# Lectures on The Many-Body Problems V1

## 1st Edition

**Editors:**E.R. Caianiello

**eBook ISBN:**9780323154475

**Imprint:**Academic Press

**Published Date:**1st January 1962

**Page Count:**356

## Description

Lectures on Field Theory and the Many-Body Problem is a 23-chapter lecture series on the developments in the understanding of the structure and axiomatics of Field Theory, which has proved to be a most useful tool in the study of many-body problems. This book starts with a brief introduction to the TCP theorem, followed by a discussion on the gauge properties of the quantum electrodynamical quantities. The subsequent chapters describe the features and applications of unstable and composite particles to quantum field theory. These topics are followed by significant chapters on other aspects of the field theory, including the configuration space method, Wightman functions, vacuum expectation value, Pais doublets, time reversal in nuclear forces, and symmetry operations in quantum mechanics. This text also covers the ground state theory of many-particle systems and the many body problems at non-zero temperature. The last chapters explore the behavior of a Boson system, the polaron model, and the mathematical aspects of the Hilbert spaces. Physicists and researchers in allied sciences will find this book of great value.

## Table of Contents

Contributors

Preface

TCP Theorem and Related Problems

I. Connection between Spin and Statistics

II. The TCP Theorem

III. Particles and Antiparticles

References

The Gauge Transformation of Propagators in Quantum Electrodynamics

I. Introduction

II. The Generating Functional

III. Quantum Electrodynamics

IV. Generalized Gauge Transformations

References

Unstable Particles and Complex Poles of the Propagators

I. Introduction

II. The Resolvent and Its Singularities

III. The Exponential Decay

IV. The Representation of Amplitudes

References

Description of Unstable Particles in Quantum Field Theory

I. Introduction

II. Properties of Scattering Amplitudes in the Lee Model

III. Definition of the Unstable V-Particles by Means of the Propagator

IV. Analytical Continuation of the Propagator in a More General Field Theory

V. Remarks on the Time-Graph of an Unstable Particle

References

The Asymptotic Condition and Dispersion Relations

I. Asymptotic Condition. Bound States

II. Reduction Technique and Applications

III. Derivation of Dispersion Relations and Î”2-Analyticity

IV. Results and Possible Causes of Limitations

V. Physical Interpretation of Dispersion Relations. Macrocausality

VI. Dyson's Theorem. High-Energy Behavior

Application of Dispersion Relations to the Determination of Coupling Constants

Second Quantization and Configuration Space Method

I. Configuration Space Wave Functions

II. Algebraic Representation of the State Vectors

III. Dual Space

IV. Linear Operators of Î›R

V. One-Particle Operators

VI. Two-Particle Operators

VII. Field Quantities and Field Equations

Regularization and Renormalization

Properties of Wightman Functions

I. Introduction

II. Some Properties of the Lorentz Group

III. The Theorem of Bargmann, Hall, and Wightman

IV. The Real Regularity Points of R'N and of the Permuted Domain

V. The OTP-Theorem and the Connection between Spin and Statistics

VI. On the Structure of R'N

Appendix to Part IV

References

Analyticity of Vacuum Expectation Values

I. Preface

II. Introduction

III. Simple Illustration of Completion of Analyticity Domains

IV. Reduction of Parameters in the DAÎAD Representation of the M3 Boundary

V. Proof that the M3 Boundary Contains a Non-Analytic Hypersurface

VI. Discussion

References

Connection between Wightman Functions and r -Functions

I. Statement of the Problem

II. The Multiple Commutator

III. The Function r(k1; k2; k3)

IV. The Existence of K

V. Concluding Remarks

References

A Note on the Transformation Ïˆ'= exp [iy5a]Ïˆ

I. Definitions

II. The Mass Theorem

III. Mass and Degeneracy

IV. Application to the Heisenberg Type Theory

V. The Significance of R

References

Pais Doublets and Weak Interactions

Time Reversal in Nuclear Forces

I. Nuclear Reactions (No Polarization)

II. Polarization Measurements

III. Beta-Decay

IV. Correlations in Successive Radiations

References

On Symmetry Operations in Quantum Mechanics

I. Fixing a Quantum Mechanical System

II. Defining Rays, Unit Rays, States, etc

III. Defining Symmetry Operations in Physical Terms

IV. Defining Transformations in Hubert Space Out of Ray Transformations

V. Several Properties of the Transformations Î¸ ~ Î¸

VI. Proof of the Main Theorem

VII. Representation up to a Factor

Remarks on the Nucleon Form Factors in the Configuration Space

Global Symmetry of the Elementary Particles

I. Fundamental Assumptions

II. Î²-Decay

References

Ground State Theory of Many-Particle Systems

I. Introduction

II. The Goldstone Approach

III. Resolvent Technique

IV. Difficulties of Perturbation Theory for the Fermi Gas

References

On the Many Body Problem at Non-Zero Temperature

I. Introduction

II. Expansion of the Grand Partition Function

III. The Generalized Theorem of Wick

IV. The Gibbs Potential

V. Further Transformations of the Expansion

VI. A Second Method of Expanding the Grand Partition Function

VII. Relation between the Two Expansions

VIII. The Shielded Potential and the Contribution of the Binary Collisions

References

Collective Behavior of a Boson System

The Theory of Superconductivity

The Polaron Model

I. Introduction

II. Pekar's Product Ansatz

III. Translational Invariance

IV. Adiabatic Approach of Bogoliubov and Tiablikov

V. A Variational Method Connecting the Strong and Weak Coupling Limits

VI. Special Cases of the Variational Ansatz

VII. Influence of a Cutoff

VIII. Final Remarks

References

Free Fields and Multilinear Algebra over Hilbert Spaces

I. Second Quantized SchrÃ¶dinger Equation

II. Scalar Neutral Field

III. Electron-Positron Field

IV. Photon Field

References

Author Index

Subject Index

## Details

- No. of pages:
- 356

- Language:
- English

- Copyright:
- © Academic Press 1962

- Published:
- 1st January 1962

- Imprint:
- Academic Press

- eBook ISBN:
- 9780323154475