# Analytic Properties of Feynman Diagrams in Quantum Field Theory

### International Series of Monographs in Natural Philosophy

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Analytic Properties of Feynman Diagrams in Quantum Field Theory deals with quantum field theory, particularly in the study of the analytic properties of Feynman graphs. This book is an elementary presentation of a self-contained exposition of the majorization method used in the study of these graphs. The author has taken the intermediate position between Eden et al. who assumes the physics of the analytic properties of the S-matrix, containing physical ideas and test results without using the proper mathematical methods, and Hwa and Teplitz, whose works are more mathematically inclined with applications of algebraic topology and homology theory. The book starts with the definition of the quadratic form of a Feynman diagram, and then explains the majorization of Feynman diagrams. The book describes the derivation of spectral representations, the dispersion relations for the nucleon-nucleon scattering amplitude, and for the corresponding partial wave amplitude. The text then analyzes the surface of singularities of a Feynman diagram with notes explaining the Cutkosky rules of the Mandelstam representation for the box diagram. This text is ideal for mathematicians, physicists dealing with quantum theory and mechanics, students, and professors in advanced mathematics.

## Table of Contents

Preface to the English Edition

Translator's Note

Foreword

Introduction

1. Dispersion Relations and Perturbation Theory

2. A Survey of Work on the Analytic Properties of S-matrix Elements in Perturbation Theory

3. Contents of the Book

Certain Notations

Chapter 1. The Quadratic Form of a Feynman Diagram

1. The Representation for the Contribution of an Arbitrary Diagram to the Scattering Matrix

1.1. The Incidence Matrix

1.2. The Feynman Amplitude Corresponding to a Diagram with Scalar Lines

1.3. The Domain of Analyticity of a Diagram

2. Properties of the Quadratic Form of a Diagram for Euclidean External Momenta

2.1. Definition of Euclidean Momenta

2.2. Extremal Property of the Quadratic form of a Diagram

2.3. Properties of the Quadratic Form Q(α,p) for Real and Euclidean Momenta

3. The Majorization of a Quadratic Form with Real Momenta by a Quadratic Form with Euclidean Momenta

3.1. The Decomposition of Real Vectors into Euclidean and Anti-Euclidean Parts

3.2. Example. Elastic Scattering of Two Particles

Appendix to Chapter 1. Calculation of the Jacobian of the Transformation (1.1.10)

Summary

Chapter 2. Majorization of Feynman Diagrams

1. Principle of Majorization. The Method for Obtaining the Primitive Diagrams

1.1. The Principle of Majorization for a Strongly Connected Diagram

1.2. Criteria for the Majorization of Diagrams in the Euclidean Region

1.3. Some Topological Properties of Strongly Connected Diagrams

2. Primitive Diagrams of the Vertex Part and of Scattering Processes

2.1. Primitive Diagrams of the Meson-nucleon Vertex Part (Scalar Mesons)

2.2. The Primitive Diagrams for Scattering Processes with Scalar Mesons and Nucleons

3. The Symanzik Theorem and its Generalization

3.1. The Conjugate Norms of a Diagram

3.2. Criteria of Majorization, Based on the Properties of the Conjugate Norms

4. Majorization of the Primitive Diagrams

4.1. The Vertex Part (Scalar Mesons). The Irreducible Diagram for the "Electromagnetic Structure" of the Nucleon

4.2. Nucleon-nucleon Scattering and Anti-nucleon-nucleon Scattering

4.3. Scalar Meson-nucleon Scattering

5. Majorization of Diagrams for Processes Involving Pseudoscalar Mesons

5.1. The Majorization of an Arbitrary Diagram by a Diagram that does not Contain Baryon Loops

5.2. The Vertex Part (Pseudoscalar Mesons)

5.3. Pion-pion Scattering

Appendix to Chapter 2. Nucleon-nucleon primitive scattering diagrams (Proof of Theorem 2.5)

Summary

Chapter 3. Derivation of Spectral Representations and of Dispersion Relations

1. Analytic Properties of the Vertex Part. The Concept of an Anomalous Threshold

1.1. Domain of Analyticity of the Vertex Part in the Space of Three Complex Variables

1.2. The Integral Representation of the Vertex Part

1.3. Pion-nucleon Vertex

1.4. Normal and Anomalous Thresholds

2. Dispersion Relations for the Nucleon-nucleon Scattering Amplitude and for the Corresponding Partial Wave Amplitud

2.1. The Domain of Analyticity of the Amplitudes for Nucleon-nucleon Scattering and for Anti-nucleon-nucleon Scattering

2.2. Spectral Representations for the Nucleon-nucleon and Meson-meson Scattering Amplitudes

3. Dispersion Relations for the Scalar Meson-nucleon Scattering Amplitude

3.1. The Domain of Analyticity of the Scalar Meson-nucleon Scattering Amplitude

3.2. Dispersion Relations for Elastic Meson-nucleon Scattering

Appendix to Chapter 3. Analyticity of Tn, and TD4 in the Domain (3.3.2)

Summary

Chapter 4. The Surface of Singularities of a Feynman Diagram. What Else Can We Learn from the Box Diagram?

1. Equations for the Singular Surface

1.1. The Singular Points of a Diagram. Proper Singularities

1.2. Parametric Equations for the Surface of Proper Angularities

2. Examples of the Application of the Parametric Equations of the Surface of Singularities

2.1. The Fourth-order Diagram of Nucleon—nucleon Scattering

2.2. The Tetrahedron Scattering Diagram

2.3. The Diagram of the Pion—nucleon Vertex Part

2.4. A Scattering Diagram with a Three-particle Intermediate State

3. Survey of Cutkosky Rules and of the Mandelstam Representation for the Box Diagram

3.1. Cutkosky Rules for the Discontinuity of a Feynman Amplitude

3.2. Cutkosky Rules in the Presence of an Anomalous Threshold

3.3. The Mandelstam Representation for the Box Diagram with Normal Thresholds

Appendix to Chapter 4. The Example of the Self-energy Diagram

Summary

References

Index

## Product details

- No. of pages: 168
- Language: English
- Copyright: © Pergamon 1971
- Published: January 1, 1971
- Imprint: Pergamon
- eBook ISBN: 9781483156323

## About the Author

### I. T. Todorov

## About the Editor

### D. Ter Haar

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