Ideas from quantum field theory and string theory have had considerable impact on mathematics over the past 20 years. Advances in many different areas have been inspired by insights from physics. In 1996-97 the Institute for Advanced Study (Princeton, NJ) organized a special year-long program designed to teach mathematicians the basic physical ideas which underlie the mathematical applications. The purpose is eloquently stated in a letter written by Robert MacPherson: ``The goal is to create and convey an understanding, in terms congenial to mathematicians, of some fundamental notions of physics ... [and to] develop the sort of intuition common among physicists for those who are used to thought processes stemming from geometry and algebra.'' These volumes are a written record of the program. They contain notes from several long and many short courses covering various aspects of quantum field theory and perturbative string theory. The courses were given by leading physicists and the notes were written either by the speakers or by mathematicians who participated in the program. The book also includes problems and solutions worked out by the editors and other leading participants. Interspersed are mathematical texts with background material and commentary on some topics covered in the lectures. These two volumes present the first truly comprehensive introduction to this field aimed at a mathematics audience. They offer a unique opportunity for mathematicians and mathematical physicists to learn about the beautiful and difficult subjects of quantum field theory and string theory.
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"An immense amount of valuable material on recent developments. The development of classical supersymmetry by Deligne and collaborators is careful and systematic ... masterful treatment ... the book is a magnificent achievement." -- SIAM Review ---- SIAM Review
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31 of 31 people found the following review helpful:
4.0 out of 5 stars
Definitely for mathematicians only,
By Dr. Lee D. Carlson (Baltimore, Maryland USA) - See all my reviews (VINE VOICE) (HALL OF FAME REVIEWER) (REAL NAME)
This review is from: Quantum Fields and Strings: A Course for Mathematicians (2 Volume Set) (v. 1 & 2) (Paperback)
This book is an excellent compliation of articles written for mathematicians who want to understand quantum field theory. It is not surprising then that the articles are very formal and there is no attempt to give any physical intuition to the subject of quantum field theory. This does not mean however that aspiring physicists who want to specialize in quantum field theory should ont take a look at the contents. The two volumes are worth reading, even if every article cannot be read because of time constraints. All of the articles are written by the some of the major players in the mathematics of quantum field theory. Volume 1 starts off with a glossary of the terms used by physicists in quantum field theory and is nicely written. The next few hundred pages are devoted to supersymmetry and supermanifolds. A very abstract approach is given to these areas, with the emphasis not on computation but on the structure of supermanifolds as they would be studied mathematically. There is an article on classical field theory put in these pages, which is written by Pierre Deligne and Daniel Freed, and discussed in the framework of fiber bundles. The discussion of topological terms in the classical Lagrangian is especially well written. There is an introduction to smooth Deligne cohomology in this article, and this is nice because of the difficulty in finding understandable literature on this subject. Part Two of Volume 1 is devoted to the formal mathematical aspects of quantum field theory. After a short introduction to canonical quantization, the Wightman approach is discussed in an article by David Kazhdan. Most refreshing is that statement of Kazhdan that the Wightman approach does not work for gauge field theories. This article is packed with interesting insights, especially the section on scattering theory, wherein Kazdan explains how the constructions in scattering theory have no finite dimensional analogs. The article by Witten on the Dirac operator in finite dimensions is fascinating and a good introduction to how powerful concepts from quantum field theory can be used to prove important results in mathematics. A fairly large collection of problems (with solutions) ends Volume 1. The first part of Volume 2 is devoted entirely to the mathematics of string theory and conformal field theory. The article by D'Hoker stands out as one that is especially readable and informative. D. Gaitsgory has a well written article on vertex algebras and defines in a very rigorous manner the constructions that occur in the subject. The last part of Volume 2 discusses the dynamics of quantum field theory and uses as much mathematical rigor as possible. One gets the impression that it this is the area where it is most difficult to proceed in an entirely rigorous way. Path integrals, not yet defined mathematically and used throughout the discussion. The best article in Volume 2, indeed of the entire two volumes is the one on N = 2 Yang-Mills theory in four dimensions. It is here that the most fascinating constructions in all of mathematics find their place. These two volumes are definitely worth having on one's shelf, and the price is very reasonable considering the expertise of the authors and considering what one will take away after reading them.
6 of 8 people found the following review helpful:
5.0 out of 5 stars
Amazingly different!,
By Cybertronian (Finland) - See all my reviews
This review is from: Quantum Fields and Strings: A Course for Mathematicians (2 Volume Set) (v. 1 & 2) (Paperback)
If you are a mathematician and want to know what QFT and string theory (ST) are about, or if you are a string theorist, but want to know more about the mathematics behind the theory, then this book is what you are looking for. This book is totally different than any other standard textbook on QFT or ST. Mathematicians will love the fact that it is written in a language that is (close to) theirs, and physicists will enjoy it (provided they have a more solid background in mathematics than the average practical physicists), because it shows QFT and ST from completely "new" angles.
The book is a collection of lectures given by the various contributors, which fit together perfectly . The first volume has a very good overview of concepts and definitions at the very beginning and serves as a more mathematically oriented introductory treatment of SUSY and QFT with many advanced topics near the end. The second volume starts with an axiomatic approach of CFT and its use in string theory and then quantises strings with the BRST method (on a more mathematical level than usual), which is done superbly. It's a great reference, a wonderful textbook and a true delicacy in modern mathematical physics. A MUST HAVE for every mathematically inclined physicst or any mathematician with interests in physics! Great value for money...
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