Knots and Feynman Diagrams

Book Description

This volume explains how knot theory and Feynman diagrams can be used to illuminate problems in quantum field theory. The author emphasizes how new discoveries in mathematics have inspired conventional calculational methods for perturbative quantum field theory to become more elegant and potentially more powerful methods. The material illustrates what may possibly be the most productive interface between mathematics and physics. As a result, it will be of interest to graduate students and researchers in theoretical and particle physics as well as mathematics.

Quantum Invariants of Knots and 3-Manifolds (De Gruyter Studies in Mathematics)

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Excellent overview of what was known at the time
an axiomatic book

Quantum Invariants of Knots and 3-Manifolds (De Gruyter Studies in Mathematics)
V. G. Turaev
Manufacturer: Walter de Gruyter
ProductGroup: Book
Binding: Hardcover

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ASIN: 3110137046

Book Description

In the last decade we have witnessed the birth of a fascinating new mathematical theory. It is often called by algebraists the theory of quantum groups and by topologists quantum topology. These terms, however, seem to be too restrictive and do not convey the breadth of this new domain which is closely related to the theory of von Neumann algebras, the theory of Hopf algebras, the theory of representations of semisimple Lie algebras, the topology of knots, etc. The most spectacular achievements in this theory are centered around quantum groups and invariants of knots and 3-dimensional manifolds.

The whole theory has been, to a great extent, inspired by ideas that arose in theoretical physics. Among the relevant areas of physics are the theory of exactly solvable models of statistical mechanics, the quantum inverse scattering method, the quantum theory of angular momentum, 2-dimensional conformal field theory, etc. The development of this subject shows once more that physics and mathematics intercommunicate and influence each other to the profit of both disciplines.

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Excellent overview of what was known at the time.......2002-07-01

Some quite amazing results have appeared in the last two decades that connect two seemingly different fields of knowledge, namely topology and quantum field theory. Topological considerations have played a role in quantum field theory for quite some time, due to the role of instantons, but quantum field theory did not shed any light on topological questions until Edward Witten, Vaughan Jones, and other physicists and mathematicians showed that it can be a powerful tool of investigation in 3-dimensional and 4-dimensional topology.

This book, written for the mathematician, does not follow the physical line of reasoning that has been employed to obtain invariants of knots and 3-manifolds. Instead, it endeavors to remain as rigorous as possible, and thus the approaches using conformal field theory or Chern-Simons field theory are not developed by the author (this is not to say that one cannot see the influence of these areas in the book). The author incorporates not only the developments from the current research literature up to the date the book was written, but also interjects some original results of his own. The book could also be viewed as a textbook, as there are exercises put in at various places in the book.

Topological quantum field theory is defined rigorously in this book, but is put in the context of what are called modular categories and modular functors by the author.Modular categories are finite dimensional modules over a Hopf algebra. Hence, one should think of the designation 'quantum field theory' in the book as being one that indicates only its historical roots. A fully operational quantum field theory always needs infinite dimensions to gain its predictive power. These modular categories are constructed from modular functors, the latter arising from closed oriented surfaces with a distinguished Lagrangian subspace and a finite set of marked points, or "colors" (the reader versed in conformal field theory will see the origins of these ideas). Choosing a particular modular category and set of colors will give the familiar Jones polynomial.

After defining an isotopy invariant for colored frame oriented links in Euclidean space, topological invariants for closed oriented 3-manifolds are defined by doing surgery on the standard 3-sphere along a framed link. The dependence on the link is removed by employing Kirby calculus, which gives the sequence of moves needed to relate one link to another. The resulting quantum invariant is thus dependent on the link diagrams, but an intrinsic definition computed from the manifold is via a state sum on a triangulation of the manifold. Most interesting is that this state sum is computed using the 6j-symbols, familiar to physicists in the quantum theory of angular momentum. The actual invariant requires the computation of a product over the manifold and one equal to it except taking the opposite orientation. In addition, the computation is done inside an arbitrary compact oriented piecewise-linear 4-manifold bounded by the manifold. This computation utilizes the concept of a "shadow" of a 4-manifold, which are topological objects related to 6j-symbols.The illumination (no pun intended) by the author of the theory of shadows is done in great detail and occupies most of the space in the book.

The existence of modular categories is related to the theory of representations of quantum groups at roots of unity, these quantum groups being Hopf algebras over the complex numbers which are constructed via 1-parameter deformations of the universal enveloping algebra of simple Lie algebras. The author though sticks with the general language of categories, and algebraic and geometric constructions of them are discussed in detail by the author.

All of the results in this book are interesting, but the author admits that their connection with low-dimensional topology and the classical invariants of 3-manifolds is not readily apparent, especially their connection with homotopy via the fundamental group. Such a connection would possibly shed light on the one of the most nagging questions in 3-dimensional topology: the Poincare conjecture.

an axiomatic book.......2000-05-01

this is an interesting book. the approach is axiomatic. it seems to lack physical motivation though. it helps if you are familiar with witten's original paper on tqft. otherwise, you might think that it is all abstract nonsense. in fact, the author himself said that the axioms of tqft might seem to be abstract nonsense, but it is powerful and rigorous.

Book Description

This is an introduction to the basic tools of mathematics needed to understand the relation between knot theory and quantum gravity. The book begins with a rapid course on manifolds and differential forms, emphasizing how these provide a proper language for formulating Maxwell's equations on arbitrary spacetimes. The authors then introduce vector bundles, connections and curvature in order to generalize Maxwell theory to the Yang-Mills equations. The relation of gauge theory to the newly discovered knot invariants such as the Jones polynomial is sketched. Riemannian geometry is then introduced in order to describe Einstein's equations of general relativity and show how an attempt to quantize gravity leads to interesting applications of knot theory.

Customer Reviews:

Fantastic Text.......2005-07-18

I really enjoyed reading this book! A must have if you are interested in mathematical physics. Every page is a pedagogical masterpiece.

My favourite text of all time (so far).......2003-09-14

This book should be at the top of anyone's reading list who is planning to get into serious mathematical physics. It deals with a good deal of complex material, but the presentation is easy to follow, and shouldn't be beyond most advanced undergraduates. There are a lot of good exercises which fill in most of the gaps. (If you want a book heavy on detail, this book may not be for you. If you want a book that gives you all the tools you're going to need to get start understanding quantum gravity and other areas in a short time, get this book immediately!) It's a shame the paperback edition doesn't seem to be available anymore; it's half the price, and checking with the publisher reveals that the paperback edition is still in print.

An excellent book !.......2002-12-21

Covers many topics in Mathematical Physics with great clarity. Highly recommended for those who are interested in a modern approach to Mathematical Physics.

Perfect.......2000-07-29

A beautifully written book which should be entitled "quantum gravity primer for the practical man". Clear and self-contained, this book will serve aa a small survey of mathematical physics, giving the reader tools in particle physics and gravity. Excellently motivated topics. Compact enough to bring with you anywhere. The only thing it fails at is dicing a proper tomato.

Worth its weight in gold!.......1999-06-24

I think the review above by J. Pullin puts it very well. This is a great book, and a good place to get started (it also provides suggestions for further reading). The authors have done a fantastic job, and I highly recommend the book!

A Course in Enumeration (Graduate Texts in Mathematics)

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A Course in Enumeration (Graduate Texts in Mathematics)
Martin Aigner
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 3540390324

Book Description

Combinatorial enumeration is a readily accessible subject full of easily stated, but sometimes tantalizingly difficult problems. This book leads the reader in a leisurely way from the basic notions to a variety of topics, ranging from algebra to statistical physics. Its aim is to introduce the student to a fascinating field, and to be a source of information for the professional mathematician who wants to learn more about the subject. The book is organized in three parts: Basics, Methods, and Topics. There are 666 exercises, and as a special feature every chapter ends with a highlight, discussing a particularly beautiful or famous result.

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math grad.

Knots and Physics
Louis H. Kauffman
Manufacturer: World Scientific Publishing Company
ProductGroup: Book
Binding: Paperback

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ASIN: 9810241127

Book Description

This book is an introductory explication on the theme of knot and link invariants as generalized amplitudes (vacuum-vacuum amplitudes) for a quasi-physical process. The demands of the knot theory, coupled with a quantum statistical frame work create a context that naturally and powerfully includes an extraordinary range of interelated topics in topology and mathematical physics. The author takes a primarily combinatorial stance toward the knot theory and its relations with these subjects. This has the advantage of providing very direct access to the algebra and to the combinatorial topology, as well as the physical ideas. This book is divided into 2 parts: Part I of the book is a systematic course in knots and physics starting from the ground up. Part II is a set of lectures on various topics related with and sometimes based on Part I. Part II also explores some side-topics such as frictional properties of knots, relations with combinatorics, knots in dynamical systems.

Customer Reviews:

math grad........2002-03-20

Overview of knots/physics. The book is fairly self-contained. It also has lots of pictures and works through the mathematics.
Introduces bracket polynomial, temperly-lieb algebra, and modeling physics ideas out of this stuff.

Quantum Invariants: A Study of Knot, 3-Manifolds, and Their Sets

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Quantum Invariants: A Study of Knot, 3-Manifolds, and Their Sets
Tomotada Ohtsuki
Manufacturer: World Scientific Publishing Company
ProductGroup: Book
Binding: Hardcover

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ASIN: 9810246757

Book Description

This book provides an extensive and self-contained presentation of quantum and related invariants of knots and 3-manifolds. Polynomial invariants of knots, such as the Jones and Alexander polynomials, are constructed as quantum invariants, i.e. invariants derived from representations of quantum groups and from the monodromy of solutions to the Knizhnik-Zamolodchikov equation. With the introduction of the Kontsevich invariant and the theory of Vassiliev invariants, the quantum invariants become well-organized. Quantum and perturbative invariants, the LMO invariant, and finite type invariants of 3-manifolds are discussed. The Chern-Simons field theory and the Wess-Zumino-Witten model are described as the physical background of the invariants.

Relativistic Reality: A Modern View (Knots and Everything, Vol 12)

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Relativistic Reality: A Modern View (Knots and Everything, Vol 12)
James D. Edmonds
Manufacturer: World Scientific Publishing Company
ProductGroup: Book
Binding: Hardcover

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ASIN: 9810228511

Loops, Knots, Gauge Theories and Quantum Gravity

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Loops, Knots, Gauge Theories and Quantum Gravity
Rodolfo Gambini , and Jorge Pullin
Manufacturer: Cambridge University Press
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Binding: Paperback

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ASIN: 0521654750

Book Description

Loop representations (and the related topic of knot theory) are of considerable current interest because they provide a unified arena for the study of the gauge invariant quantization of Yang-Mills theories and gravity, and suggest a promising approach to the eventual unification of the four fundamental forces. This text provides a self-contained introduction to applications of loop representations and knot theory in particle physics and quantum gravity. The book begins with a detailed review of loop representation theory. It then goes on to describe loop representations in Maxwell theory, Yang-Mills theories, as well as lattice techniques. The authors then discuss applications in quantum gravity in detail. Following chapters consider knot theories, braid theories and extended loop representations in quantum gravity. A final chapter assesses the current status of the theory and points out possible directions for future research.

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The Geometry and Physics of Knots (Lezioni Lincee)

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Quick overview of TQFT and knot invariants
Good intro to topological quantum field theory
NOT a book about KNOTS!

The Geometry and Physics of Knots (Lezioni Lincee)
Michael Atiyah
Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Paperback

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ASIN: 0521395542

Book Description

Deals with an area of research that lies at the crossroads of mathematics and physics. The material presented here rests primarily on the pioneering work of Vaughan Jones and Edward Witten relating polynomial invariants of knots to a topological quantum field theory in 2+1 dimensions. Professor Atiyah presents an introduction to Witten's ideas from the mathematical point of view. The book will be essential reading for all geometers and gauge theorists as an exposition of new and interesting ideas in a rapidly developing area.

Customer Reviews:

Quick overview of TQFT and knot invariants.......2001-05-28

This book is a very quick overview of what was known at the time (1989) about the connection between quantum field theory and knot theory. The subject of topological quantum field theories and their connection with knot invariants was at that time just beginning thanks to the work of Edward Witten on the Jones polynomial.

The approach that the author takes in the book is very formal and not for the beginner who is looking to learn about these results. Readers with enough background to read it will no doubt want to read more up-to-date treatments of the subject. The book however does give an indication of how Feynman path integrals are used to define the invariants. The use of these is not rigorous mathematics and this has not changed at the present day.

Good intro to topological quantum field theory.......1998-12-23

As the above review indicates, this is not an introduction to knot theory. Those interested should first read Colin Adams "The Knot Book" and Kauffman's "Knots and Physics". This instead introduces the reader to the subject of topological quantum field theory, and shows how a Feynmann Path integral approach gives rise to the Jones polynomial knot invariant (Witten's approach).

The first two chapters are accessible to those who have had graduate-level abstract algebra and some topology. After that, a good familiarity with quantum field theory and quantization of symplectic manifolds (although not strictly speaking necessary) makes the subjects clearer.

Those who are working in Donaldson and Seiberg-Witten approaches to four-dimensional topology will want to read this book.

NOT a book about KNOTS!.......1998-12-08

If you are seeking a graduate level discussion on quantum theory and knot theory, this is your book. Definitely not for the amateur user...

Quantum Topology (Series on Knots & Everything)

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Quantum Topology (Series on Knots & Everything)
Louis H. Kauffman , and Randy A. Baadhio
Manufacturer: World Scientific Pub Co Inc
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Binding: Paperback

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ASIN: 981022575X

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