carried out along the lines that were put forward by the authors in a previous work for spherical topology. The action of the system is truncated at second order in perturbations. The local gauge freedom is fixed at the classical level, although different gauges are discussed and shown to lead to equivalent conclusions. Moreover, descriptions in terms of gauge-invariant quantities are considered. The reduced system is proven to admit a symplectic structure, and its dynamical evolution is dictated by a Hamiltonian constraint. Then, the background geometry is polymerically quantized, while a Fock representation is adopted for the inhomogeneities. The latter is selected by uniqueness criteria adapted from quantum field theory in curved spacetimes, which determine a specific scaling of the perturbations. In our hybrid quantization, we promote the Hamiltonian constraint to an operator on the kinematical Hilbert space. If the zero mode of the scalar field is interpreted as a relational time, a suitable ansatz for the dependence of the physical states on the polymeric degrees of freedom leads to a quantum wave equation for the evolution of the perturbations. Alternatively, the solutions to the quantum constraint can be characterized by their initial data on the minimum-volume section of each superselection sector. The physical implications of this model will be addressed in a future work, in order to check whether they are compatible with observations.Comment: 20 pages, no figures. v2: minor changes, in particular, abstract shortened, final discussion improve