Renormalization (eBook)

Front Cover 1
Renormalization 4
Copyright Page 5
Contents 8
Preface 10
Acknowledgments 12
Interoduction 14
Chapter 1. Basic Analysis 18
1.1 Sets and the Heine–Borel Theorem 18
1.2 Measure, Integration, and Fubini’s Theorem 25
1.3 Geometry in Rk 39
Notes 41
Chapter 2. Class Bn Functions and Feynman Integrals 42
2.1 Definition of Class Bn Functions 43
2.2 Structure of Feynman Integrals 43
Notes 55
Chapter 3. The Power-Counting Theorem and More 56
3.1 Statement of the Theorem 56
3.2 Proof of the Theorem in One Dimension 59
3.3 Proof of the Theorem in an Arbitrary Finite Number of Dimensions 67
Notes 75
Chapter 4. / e . + 0 Limit of Feynman Integrals and Lorentz Covariance 76
4.1 Class Y(Rk) Functions (Test Functions) 76
4.2 Basic Estimates, e . + 0 Limit,and Lorentz covariance 79
Notes 89
Chapter 5. The Subtraction Formalism 90
5.1 Basic Definitions and Examples 90
5.2 The Subtraction Scheme 115
5.3 Convergence of the Subtraction Scheme 120
5.4 The Unifying Theorem of Renormalization and Basic Identities 141
Notes 145
Chapter 6. Asymptotic Behavior in Quantum Field Theory 148
6.1 Preliminary Asymptotic Estimates 149
6.2 General Dimensional Analysis of Subtracted-Out Feynman Integrands 152
6.3 High-Energy Behavior 164
6.4 General Asymptotic Behavior I 175
6.5 Zero-Mass Behavior 177
6.6 Low-Energy Behavior 184
6.7 General Asymptotic Behavior II 188
6.8 Generalized Decoupling Theorem 191
Notes 194
Appendix. Subtractions versus Counterterms 196
A.1 The Formal Unrenormalized Theory 196
A.2 Equivalence 200
Notes 205
References 206
List of Symbols 214
Index 216