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Dynamical black-hole scenarios have been developed in loop quantum gravity in various ways, combining results from mini and midisuperspace models. In the past, the underlying geometry of space-time has often been expressed in terms of line elements with metric components that differ from the classical solutions of general relativity, motivated by modified equations of motion and constraints. However, recent results have shown by explicit calculations that most of these constructions violate general covariance and slicing independence. The proposed line elements and black-hole models are therefore ruled out. The only known possibility to escape this sentence is to derive not only modified metric components but also a new space-time structure which is covariant in a generalized sense. Formally, such a derivation is made available by an analysis of the constraints of canonical gravity, which generate deformations of hypersurfaces in space-time, or generalized versions if the constraints are consistently modified. A generic consequence of consistent modifications in effective theories suggested by loop quantum gravity is signature change at high density. Signature change is an important ingredient in long-term models of black holes that aim to determine what might happen after a black hole has evaporated. Because this effect changes the causal structure of space-time, it has crucial implications for black-hole models that have been missed in several older constructions, for instance in models based on bouncing black-hole interiors. Such models are ruled out by signature change even if their underlying space-times are made consistent using generalized covariance. The causal nature of signature change brings in a new internal consistency condition, given by the requirement of deterministic behavior at low curvature. Even a causally disconnected interior transition, opening back up into the former exterior as some kind of astrophysical white hole, is then ruled out. New versions consistent with both generalized covariance and low-curvature determinism are introduced here, showing a remarkable similarity with models developed in other approaches, such as the final-state proposal or the no-transition principle obtained from the gauge-gravity correspondence.

A large number of models have been analyzed in loop quantum cosmology, using mainly minisuperspace constructions and perturbations. At the same time, general physics principles from effective field theory and covariance have often been ignored. A consistent introduction of these ingredients requires substantial modifications of existing scenarios. As a consequence, none of the broader claims made mainly by the Ashtekar school—such as the genericness of bounces with astonishingly semiclassical dynamics, robustness with respect to quantization ambiguities, the realization of covariance, and the relevance of certain technical results for potential observations—hold up to scrutiny. Several useful lessons for a sustainable version of quantum cosmology can be drawn from this evaluation.

Quantum gravity is expected to be necessary in order to understand situations in which classical general relativity breaks down. In particular in cosmology one has to deal with initial singularities, i.e., the fact that the backward evolution of a classical spacetime inevitably comes to an end after a finite amount of proper time. This presents a breakdown of the classical picture and requires an extended theory for a meaningful description. Since small length scales and high curvatures are involved, quantum effects must play a role. Not only the singularity itself but also the surrounding spacetime is then modified. One particular theory is loop quantum cosmology, an application of loop quantum gravity to homogeneous systems, which removes classical singularities. Its implications can be studied at different levels. The main effects are introduced into effective classical equations, which allow one to avoid the interpretational problems of quantum theory. They give rise to new kinds of early-universe phenomenology with applications to inflation and cyclic models. To resolve classical singularities and to understand the structure of geometry around them, the quantum description is necessary. Classical evolution is then replaced by a difference equation for a wave function, which allows an extension of quantum spacetime beyond classical singularities. One main question is how these homogeneous scenarios are related to full loop quantum gravity, which can be dealt with at the level of distributional symmetric states. Finally, the new structure of spacetime arising in loop quantum gravity and its application to cosmology sheds light on more general issues, such as the nature of time.

In canonical gravity, general covariance is implemented by hypersurface-deformation symmetries on thephase space. The different versions of hypersurface deformations required for full covariance have complicated interplays with one another, governed by non-Abelian brackets with structure functions. For spherically symmetric space-times, it is possible to identify a certain Abelian substructure within general hypersurface deformations, which suggests a simplified realization as a Lie algebra. The generators of this substructure can be quantized more easily than full hypersurface deformations, but the symmetries they generate do not directly correspond to hypersurface deformations. The availability of consistent quantizations therefore does not guarantee general covariance or a meaningful quantum notion thereof. In addition to placing the Abelian substructure within the full context of spherically symmetric hypersurface deformation, this paper points out several subtleties relevant for attempted applications in quantized space-time structures. In particular, it follows that recent constructions by Gambini, Olmedo, and Pullin in an Abelianized setting fail to address the covariance crisis of loop quantum gravity.

Quantum gravity is expected to be necessary in order to understand situations where classical general relativity breaks down. In particular in cosmology one has to deal with initial singularities, i.e., the fact that the backward evolution of a classical space-time inevitably comes to an end after a finite amount of proper time. This presents a breakdown of the classical picture and requires an extended theory for a meaningful description. Since small length scales and high curvatures are involved, quantum effects must play a role. Not only the singularity itself but also the surrounding space-time is then modified. One particular realization is loop quantum cosmology, an application of loop quantum gravity to homogeneous systems, which removes classical singularities. Its implications can be studied at different levels. Main effects are introduced into effective classical equations which allow to avoid interpretational problems of quantum theory. They give rise to new kinds of early universe phenomenology with applications to inflation and cyclic models. To resolve classical singularities and to understand the structure of geometry around them, the quantum description is necessary. Classical evolution is then replaced by a difference equation for a wave function which allows to extend space-time beyond classical singularities. One main question is how these homogeneous scenarios are related to full loop quantum gravity, which can be dealt with at the level of distributional symmetric states. Finally, the new structure of space-time arising in loop quantum gravity and its application to cosmology sheds new light on more general issues such as time.

Background independence is often emphasized as an important property of a quantum theory of gravity that takes seriously the geometrical nature of general relativity. In a background-independent formulation, quantum gravity should determine not only the dynamics of space–time but also its geometry, which may have equally important implications for claims of potential physical observations. One of the leading candidates for background-independent quantum gravity is loop quantum gravity. By combining and interpreting several recent results, it is shown here how the canonical nature of this theory makes it possible to perform a complete space–time analysis in various models that have been proposed in this setting. In spite of the background-independent starting point, all these models turned out to be non-geometrical and even inconsistent to varying degrees, unless strong modifications of Riemannian geometry are taken into account. This outcome leads to several implications for potential observations as well as lessons for other background-independent approaches.

Loop quantum cosmology is a conflicted field in which exuberant claims of observability coexist with serious objections against the conceptual and physical viability of its current formulations. This contribution presents a non-technical case study of the recent claim that loop quantum cosmology might alleviate anomalies in the observations of the cosmic microwave background.

A bstract According to the pioneering work of Nielsen and collaborators, the length of the minimal geodesic in a geometric realization of a suitable operator space provides a measure of the quantum complexity of an operation. Compared with the original concept of complexity based on the minimal number of gates required to construct the desired operation as a product, this geometrical approach amounts to a more concrete and computable definition, but its evaluation is nontrivial in systems with a high-dimensional Hilbert space. The geometrical formulation can more easily be evaluated by considering the geometry associated with a suitable finite-dimensional group generated by a small number of relevant operators of the system. In this way, the method has been applied in particular to the harmonic oscillator, which is also of interest in the present paper. However, subtle and previously unrecognized issues of group theory can lead to unforeseen complications, motivating a new formulation that remains on the level of the underlying Lie algebras for most of the required steps. Novel insights about complexity can thereby be found in a low-dimensional setting, with the potential of systematic extensions to higher dimensions as well as interactions. Specific examples include the quantum complexity of various target unitary operators associated with a harmonic oscillator, inverted harmonic oscillator, and coupled harmonic oscillators. The generality of this approach is demonstrated by an application to an anharmonic oscillator with a cubic term.

Quantum cosmology is traditionally formulated in a minisuperspace setting, implicitly averaging fields over space to obtain homogeneous models. For universal reasons related to the uncertainty principle, quantum corrections then depend on the size of the averaging volume. In minisuperspace truncations, the value of this volume remains an arbitrary parameter devoid of physical meaning, but in an effective field theory it is identified with the infrared scale of inhomogeneous modes. Moreover, the infrared scale is running during gravitational collapse, when regions in which homogeneity remains approximately valid shrink to increasingly smaller co-moving sizes. Conceptual implications of this infrared renormalization for perturbative inhomogeneity in quantum cosmology are presented here, mainly for the example of loop quantum cosmology. Several claims made in this framework are altered by infrared renormalization.