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