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Rating: 
Summary: Lots to learn...
Review: These articles are great. Fills the ubiquitous need to retract the gap between then conceptual and rigorous framework of the subjects.Physicists interested in the mathematical aspects of quantum field/string theory would do well to read these volumes as well. Deserving, in my opinion, more than 5 stars -- many more!!
Rating:
Summary: Lots to learn...
Review: These articles are great. Fills the ubiquitous need to retract the gap between then conceptual and rigorous framework of the subjects. Physicists interested in the mathematical aspects of quantum field/string theory would do well to read these volumes as well. Deserving, in my opinion, more than 5 stars -- many more!!
Rating: 
Summary: Definitely for mathematicians only
Review: 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.
Rating:
Summary: Definitely for mathematicians only
Review: 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.
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