CONTENTS

0. Introduction and Preliminaries 1

0.1. Introduction 1

0.2. Notation and Preliminaries 6

1. A Simple Generalized Dyson Series (n = y + w6 ) 15

2. The Generalized Dyson Series 26

3. Examples of Perturbation Expansions 35

3.1. A Single Measure and Potential 35

3.2. Several Measures and Potentials 41

4. Stability Results 47

4.1. Stability in the Potentials 47

4.2. Stability in the Measures 49

5. Generalized Feynman Diagrams . 62

6. The Banach Algebra of Functionals.

Connections with Feynman's Operational Calculus 69

Acknowledgments 75

References 76

LIST OF FIGURES

Figure 1.1. The regions A, . , k = 2 16

Figure 5.1. n = y 62

Figure 5.2. r\ = y + o) 5 64

Figure 5.3. n = y + w 6 , n = 3 65

Figure 5.4. T in Figure 5.2 under an infinitesimal microscope . 66

h

Figure 5.5. n = y + £ w 5 67

p-i p T p

Figure 5.6. An example with multiple 6's 68

iii