Renormalization and Effective Field Theory

Mathematical Surveys and Monographs Volume 170 Renormalization and Effective Field Theory Kevin Costello American Mathematical Society surv-170-costello-cov.indd 1 1/28/11 8:15 AM Renormalization and Effective Field Theory Mathematical Surveys and Monographs Volume 170 Renormalization and Effective Field Theory Kevin Costello American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair MichaelA.Singer Eric M. Friedlander Benjamin Sudakov MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 81T13, 81T15, 81T17, 81T18, 81T20, 81T70. The author was partially supported by NSF grant 0706954 and an Alfred P. Sloan Fellowship. For additional information and updates on this book, visit www.ams.org/bookpages/surv-170 Library of Congress Cataloging-in-Publication Data Costello, Kevin. Renormalization and effective fieldtheory/KevinCostello. p. cm. — (Mathematical surveys and monographs ; v. 170) Includes bibliographical references. ISBN 978-0-8218-5288-0 (alk. paper) 1. Renormalization (Physics) 2. Quantum field theory. I. Title. QC174.17.R46C67 2011 530.143—dc22 2010047463 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2011 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 161514131211 Contents Chapter 1. Introduction 1 1. Overview 1 2. Functional integrals in quantum field theory 4 3. Wilsonian low energy theories 6 4. A Wilsonian definition of a quantum field theory 13 5. Locality 13 6. The main theorem 16 7. Renormalizability 19 8. Renormalizable scalar field theories 21 9. Gauge theories 22 10. Observables and correlation functions 27 11. Other approaches to perturbative quantum field theory 27 Acknowledgements 28 Chapter 2. Theories, Lagrangians and counterterms 31 1. Introduction 31 2. The effective interaction and background field functional integrals 32 3. Generalities on Feynman graphs 34 4. Sharp and smooth cut-offs 42 5. Singularities in Feynman graphs 44 6. The geometric interpretation of Feynman graphs 47 7. A definition of a quantum field theory 53 8. An alternative definition 55 9. Extracting the singular part of the weights of Feynman graphs 57 10. Constructing local counterterms 61 11. Proof of the main theorem 67 12. Proof of the parametrix formulation of the main theorem 69 13. Vector-bundle valued field theories 71 14. Field theories on non-compact manifolds 80 Chapter 3. Field theories on Rn 91 1. Some functional analysis 92 2. The main theorem on Rn 99 3. Vector-bundle valued field theories on Rn 104 4. Holomorphic aspects of theories on Rn 107 Chapter 4. Renormalizability 113 v vi CONTENTS 1. The local renormalization group flow 113 2. The Kadanoff-Wilson picture and asymptotic freedom 122 3. Universality 125 4. Calculations in φ4 theory 126 5. Proofs of the main theorems 131 6. Generalizations of the main theorems 135 Chapter 5. Gauge symmetry and the Batalin-Vilkovisky formalism 139 1. Introduction 139 2. A crash course in the Batalin-Vilkovisky formalism 141 3. The classical BV formalism in infinite dimensions 155 4. Example: Chern-Simons theory 160 5. Example : Yang-Mills theory 162 6. D-modules and the classical BV formalism 164 7. BV theories on a compact manifold 170 8. Effective actions 173 9. The quantum master equation 175 10. Homotopies between theories 178 11. Obstruction theory 186 12. BV theories on Rn 189 13. The sheaf of BV theories on a manifold 196 14. Quantizing Chern-Simons theory 203 Chapter 6. Renormalizability of Yang-Mills theory 207 1. Introduction 207 2. First-order Yang-Mills theory 207 3. Equivalence of first-order and second-order formulations 210 4. Gauge fixing 213 5. Renormalizability 214 6. Universality 217 7. Cohomology calculations 218 Appendix 1: Asymptotics of graph integrals 227 1. Generalized Laplacians 227 2. Polydifferential operators 229 3. Periods 229 4. Integrals attached to graphs 230 5. Proof of Theorem 4.0.2 233 Appendix 2 : Nuclear spaces 243 1. Basic definitions 243 2. Examples 244 3. Subcategories 245 4. Tensor products of nuclear spaces from geometry 247 5. Algebras of formal power series on nuclear Fréchet spaces 247 CONTENTS vii Bibliography 249 CHAPTER 1 Introduction 1. Overview Quantum field theory has been wildly successful as a framework for the study of high-energy particle physics. In addition, the ideas and techniques of quantum field theory have had a profound influence on the development of mathematics. There is no broad consensus in the mathematics community, however, as to what quantum field theory actually is. This book develops another point of view on perturbative quantum field theory, based on a novel axiomatic formulation. Most axiomatic formulations of quantum field theory in the literature start from the Hamiltonian formulation of field theory. Thus, the Segal (Seg99) axioms for field theory propose that one assigns a Hilbert space of states to a closed Riemannian manifold of dimension d − 1, and a unitary operator between Hilbert spaces to a d-dimensional manifold with boundary. In the case when the d- dimensional manifold is of the form M × [0,t], we should view the corresponding operator as time evolution. The Haag-Kastler (Haa92) axioms also start from the Hamiltonian formulation, but in a slightly different way. They take as the primary object not the Hilbert space, but rather a C algebra, which will act on a vacuum Hilbert space. I believe that the Lagrangian formulation of quantum field theory, using Feynman’s sum over histories, is more fundamental. The axiomatic framework developed in this book is based on the Lagrangian formalism, and on the ideas of low-energy effective field theory developed by Kadanoff (Kad66), Wilson (Wil71), Polchinski (Pol84) and others. 1.1. The idea of the definition of quantum field theory I use is very simple. Let us assume that we are limited, by the power of our detectors, to studying physical phenomena that occur below a certain energy, say Λ. The part of physics that is visible to a detector of resolution Λ we will call the low-energy effective field theory. This low-energy effective field theory is succinctly encoded by the energy Λ version of the Lagrangian, which is called the low-energy effective action Seff [Λ]. The notorious infinities of quantum field theory only occur if we con- sider phenomena of arbitrarily high energy. Thus, if we restrict attention to 1 2 1. INTRODUCTION phenomena occurring at energies less than Λ, we can compute any quantity we would like in terms of the effective action Seff [Λ]. If Λ < Λ, then the energy Λ effective field theory can be deduced from knowledge of the energy Λ effective field theory. This leads to an equation expressing the scale Λ effective action Seff [Λ]intermsofthe scale Λ effective action Seff [Λ]. This equation is called the renormalization group equation. If we do have a continuum quantum field theory (whatever that is!) we should, in particular, have a low-energy effective field theory for every energy. This leads to our definition : a continuum quantum field theory is a sequence of low-energy effective actions Seff [Λ], for all Λ < ∞, which are related by the renormalization group flow. In addition, we require that the Seff [Λ] satisfy a locality axiom, which says that the effective actions Seff [Λ] become more and more local as Λ →∞. This definition aims to be as parsimonious as possible. The only as- sumptions I am making about the nature of quantum field theory are the following: (1) The action principle: physics at every energy scale is described by a Lagrangian, according to Feynman’s sum-over-histories philosophy. (2) Locality: in the limit as energy scales go to infinity, interactions between fields occur at points. 1.2. In this book, I develop complete foundations for perturbative quantum field theory in Riemannian signature, on any manifold, using this definition. The first significant theorem I prove is an existence result: there are as many quantum field theories, using this definition, as there are Lagrangians. Let me state this theorem more precisely. Throughout the book, I will treat as a formal parameter; all quantities will be formal power series in . Setting to zero amounts to passing to the classical limit. Let us fix a classical action functional Scl on some space of fields E , which is assumed to be the space of global sections of a vector bundle on a manifold M 1.LetT (n)(E ,Scl) be the space of quantizations of the classical theory that are defined modulo n+1. Then, Theorem 1.2.1. T (n+1)(E ,Scl) → T (n)(E ,Scl) is a torsor for the abelian group of Lagrangians under addition (modulo those Lagrangians which are a total derivative). Thus, any quantization defined to order n in can be lifted to a quantization defined to order n +1in, but there is no canonical lift; any two lifts differ by the addition of a Lagrangian.

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