Conformal Field Theory for Particle Physicists : From QFT Axioms to the Modern Conformal Bootstrap

This book is a set of introductory lecture notes on Conformal Field Theory (CFT). Unlike most existing reviews on the subject, CFT is presented here from the perspective of a unitary quantum field theory in Minkowski space-time. The book starts with a non-perturbative formulation of quantum field theory (Wightman axioms) and then, gradually, focuses on the implications of scale and special conformal symmetry, all the way to the modern conformal bootstrap. This approach includes topics such as subtleties of conformal transformations in Minkowski space-time, the construction of Wightman functions and time-ordered correlators both in position- and momentum-space, unitarity bounds derived from the spectral representation, and the appearance of UV and IR divergences. In each chapter, the reader finds useful exercises to master the subject. This book is meant for graduate students in theoretical physics and for more advanced researchers working in high-energy physics who are not necessarily familiar with the concepts of conformal field theory. Prior knowledge of quantum field theory is needed to master the arguments. Preface 7 Contents 9 Acronyms 11 1 Introduction 12 References 13 2 Classical Conformal Transformations 14 2.1 Infinitesimal Transformations 14 2.2 The Conformal Algebra 18 2.3 Finite Transformations 20 2.4 Compactifications 23 2.5 Minkowski Space-Time 25 2.6 Conformal Symmetry in Classical Field Theory 27 Reference 32 3 Conformal Quantum Field Theory 33 3.1 Non-Perturbative Quantum Field Theory 33 3.2 Wightman Functions 36 3.3 Spectral Representation 39 3.4 Scale Symmetry 41 3.5 Special Conformal Symmetry 45 3.6 UV/IR Divergences and Anomalies 51 Reference 54 4 Conformal Correlation Functions 55 4.1 From Minkowksi Space-Time to Euclidean Space 55 4.2 From Euclidean Space to Embedding Space 58 4.3 3-Point Functions 60 4.4 4-Point Functions 63 References 65 5 State-Operator Correspondence and OPE 67 5.1 The OPE in Quantum Field Theory 67 5.2 The State/Operator Correspondence 68 5.3 The Conformal OPE 70 6 The Conformal Bootstrap 73 6.1 Conformal Blocks 74 6.2 OPE Convergence 77 6.3 The Crossing Equation and Simple Solutions 79 6.4 The Numerical Bootstrap 82 6.5 Example: The Ising Model in 3 Dimensions 83 6.6 Other Conformal Bootstrap Results 87 References 88 7 Outlook 90 References 92