Tensorial Methods And Renormalization In Group Field Theories

The main focus of this thesis is the mathematical structure of Group Field Theories (GFTs) from the point of view of renormalization theory. Such quantum field theories are found in approaches to quantum gravity related, on the one hand, to Loop Quantum Gravity (LQG) and on the other, to matrix- and tensor models. Background material on these topics, including conceptual and technical aspects, are introduced in the first chapters. The work then goes on to explain how the standard tools of Quantum Field Theory can be generalized to GFTs and exploited to study the large cut-off behaviour and renormalization group transformations of the latter. Among the new results derived in this context are a proof of renormalizability of a three-dimensional GFT with gauge group SU(2), which opens the way to applications of the formalism to quantum gravity.

Tensor network is a fundamental mathematical tool with a huge range of applications in physics, such as condensed matter physics, statistic physics, high energy physics, and quantum information sciences. This open access book aims to explain the tensor network contraction approaches in a systematic way, from the basic definitions to the important applications. This book is also useful to those who apply tensor networks in areas beyond physics, such as machine learning and the big-data analysis. Tensor network originates from the numerical renormalization group approach proposed by K. G. Wilson in 1975. Through a rapid development in the last two decades, tensor network has become a powerful numerical tool that can efficiently simulate a wide range of scientific problems, with particular success in quantum many-body physics. Varieties of tensor network algorithms have been proposed for different problems. However, the connections among different algorithms are not well discussed or reviewed. To fill this gap, this book explains the fundamental concepts and basic ideas that connect and/or unify different strategies of the tensor network contraction algorithms. In addition, some of the recent progresses in dealing with tensor decomposition techniques and quantum simulations are also represented in this book to help the readers to better understand tensor network. This open access book is intended for graduated students, but can also be used as a professional book for researchers in the related fields. To understand most of the contents in the book, only basic knowledge of quantum mechanics and linear algebra is required. In order to fully understand some advanced parts, the reader will need to be familiar with notion of condensed matter physics and quantum information, that however are not necessary to understand the main parts of the book. This book is a good source for non-specialists on quantum physics to understand tensor network algorithms and the related mathematics.

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.

The subject of the exact renormalization group started from pioneering work by Wegner and Houghton in the early seventies and, a decade later, by Polchinski, who formulated the Wilson renormalization group for field theory. In the past decade considerable progress has been made in this field, which includes the development of alternative formulations of the approach and of powerful techniques for solving the exact renormalization group equations, as well as widening of the scope of the exact renormalization group method to include fermions and gauge fields. In particular, two very recent results, namely the manifestly gauge-invariant formulation of the exact renormalization group equation and the proof of the c-theorem in four dimensions, are presented in this volume.

Renormalization group theory of tensor network states provides a powerful tool for studying quantum many-body problems and a new paradigm for understanding entangled structures of complex systems. In recent decades the theory has rapidly evolved into a universal framework and language employed by researchers in fields ranging from condensed matter theory to machine learning. This book presents a pedagogical and comprehensive introduction to this field for the first time. After an introductory survey on the major advances in tensor network algorithms and their applications, it introduces step-by-step the tensor network representations of quantum states and the tensor-network renormalization group methods developed over the past three decades. Basic statistical and condensed matter physics models are used to demonstrate how the tensor network renormalization works. An accessible primer for scientists and engineers, this book would also be ideal as a reference text for a graduate course in this area.

This book is a broad-based text intended to help the growing student body interested in constructing and applying methods of effective field theory to solve problems in their research. It begins with a review of using symmetries to identify the relevant degrees of freedom in a problem, and then presents a variety of methods that can be used to construct various effective theories. A detailed discussion of canonical applications of effective field theory techniques with increasing complexity is given, including Fermi's weak interaction, heavy-quark effective theory, and soft-collinear effective theory. Applications of these techniques to study physics beyond the standard model, dark matter, and quantum and classical gravity are explored. Although most examples come from questions in high-energy physics, many of the methods can also be applied in condensed-matter settings. Appendices include various factoids from group theory and other topics that are used throughout the text, in an attempt to make the book self-contained.

This book introduces "Random Tensors," a framework for studying random geometries in any dimension. It provides a complete derivation of the key results in the field. Whatever form a theory of Quantum Gravity may take, it must incorporate random geometry.

This volume of lecture notes briefly introduces the basic concepts needed in any computational physics course: software and hardware, programming skills, linear algebra, and differential calculus. It then presents more advanced numerical methods to tackle the quantum many-body problem: it reviews the numerical renormalization group and then focuses on tensor network methods, from basic concepts to gauge invariant ones. Finally, in the last part, the author presents some applications of tensor network methods to equilibrium and out-of-equilibrium correlated quantum matter. The book can be used for a graduate computational physics course. After successfully completing such a course, a student should be able to write a tensor network program and can begin to explore the physics of many-body quantum systems. The book can also serve as a reference for researchers working or starting out in the field.

This book is one of the most important reference books in Field Theory with permanent value. To enable wider access by students, researchers and libraries of developing countries, this valuable volume has been reprinted and is sold at a much lower price than before.