Towards the Mathematics of Quantum Field Theory (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 59) 2014th Edition by Frédéric Paugam (PDF)

Description

This ambitious and original book sets out to introduce to mathematicians (even including graduate students ) the mathematical methods of theoretical and experimental quantum field theory, with an emphasis on coordinate-free presentations of the mathematical objects in use. This in turn promotes the interaction between mathematicians and physicists by supplying a common and flexible language for the good of both communities, though mathematicians are the primary target. This reference work provides a coherent and complete mathematical toolbox for classical and quantum field theory, based on categorical and homotopical methods, representing an original contribution to the literature.The first part of the book introduces the mathematical methods needed to work with the physicists’ spaces of fields, including parameterized and functional differential geometry, functorial analysis, and the homotopical geometric theory of non-linear partial differential equations, with applications to general gauge theories. The second part presents a large family of examples of classical field theories, both from experimental and theoretical physics, while the third part provides an introduction to quantum field theory, presents various renormalization methods, and discusses the quantization of factorization algebras.

User’s Reviews

Editorial Reviews: Review “The book under review is a valuable witness to a wide range of recent developments that aim at unraveling the mathematics of quantum field theory. … the book is well written with an original organization of the material. Its reductionist viewpoint makes it a valuable textbook, especially for the more experienced practitioners in this domain of mathematical physics.” (Walter D. van Suijlekom, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 117, 2015)“This is a highly specialized monograph exposing current research in a systematic way, aimed at researchers and graduate students in this area. The author has tried to include several examples to illustrate the general methods being treated and ease the reader through the journey.” (Felipe Zaldivar, MAA Reviews, July, 2014)“This is a complete guide to the mathematics underlying the laws of physics in general and quantum physics in particular. … Frederic Paugam has given us a work of enormous scope and shows students a road to a coordinate-free presentation of mathematical objects underlying classical and quantum field theories. … The book will be useful to graduate student, but also to pure and applied mathematicians interested in modern mathematical developments related to quantum theory.” (Gert Roepstorff, zbMATH, Vol. 1287, 2014) From the Back Cover The aim of this book is to introduce mathematicians (and, in particular, graduate students) to the mathematical methods of theoretical and experimental quantum field theory, with an emphasis on coordinate-free presentations of the mathematical objects in play. This should in turn promote interaction between mathematicians and physicists by supplying a common and flexible language for the good of both communities, even if the mathematical one is the primary target. This reference work provides a coherent and complete mathematical toolbox for classical and quantum field theory, based on categorical and homotopical methods, representing an original contribution to the literature.The first part of the book introduces the mathematical methods needed to work with the physicists’ spaces of fields, including parameterized and functional differential geometry, functorial analysis, and the homotopical geometric theory of non-linear partial differential equations, with applications to general gauge theories. The second part presents a large family of examples of classical field theories, both from experimental and theoretical physics, while the third part provides an introduction to quantum field theory, presents various renormalization methods, and discusses the quantization of factorization algebras. The book is primarily intended for pure mathematicians (and in particular graduate students) who would like to learn about the mathematics of quantum field theory. About the Author Frédéric Paugam is a pure mathematician working at the University Pierre et Marie Curie. He started his career in arithmetic geometry, working on Galois representations and abelian varieties. He first became interested in the mathematics of quantum physics through the study of quantum statistical mechanics. He then approached quantum field theory with the categorical methods that he had learned from the work of Grothendieck’s school. He has since held various courses on this subject, allowing him to develop the tools and content of this book with the aim of teaching in mind. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Readers who decide to commit to this book should not expect to find fresh approaches to the calculation of cross sections or a solution of the bound state problem in quantum field theory. Rather, such readers will be exposed to a formulation of quantum field theory in terms of category theory and higher category theory, and as such will be confronted with a formidable array of new concepts and terminology that is not to be found in a standard textbook or treatise on quantum field theory. That is not to say that a categorical approach to quantum field theory is without merits, but rather that the mathematical background needed to understand such an approach may not be in the repertoire of even the most mathematically astute reader. It is assumed that readers have this background, and most importantly understand the meaning and intuition behind the mathematical constructions. The author for the most part does not give the reader any special insights into these constructions, and admits that his account is incomplete and was chosen because of his level of comfort with the mathematical constructions.As evidenced by the title of this book, the author does not assert that his approach is the final mathematical framework for quantum field theory. The ideas that are discussed are supposed to cover the case of gauge theories and this is done to a large degree using categorical formulations of differential geometry. Readers who are familiar with topological quantum field theory will have an edge over readers who do not, since the former class will appreciate the fact that ideas from cobordism theory and symmetric monoidal categories seem very natural in the context of topological quantum field theory. Mathematicians who are experts in higher category theory will no doubt find the order of presentation very palatable, since the emphasis is not on the physics of quantum field theory. If both physicists and mathematicians could get together and jointly study this book, this would fulfill the author’s stated goal of promoting interactions between these two groups.The reviewer read and studied this book in its entirety, but was particularly interested in the chapter on nonperturbative quantum field theory and how it would be formulated in the categorical language chosen by the author. The formulation involves representing the effective field theory approach of Kenneth Wilson in terms of jet bundles. The formulation is viewed as “nonperturbative” since it does not involve doing expansions around a (Gaussian) saddle point or the use of background field expansions. The author does not discuss effective field theory at all but instead refers to the references, and so readers not familiar with it will have to consult these references. In a nutshell the author views the renormalization group as a vector field on the space of theories the solution of which begins with a specification of an initial theory. Since regularization is an inevitable requirement the vector field (essentially the effective action) depends on a regularization operator, and the author very concisely derives a coordinate-free nonperturbative renormalization group equation that shows succinctly how the effective action evolves with respect to the regularization operator and a regularized nonperturbative propagator. Functional derivation of this equation gives the famous Schwinger-Dyson equations. Unfortunately the author does not show any advantages in casting the Schwinger-Dyson equations in this way to the actual solution of these equations.Another topic that the reviewer was hoping for more explanation was that of “delooping” and its different manifestations in algebraic topology, such as the representations of vector bundles as pullbacks of a universal bundle. The author does discuss it at various places in the book, particularly in the chapter on homotopical geometry, but since it has such a major role to play in homotopy theory historically, some motivation for the need for it seems necessary. This is particularly true also given the author’s predilection for using ideas from derived categories in the formulation of quantum field theory. Indeed, the author speaks of the “categorification” of theories by using a “doctrine machine”, which will give a homotopical version of any theory of interest. A natural question to ask here is whether quantum field theory finds its best categorical/mathematical description in terms of derived categories or derived algebraic geometry. Whether this is the case or not, the author’s discussion of derived categories and homotopical algebra is the most interesting one in the book, and in fact could be said to predominant its theme.

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Free Download Towards the Mathematics of Quantum Field Theory (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 59) 2014th Edition in PDF format
Towards the Mathematics of Quantum Field Theory (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 59) 2014th Edition PDF Free Download
Download Towards the Mathematics of Quantum Field Theory (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 59) 2014th Edition 2014 PDF Free
Towards the Mathematics of Quantum Field Theory (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 59) 2014th Edition 2014 PDF Free Download
Download Towards the Mathematics of Quantum Field Theory (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 59) 2014th Edition PDF
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