Softcover ISBN:  9781470470081 
Product Code:  SURV/170.S 
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Electronic ISBN:  9781470413972 
Product Code:  SURV/170.E 
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Book DetailsMathematical Surveys and MonographsVolume: 170; 2011; 251 ppMSC: Primary 81;
This book tells mathematicians about an amazing subject invented by physicists and it tells physicists how a master mathematician must proceed in order to understand it. Physicists who know quantum field theory can learn the powerful methodology of mathematical structure, while mathematicians can position themselves to use the magical ideas of quantum field theory in “mathematics” itself. The retelling of the tale mathematically by Kevin Costello is a beautiful tour de force.
—Dennis Sullivan
This book is quite a remarkable contribution. It should make perturbative quantum field theory accessible to mathematicians. There is a lot of insight in the way the author uses the renormalization group and effective field theory to analyze perturbative renormalization; this may serve as a springboard to a wider use of those topics, hopefully to an eventual nonperturbative understanding.
—Edward Witten
Quantum field theory has had a profound influence on mathematics, and on geometry in particular. However, the notorious difficulties of renormalization have made quantum field theory very inaccessible for mathematicians. This book provides complete mathematical foundations for the theory of perturbative quantum field theory, based on Wilson's ideas of lowenergy effective field theory and on the Batalin–Vilkovisky formalism. As an example, a cohomological proof of perturbative renormalizability of Yang–Mills theory is presented.
An effort has been made to make the book accessible to mathematicians who have had no prior exposure to quantum field theory. Graduate students who have taken classes in basic functional analysis and homological algebra should be able to read this book.ReadershipGraduate students and research mathematicians interested in quantum field theory and mathematical physics.

Table of Contents

Chapters

1. Introduction

2. Theories, Lagrangians and counterterms

3. Field theories on $\mathbb {R}^n$

4. Renormalizability

5. Gauge symmetry and the BatalinVilkovisky formalism

6. Renormalizability of YangMills theory

Appendix 1. Asymptotics of graph integrals

Appendix 2. Nuclear spaces


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This book tells mathematicians about an amazing subject invented by physicists and it tells physicists how a master mathematician must proceed in order to understand it. Physicists who know quantum field theory can learn the powerful methodology of mathematical structure, while mathematicians can position themselves to use the magical ideas of quantum field theory in “mathematics” itself. The retelling of the tale mathematically by Kevin Costello is a beautiful tour de force.
—Dennis Sullivan
This book is quite a remarkable contribution. It should make perturbative quantum field theory accessible to mathematicians. There is a lot of insight in the way the author uses the renormalization group and effective field theory to analyze perturbative renormalization; this may serve as a springboard to a wider use of those topics, hopefully to an eventual nonperturbative understanding.
—Edward Witten
Quantum field theory has had a profound influence on mathematics, and on geometry in particular. However, the notorious difficulties of renormalization have made quantum field theory very inaccessible for mathematicians. This book provides complete mathematical foundations for the theory of perturbative quantum field theory, based on Wilson's ideas of lowenergy effective field theory and on the Batalin–Vilkovisky formalism. As an example, a cohomological proof of perturbative renormalizability of Yang–Mills theory is presented.
An effort has been made to make the book accessible to mathematicians who have had no prior exposure to quantum field theory. Graduate students who have taken classes in basic functional analysis and homological algebra should be able to read this book.
Graduate students and research mathematicians interested in quantum field theory and mathematical physics.

Chapters

1. Introduction

2. Theories, Lagrangians and counterterms

3. Field theories on $\mathbb {R}^n$

4. Renormalizability

5. Gauge symmetry and the BatalinVilkovisky formalism

6. Renormalizability of YangMills theory

Appendix 1. Asymptotics of graph integrals

Appendix 2. Nuclear spaces