State Sum Models Of Piecewise Linear Quantum Gravity
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| Author | : Aleksandar Mikovic |
| Publisher | : World Scientific |
| Total Pages | : 184 |
| Release | : 2023-05-03 |
| Genre | : Science |
| ISBN | : 9811269335 |
This book gives a description of state-sum quantum gravity models which are based on triangulations of a smooth spacetime manifold. It contains detailed descriptions of Regge quantum gravity, spin-foam models and spin-cube models. Some other similar models, like the dynamical triangulations models, are only briefly described, since the sum over the spacetime triangulations is outside the scope of this book.The book also contains a detailed description of the approach where the piecewise linear (PL) manifold corresponding to a smooth manifold triangulation is considered as the basic structure of the spacetime. Hence the PL structure is not an auxiliary tool used to define the gravitational path integral for a smooth spacetime, but it is taken as a physical property of the spacetime. Consequently, it is straightforward to construct a finite gravitational path integral. Another consequence is that the problems of determination of the classical limit and how to calculate the quantum corrections can be solved by using the effective action method. The smooth manifold limit problem is then replaced by the problem of a smooth manifold approximation for the effective action, which can be obtained by using the standard quantum field theory with a physical cutoff.Some physical effects of a PL spacetime quantum gravity theory are also described, one of which is that the cosmological constant spectrum contains the observed value.A short exposition of higher gauge theory is also given, which is a promising way to generalize a gauge symmetry by using the concept of a 2-group. A 2-group is a categorical generalization of a group, and by using this approach one can construct the spin-cube models of quantum gravity.
| Author | : Aleksandar Miković |
| Publisher | : World Scientific Publishing Company |
| Total Pages | : 0 |
| Release | : 2023 |
| Genre | : Quantum gravity |
| ISBN | : 9789811269318 |
"It describes a novel approach in the study of quantum gravity (QG) state-sum models, which is based on the application of the effective action method from quantum field theory. Related to that is a study of the effect of a non-trivial path-integral (PI) measure on the PI finiteness, as well as a study on the dependence of the semi-classical expansion of the effective action on the PI measure. Detailed study of the idea that the spacetime at small distances is not a smooth manifold but a piecewise linear (PL) manifold corresponding to a triangulation of a smooth manifold. One consequence of a QG theory on a PL spacetime is that the cosmological constant has a continuous spectrum, and the spectrum contains the observed value of the cosmological constant. A clear explanation of the relationship between a higher gauge theory and general relativity, and how new state-sum models of QG are then constructed. These new state-sum models are called spin-cube models, which are categorical generalizations of the spin-foam models, since one labels the edges, triangles and tetrahedra with representations of 2-groups, which are categorical generalizations of groups"--
| Author | : R. Cianci |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 453 |
| Release | : 2013-04-17 |
| Genre | : Science |
| ISBN | : 8847021014 |
A survey of the most recent developments in general relativity and in the theory of the unification of Fundamental Interactions is presented in this book. The theoretical results, the cosmological and astrophysical aspects, the experimental and observational programs are shown in 26 general talks by renowned scientists active in this field.
| Author | : H. Gausterer |
| Publisher | : Springer |
| Total Pages | : 413 |
| Release | : 2008-01-11 |
| Genre | : Science |
| ISBN | : 3540465529 |
In modern mathematical physics, classical together with quantum, geometrical and functional analytic methods are used simultaneously. Non-commutative geometry in particular is becoming a useful tool in quantum field theories. This book, aimed at advanced students and researchers, provides an introduction to these ideas. Researchers will benefit particularly from the extensive survey articles on models relating to quantum gravity, string theory, and non-commutative geometry, as well as Connes' approach to the standard model.
| Author | : Pavel Mnev |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 200 |
| Release | : 2019-08-20 |
| Genre | : Mathematics |
| ISBN | : 1470452715 |
This book originated from lecture notes for the course given by the author at the University of Notre Dame in the fall of 2016. The aim of the book is to give an introduction to the perturbative path integral for gauge theories (in particular, topological field theories) in Batalin–Vilkovisky formalism and to some of its applications. The book is oriented toward a graduate mathematical audience and does not require any prior physics background. To elucidate the picture, the exposition is mostly focused on finite-dimensional models for gauge systems and path integrals, while giving comments on what has to be amended in the infinite-dimensional case relevant to local field theory. Motivating examples discussed in the book include Alexandrov–Kontsevich–Schwarz–Zaboronsky sigma models, the perturbative expansion for Chern–Simons invariants of 3-manifolds given in terms of integrals over configurations of points on the manifold, the BF theory on cellular decompositions of manifolds, and Kontsevich's deformation quantization formula.
| Author | : Vladimir G. Turaev |
| Publisher | : Walter de Gruyter GmbH & Co KG |
| Total Pages | : 620 |
| Release | : 2016-07-11 |
| Genre | : Mathematics |
| ISBN | : 3110434563 |
Due to the strong appeal and wide use of this monograph, it is now available in its third revised edition. The monograph gives a systematic treatment of 3-dimensional topological quantum field theories (TQFTs) based on the work of the author with N. Reshetikhin and O. Viro. This subject was inspired by the discovery of the Jones polynomial of knots and the Witten-Chern-Simons field theory. On the algebraic side, the study of 3-dimensional TQFTs has been influenced by the theory of braided categories and the theory of quantum groups. The book is divided into three parts. Part I presents a construction of 3-dimensional TQFTs and 2-dimensional modular functors from so-called modular categories. This gives a vast class of knot invariants and 3-manifold invariants as well as a class of linear representations of the mapping class groups of surfaces. In Part II the technique of 6j-symbols is used to define state sum invariants of 3-manifolds. Their relation to the TQFTs constructed in Part I is established via the theory of shadows. Part III provides constructions of modular categories, based on quantum groups and skein modules of tangles in the 3-space. This fundamental contribution to topological quantum field theory is accessible to graduate students in mathematics and physics with knowledge of basic algebra and topology. It is an indispensable source for everyone who wishes to enter the forefront of this fascinating area at the borderline of mathematics and physics. Contents: Invariants of graphs in Euclidean 3-space and of closed 3-manifolds Foundations of topological quantum field theory Three-dimensional topological quantum field theory Two-dimensional modular functors 6j-symbols Simplicial state sums on 3-manifolds Shadows of manifolds and state sums on shadows Constructions of modular categories
| Author | : Daniele Oriti |
| Publisher | : Cambridge University Press |
| Total Pages | : 605 |
| Release | : 2009-03-05 |
| Genre | : Science |
| ISBN | : 0521860458 |
Containing contributions from leading researchers in this field, this book provides a complete overview of this field from the frontiers of theoretical physics research for graduate students and researchers. It introduces the most current approaches to this problem, and reviews their main achievements.
| Author | : Steven H. Simon |
| Publisher | : Oxford University Press |
| Total Pages | : 641 |
| Release | : 2023-09-29 |
| Genre | : Science |
| ISBN | : 0198886772 |
At the intersection of physics, mathematics, and computer science, an exciting new field of study has formed, known as “Topological Quantum.” This research field examines the deep connections between the theory of knots, special types of subatomic particles known as anyons, certain phases of matter, and quantum computation. This book elucidates this nexus, drawing in topics ranging from quantum gravity to topology to experimental condensed matter physics. Topological quantum has increasingly been a focus point in the fields of condensed matter physics and quantum information over the last few decades, and the forefront of research now builds on the basic ideas presented in this book. The material is presented in a down-to-earth and entertaining way that is far less abstract than most of what is in the literature. While introducing the crucial concepts and placing them in context, the subject is presented without resort to the highly mathematical category theory that underlies the field. Requiring only an elementary background in quantum mechanics, this book is appropriate for all readers, from advanced undergraduates to the professional practitioner. This book will be of interest to mathematicians and computer scientists as well as physicists working on a wide range of topics. Those interested in working in these field will find this book to be an invaluable introduction as well as a crucial reference.
| Author | : B. Casciaro |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 511 |
| Release | : 2011-06-28 |
| Genre | : Science |
| ISBN | : 8847021138 |
The 13th Italian Conference on General Relativity and Gravitational Physics was held in Cala Corvino-Monopoli (Bari) from September 21to September 25, 1998. The Conference, which is held every other year in different Italian locations, has brought together, as in the earlier conferences in this series, those scientists who are interested and actively work in all aspects of general relativity, from both the mathematical and the physical points of view: from classical theories of gravitation to quantum gravity, from relativistic astrophysics and cosmology to experiments in gravitation. About 70 participants came from Departments of Astronomy and Astrophysics, Departments of Mathematics and Departments of Experimental and Theoretical Physics from all over the Country; in addition a few Italian scientists working abroad kindly accepted invitations from the Scientific Committee. The good wishes of the University and of the Politecnico di Bari were conveyed by the director of Diparti mento Interuniversitario di Matematica, Prof. Franco Altomare. These proceedings contain the contributions of the two winners of the SIGRAV prizes, the invited talks presented at the Conference and most of the contributed talks. We thank all of our colleagues, who did their best to prepare their manuscripts. The pleasant atmosphere induced by the beauty of the place was greatlyenhanced not only by the participation of so many colleagues, who had lively discussions about science well beyond Conference hours, but also by the feeling of hospitalityextended to the participants by the staff of the Cala Corvino Hotel, where the Conference was held.
| Author | : Mauro Carfora |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 298 |
| Release | : 2012-01-14 |
| Genre | : Science |
| ISBN | : 3642244394 |
Research on polyhedral manifolds often points to unexpected connections between very distinct aspects of Mathematics and Physics. In particular triangulated manifolds play quite a distinguished role in such settings as Riemann moduli space theory, strings and quantum gravity, topological quantum field theory, condensed matter physics, and critical phenomena. Not only do they provide a natural discrete analogue to the smooth manifolds on which physical theories are typically formulated, but their appearance is rather often a consequence of an underlying structure which naturally calls into play non-trivial aspects of representation theory, of complex analysis and topology in a way which makes manifest the basic geometric structures of the physical interactions involved. Yet, in most of the existing literature, triangulated manifolds are still merely viewed as a convenient discretization of a given physical theory to make it more amenable for numerical treatment. The motivation for these lectures notes is thus to provide an approachable introduction to this topic, emphasizing the conceptual aspects, and probing, through a set of cases studies, the connection between triangulated manifolds and quantum physics to the deepest. This volume addresses applied mathematicians and theoretical physicists working in the field of quantum geometry and its applications.
