The open problem of constructing a consistent and experimentally tested quantum theory of the gravitational field has its place at the heart of fundamental physics. The main approaches can be roughly divided into two classes: either one seeks a unified quantum framework of all interactions or one starts with a direct quantization of general relativity. In the first class, string theory (M-theory) is the only known example. In the second class, one can make an additional methodological distinction: while covariant approaches such as path-integral quantization use the four-dimensional metric as an essential ingredient of their formalism, canonical approaches start with a foliation of spacetime into spacelike hypersurfaces in order to arrive at a Hamiltonian formulation. The present book is devoted to one of the canonical approaches—loop quantum gravity. It is named modern canonical quantum general relativity by the author because it uses connections and holonomies as central variables, which are analogous to the variables used in Yang–Mills theories. In fact, the canonically conjugate variables are a holonomy of a connection and the flux of a non-Abelian electric field. This has to be contrasted with the older geometrodynamical approach in which the metric of three-dimensional space and the second fundamental form are the fundamental entities, an approach which is still actively being pursued. It is the author's ambition to present loop quantum gravity in a way in which every step is formulated in a mathematically rigorous form. In his own words: 'loop quantum gravity is an attempt to construct a mathematically rigorous, background-independent, non-perturbative quantum field theory of Lorentzian general relativity and all known matter in four spacetime dimensions, not more and not less'. The formal Leitmotiv of loop quantum gravity is background independence. Non-gravitational theories are usually quantized on a given non-dynamical background. In contrast, due to the geometrical nature of gravity, no such background exists in quantum gravity. Instead, the notion of a background is supposed to emerge a posteriori as an approximate notion from quantum states of geometry. As a consequence, the standard ultraviolet divergences of quantum field theory do not show up because there is no limit of Δx → 0 to be taken in a given spacetime. On the other hand, it is open whether the theory is free of any type of divergences and anomalies. A central feature of any canonical approach, independent of the choice of variables, is the existence of constraints. In geometrodynamics, these are the Hamiltonian and diffeomorphism constraints. They also hold in loop quantum gravity, but are supplemented there by the Gauss constraint, which emerges due to the use of triads in the formalism. These constraints capture all the physics of the quantum theory because no spacetime is present anymore (analogous to the absence of trajectories in quantum mechanics), so no additional equations of motion are needed. This book presents a careful and comprehensive discussion of these constraints. In particular, the constraint algebra is calculated in a transparent and explicit way. The author makes the important assumption that a Hilbert-space structure is still needed on the fundamental level of quantum gravity. In ordinary quantum theory, such a structure is needed for the probability interpretation, in particular for the conservation of probability with respect to external time. It is thus interesting to see how far this concept can be extrapolated into the timeless realm of quantum gravity. On the kinematical level, that is, before the constraints are imposed, an essentially unique Hilbert space can be constructed in terms of spin-network states. Potentially problematic features are the implementation of the diffeomorphism and Hamiltonian constraints. The Hilbert space Hdiff defined on the diffeomorphism subspace can throw states out of the kinematical Hilbert space and is thus not contained in it. Moreover, the Hamiltonian constraint does not seem to preserve Hdiff, so its implementation remains open. To avoid some of these problems, the author proposes his 'master constraint programme' in which the infinitely many local Hamiltonian constraints are combined into one master constraint. This is a subject of his current research. With regard to this situation, it is not surprising that the main results in loop quantum gravity are found on the kinematical level. An especially important feature are the discrete spectra of geometric operators such as the area operator. This quantifies the earlier heuristic ideas about a discreteness at the Planck scale. The hope is that these results survive the consistent implementation of all constraints. The status of loop quantum gravity is concisely and competently summarized in this volume, whose author is himself one of the pioneers of this approach. What is the relation of this book to the other monograph on loop quantum gravity, written by Carlo Rovelli and published in 2004 under the title Quantum Gravity with the same company? In the words of the present author: 'the two books are complementary in the sense that they can be regarded almost as volume I ('introduction and conceptual framework') and volume II ('mathematical framework and applications') of a general presentation of quantum general relativity in general and loop quantum gravity in particular'. In fact, the present volume gives a complete and self-contained presentation of the required mathematics, especially on the approximately 200 pages of chapters 18–33. As for the physical applications, the main topic is the microscopic derivation of the black-hole entropy. This is presented in a clear and detailed form. Employing the concept of an isolated horizon (a local generalization of an event horizon), the counting of surface states gives an entropy proportional to the horizon area. It also contains the Barbero–Immirzi parameter β, which is a free parameter of the theory. Demanding, on the other hand, that the entropy be equal to the Bekenstein–Hawking entropy would fix this parameter. Other applications such as loop quantum cosmology are only briefly touched upon. Since loop quantum gravity is a very active field of research, the author warns that the present book can at best be seen as a snapshot. Part of the overall picture may thus in the future be subject to modifications. For example, recent work by the author using a concept of dust time is not yet covered here. Nevertheless, I expect that this volume will continue to serve as a valuable introduction and reference book. It is essential reading for everyone working on loop quantum gravity.

Modern Canonical Quantum General Relativity

It has been suggested that the homogeneous black hole interior spacetime, when quantized following the techniques of loop quantum cosmology, has a resolved singularity replaced by a black-to-white hole transition. This result has however been derived so far only using effective classical evolution equations, and depends on details of the so-called polymerization scheme for the Hamiltonian constraint. Here we propose to use the unimodular formulation of general relativity to study the full quantum dynamics of this mini-superspace model. When applied to such cosmological models, unimodular gravity has the advantage of trivializing the problem of time by providing a true Hamiltonian which follows a Schr\"odinger evolution equation. By choosing variables adapted to this setup, we show how to write semiclassical states agreeing with that of the Wheeler--DeWitt theory at late times, and how in loop quantum cosmology they evolve through the would-be singularity while remaining sharply peaked. This provides a very simple setup for the study of the full quantum dynamics of these models, which can hopefully serve to tame regularization ambiguities.

Quantum dynamics of the black hole interior in loop quantum cosmology

We propose a new model of the spherical symmetric quantum black hole in the reduced phase space formulation. We deparametrize gravity by coupling to the Gaussian dust which provides the material coordinates. The foliation by dust coordinates covers both the interior and exterior of the black hole. After the spherical symmetry reduction, our model is a 1+1 dimensional field theory containing infinitely many degrees of freedom. The effective dynamics of the quantum black hole is generated by an improved physical Hamiltonian ${\bf H}_\Delta$ where the holonomy correction is implemented by the $\bar{\mu}$-scheme regularization with a Planckian area scale $\Delta$. The effective dynamics recovers the semiclassical Schwarzschild geometry at low curvature regime and resolves the black hole singularity with Planckian curvature. Our model predicts that the evolution of the black hole at late time reaches the charged Nariai geometry ${\rm dS}_2\times S^2$ with Planckian radii. The Nariai geometry is stable under linear perturbations but may be unstable by nonperturbative quantum effects. Our model suggests the existence of quantum tunneling of the Nariai geometry and a scenario of black-hole-to-white-hole transition. During the transition, the linear perturbations exhibit chaotic dynamics with Lyapunov exponent $\lambda=2\pi T_{\rm dS}\sim \Delta^{-1/2}$ relating to the Hawking temperature $T_{\rm dS}$ of ${\rm dS}_2$. In addition, the Nariai geometry in our model provides an interesting example of Wheeler's bag of gold and contains infinitely many infrared soft modes with zero energy density. These infrared modes span a Hilbert space carrying a representation of 1-dimensional spatial diffeomorphisms (or the Witt/Virasoro algebra) which are conserved charges of the effective dynamics by ${\bf H}_\Delta$.

Improved Effective Dynamics of Loop-Quantum-Gravity Black Hole and Nariai Limit

The tension between the value of the Hubble constant $$H_0$$
 determined from local supernovae data and the one inferred from the cosmic microwave background based on the $$\Lambda $$
 CDM cosmological model may indicate the need for new physics. Here, we show that this ‘Hubble tension’ can be resolved in models involving an effective energy flux from the matter sector into dark energy resulting naturally from a combination of unimodular gravity and an energy diffusion process. The scheme is one where dark energy has the standard equation of state $$w=-1$$
 . This proposal provides an alternative phenomenological paradigm accounting for the observations, while offering a general framework to study diffusion effects coming from novel fundamental physical processes.

Resolving the $H_0$ tension with diffusion

In approaches to quantum gravity, where smooth spacetime is an emergent approximation of a discrete Planckian fundamental structure, any standard effective field theoretical description will miss part of the degrees of freedom and thus break unitarity. Here we show that these expectations can be made precise in loop quantum cosmology. Concretely, even when loop quantum cosmology is unitary at the fundamental level, when microscopic degrees of freedom, irrelevant to low-energy cosmological observers, are suitably ignored, pure states in the effective description evolve into mixed states due to decoherence with the Planckian microscopic structure. When extrapolated to black hole formation and evaporation, this concrete example provides a key physical insight for a natural resolution of Hawking's information paradox.

Hawking's information puzzle: a solution realized in loop quantum cosmology

In approaches to quantum gravity, where smooth spacetime is an emergent approximation of a discrete Planckian fundamental structure, any effective smooth field theoretical description would miss part of the fundamental degrees of freedom and thus break unitarity. This is applicable also to trivial gravitational field (low energy) idealizations realized by the use of the Minkowski background geometry which, as any other spacetime geometry, corresponds, in the fundamental description, to infinitely many different and closely degenerate discrete microstates. The existence of such microstates provides a large reservoir for information to be coded at the end of black hole evaporation and thus opens the way to a natural resolution of the black hole evaporation information puzzle. 
In this paper we show that these expectations can be made precise in a simple quantum gravity model for cosmology motivated by loop quantum gravity. Concretely, even when the model is fundamentally unitary, when microscopic degrees of freedom irrelevant to low-energy cosmological observers are suitably ignored, pure states in the effective description evolve into mixed states due to decoherence with the Planckian microscopic structure. Moreover, in the relevant physical regime these hidden degrees freedom do not carry any `energy' and thus realize in a fully quantum gravitational context the idea (emphasized before by Unruh and Wald) that decoherence can take place without dissipation, now in a concrete gravitational model strongly motivated by quantum gravity. All this strengthen the perspective of a quite conservative and natural resolution of the black hole evaporation puzzle where information is not destroyed but simply degraded (made unavailable to low energy observers) into correlations with the microscopic structure of the quantum geometry at the Planck scale.

Unitarity and information in quantum gravity: a simple example

In approaches to quantum gravity, where smooth spacetime is an emergent approximation of a discrete Planckian fundamental structure, any effective smooth field theoretical description would miss part of the fundamental degrees of freedom and thus break unitarity. This is applicable also to trivial gravitational field (low energy) idealizations realized by the use of the Minkowski background geometry which, as any other spacetime geometry, corresponds, in the fundamental description, to infinitely many different and closely degenerate discrete microstates. The existence of such microstates provides a large reservoir q-bits for information to be coded at the end of black hole evaporation and thus opens the way to a natural resolution of the black hole evaporation information puzzle. In this paper we show that these expectations can be made precise in a simple quantum gravity model for cosmology motivated by loop quantum gravity. Concretely, even when the model is fundamentally unitary, when microscopic degrees of freedom irrelevant to low-energy cosmological observers are suitably ignored, pure states in the effective description evolve into mixed states due to decoherence with the Planckian microscopic structure. Moreover, in the relevant physical regime these hidden degrees of freedom do not carry any `energy' and thus realize in a fully quantum gravitational context the idea (emphasized before by Unruh and Wald) that decoherence can take place without dissipation, now in a concrete gravitational model strongly motivated by quantum gravity. All this strengthens the perspective of a quite conservative and natural resolution of the black hole evaporation puzzle where information is not destroyed but simply degraded (made unavailable to low energy observers) into correlations with the microscopic structure of the quantum geometry at the Planck scale.

/pdf/unitarity-and-information-in-quantum-gravity-a-simple-12nqwocikb.pdf

Hongguang Liu

Papers

Unitarity and information in quantum gravity: a simple example

Unitarity and Information in Quantum Gravity: A Simple Example