Absolute concepts of time and energy are integral to standard thermal physics, but background independent systems are devoid of them. This thesis explores the foundational question of generalising equilibrium statistical mechanics to background independent systems, and applies the resulting formulation to non-perturbative discrete quantum gravity. It offers a discussion of the rich interface of thermal physics and quantum gravity in this context, providing a broad contextualisation of such topics. This work details how standard tools from many-body physics can be generalised to discrete quantum gravity labelled with graph-based, or dual polyhedral, quanta that are common to various approaches like tensorial group field theory, loop quantum gravity, dynamical triangulations and lattice quantum gravity. It provides a systematic investigation, both conceptual and technical, of quantum gravitational Gibbs states, thermofield double vacuum states, and their application in condensate cosmology to probe questions of Big Bang singularity resolution and geometric inflation.

Following the fundamental insights from quantum mechanics and general relativity, geometry itself should have a quantum description; the search for a complete understanding of this description is what drives the field of quantum gravity. Group field theory is an ambitious framework in which theories of quantum geometry are formulated, incorporating successful ideas from the fields of matrix models, ten-sor models, spin foam models and loop quantum gravity, as well as from the broader areas of quantum field theory and mathematical physics. This special issue collects recent work in group field theory and these related approaches, as well as other neighbouring fields (e.g., cosmology, quantum information and quantum foundations, statistical physics) to the extent that these are directly relevant to quantum gravity research.

This book constitutes the proceedings of the 4th International Conference on Geometric Science of Information, GSI 2019, held in Toulouse, France, in August 2019. The 79 full papers presented in this volume were carefully reviewed and selected from 105 submissions. They cover all the main topics and highlights in the domain of geometric science of information, including information geometry manifolds of structured data/information and their advanced applications.

This is a textbook which gradually introduces the student to the statistical mechanical study of the different phases of matter and to the phase transitions between them. Throughout, only simple models of both ordinary and soft matter are used but these are studied in full detail. The subject is developed in a pedagogical manner, starting from the basics, going from the simple ideal systems to the interacting systems, and ending with the more modern topics. The textbook provides the student with a complete overview, intentionally at an introductory level, of the theory of phase transitions. All equations and deductions are included.

Quantum gravity has developed into a fast-growing subject in physics and it is expected that probing the high-energy and high-curvature regimes of gravitating systems will shed some light on how to eventually achieve an ultraviolet complete quantum theory of gravity. Such a theory would provide the much needed information about fundamental problems of classical gravity, such as the initial big-bang singularity, the cosmological constant problem, Planck scale physics and the early-time inflationary evolution of our Universe. While in the first part of this book concepts of quantum gravity are introduced and approached from different angles, the second part discusses these theories in connection with cosmological models and observations, thereby exploring which types of signatures of modern and mathematically rigorous frameworks can be detected by experiments. The third and final part briefly reviews the observational status of dark matter and dark energy, and introduces alternative cosmological models. Edited and authored by leading researchers in the field and cast into the form of a multi-author textbook at postgraduate level, this volume will be of benefit to all postgraduate students and newcomers from neighboring disciplines wishing to find a comprehensive guide for their future research.

In this thesis some equilibrium and non-equilibrium statistical methods are implied on two different versions of non-perturbative quantum gravity. Firstly, we report a novel statistical mechanics in which a class of evolutionary maps act on trivalent spin network in randomly chosen initial states and give rise to Self-organized Criticality. The result of continuously applying these maps indicate an expansion in the space-time area associated. Secondly, a previously unknown statistical mechanics in quantum gravity is introduced in the framework of two dimensional Causal Dynamical Triangulations. This provides us a useful and new tools to understand this quantum gravity in terms of effective spins. This study reveals a correspondence between the statistics of Anti-ferromagnetic systems and Causal Dynamical quantum gravity. More importantly, it provides a basis for studying anti-ferromagnetic systems in a background independent way. Thirdly, two novel properties of area operator in Loop Quantum Gravity are reported: 1) the generic degeneracy and 2) the ladder symmetry. These were not known previously for years. The first one indicates that corresponding to any eigenvalue of area operator in loop quantum gravity there exists a finite number of degenerate eigenstates. This degeneracy is shown to be one way for the explanation of black hole entropy in a microscopic way. More importantly, we reproduce Bekenstein-Hawking entropy of black hole by comparing the minimal energy of a decaying frequency from a loop quantum black hole and the extracted energy from a perturbed black hole in the highly damping mode. This consistency reveals a treasure model for describing a black hole in loop quantum gravity that does nor suffer from the restrictions of an isolated horizon. The second property indicates there exists a ladder symmetry unexpectedly in the complete spectrum of area eigenvalues. This symmetry suggests the eigenvalues of area could be classified into different evenly spaced subsets, each called a 'generation.' All generations are evenly spaced; but the gap between the levels in any every generation is unique. One application of the two new properties of area operator have been considered here for introducing a generalized picture of horizon whose area cells are not restricted to the subset considered in quantum isolated horizon theory. Instead, the area cells accepts values from the complete spectrum. Such horizon in the presence of all elements of diffeomorphism group contains a number of degrees of freedom independently from the bulk freedom whose logarithm scales with the horizon area. Note that this is not the case in quantum isolated horizon when the complete elements of diffeomorphism applies. Finally, we use a simple statistical method in which no pre-assumption is made for the essence of the energy quanta radiated from the hole. We derive the effects of the black hole horizon fluctuations and reveal a new phenomenon called "quantum amplification effects" affecting black hole radiation. This effect causes unexpectedly a few un-blended radiance modes manifested in spectrum as discrete brightest lines. The frequency of these modes scales with the mass of black hole. This modification to Hawking's radiation indicates a window at which loop quantum gravity can be observationally tested at least for primordial black holes.

Introduction to Quantum Statistical Mechanics (2nd Edition) may be used as an advanced textbook by graduate students, even ambitious undergraduates in physics. It is also suitable for non experts in physics who wish to have an overview of some of the classic and fundamental quantum models in the subject. The explanation in the book is detailed enough to capture the interest of the reader, and complete enough to provide the necessary background material needed to dwell further into the subject and explore the research literature.

Encapsulates the latest debates on this topic, giving researchers and graduate students an up-to-date view of the field.

This book provides a clear and self-contained exposition of quantum statistical mechanics, focussing on the foundations. The unifying theme is the statistical entropy, as modified for quantum systems. From this is derived the conventional expressions for equilibrium quantum statistical mechanics, and, most exciting, their extension to non-equilibrium, time-dependent systems. A unified treatment of the equilibrium and non-equilibrium fields is given based upon the conservation laws, time symmetries, and the second law of thermodynamics. One theme of the book is the collapse of the wave function of an open quantum system, which leads to the Maxwell-Boltzmann probability operator, its relationship to the density matrix, and the von Neumann trace expression for a statistical average. A second theme is the development of the appropriate entropy for quantum systems, which, in conjunction with the second law, gives the stochastic, dissipative Schrdinger equation for an open quantum system and the fluctuation-dissipation theorem for the time propagator. A final theme is the derivation of the probability operator for non-equilibrium systems and irreversible processes, which lies at the cutting edge of modern research.

This book provides an introduction to topics in non-equilibrium quantum statistical physics for both mathematicians and theoretical physicists. The first part introduces a kinetic equation, of Kolmogorov type, which is needed to describe an isolated atom (actually, in experiments, an ion) under the effect of a classical pumping electromagnetic field which keeps the atom in its excited state(s) together with the random emission of fluorescence photons which put it back into its ground state. The quantum kinetic theory developed in the second part is an extension of Boltzmann's classical (non-quantum) kinetic theory of a dilute gas of quantum bosons. This is the source of many interesting fundamental questions, particularly because, if the temperature is low enough, such a gas is known to have at equilibrium a transition, the Bose–Einstein transition, where a finite portion of the particles stay in the quantum ground state. An important question considered is how a Bose gas condensate develops in time if its energy is initially low enough.