Renormalization and Effective Field Theory (Mathematical Surveys and Monographs)

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 low-energy 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. Machine Generated Contents Note: 1. Overview -- 2. Functional Integrals In Quantum Field Theory -- 3. Wilsonian Low Energy Theories -- 4. A Wilsonian Definition Of A Quantum Field Theory -- 5. Locality -- 6. The Main Theorem -- 7. Renormalizability -- 8. Renormalizable Scalar Field Theories -- 9. Gauge Theories -- 10. Observables And Correlation Functions -- 11. Other Approaches To Perturbative Quantum Field Theory -- Acknowledgements -- 1. Introduction -- 2. The Effective Interaction And Background Field Functional Integrals -- 3. Generalities On Feynman Graphs -- 4. Sharp And Smooth Cut-offs -- 5. Singularities In Feynman Graphs -- 6. The Geometric Interpretation Of Feynman Graphs -- 7. A Definition Of A Quantum Field Theory -- 8. Au Alternative Definition -- 9. Extracting The Singular Part Of The Weights Of Feynman Graphs. 10. Constructing Local Counterterms -- 11. Proof Of The Main Theorem -- 12. Proof Of The Parametrix Formulation Of The Main Theorem -- 13. Vector-bundle Valued Field Theories -- 14. Field Theories On Non-compact Manifolds -- 1. Some Functional Analysis -- 2. The Main Theorem On Rn -- 3. Vector-bundle Valued Field Theories On Rn -- 4. Holoniorphic Aspects Of Theories On Rn -- 1. The Local Renormalization Group Flow -- 2. The Kadanoff-wilson Picture And Asymptotic Freedom -- 3. Universality -- 4. Calculations In & Phi;4 Theory -- 5. Proofs Of The Main Theorems -- 6. Generalizations Of The Main Theorems -- 1. Introduction -- 2. A Crash Course In The Batalin-vilkovisky Formalism -- 3. The Classical Bv Formalism In Infinite Dimensions -- 4. Example: Chern-simons Theory -- 5. Example: Yang-mills Theory -- 6. D-modules And The Classical Bv Formalism -- 7. Bv Theories On A Compact Manifold. 8. Effective Actions -- 9. The Quantum Master Equation -- 10. Homotopies Between Theories -- 11. Obstruction Theory -- 12. Bv Theories On Rn -- 13. The Sheaf Of Bv Theories On A Manifold -- 14. Quantizing Chern-simons Theory -- 1. Introduction -- 2. First-order Yang-mills Theory -- 3. Equivalence Of First-order And Second-order Formulations -- 4. Gauge Fixing -- 5. Renormalizability -- 6. Universality -- 7. Cohornology Calculations -- 1. Generalized Laplacians -- 2. Polydifferential Operators -- 3. Periods -- 4. Integrals Attached To Graphs -- 5. Proof Of Theorem 4.0.2 -- 1. Basic Definitions -- 2. Examples -- 3. Subcategories -- 4. Tensor Products Of Nuclear Spaces From Geometry -- 5. Algebras Of Formal Power Series On Nuclear Fréchet Spaces. Kevin Costello. Includes Bibliographical References (p. 249-251). Cover 1 Title page 4 Contents 6 Introduction 10 Theories, Lagrangians and counterterms 40 Field theories on Rn 100 Renormalizability 122 Gauge symmetry and the Batalin-Vilkovisky formalism 148 Renormalizability of Yang-Mills theory 216 Asymptotics of graph integrals 236 Nuclear spaces 252 Bibliography 258 Back Cover 264