An Introduction to Field Quantization

International Series of Monographs in Natural Philosophy

1st Edition - January 1, 1969
Author: Y. Takahashi
Editor: D. ter Haar
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 1 - 8 6 2 7 - 6

An Introduction to Field Quantization is an introductory discussion of field quantization and problems closely related to it. Field quantization establishes a commutation relation… Read more

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An Introduction to Field Quantization is an introductory discussion of field quantization and problems closely related to it. Field quantization establishes a commutation relation of the field and finds an operator in such a manner that the Heisenberg equation of motion is satisfied. This book contains eight chapters and begins with a review of the quantization of the Schroedinger field and the close relation between quantized field theory and the many-body theory in quantum mechanics. These topics are followed by discussions of the quantization of the radiation field and the field of lattice vibrations in a solid. The succeeding chapter deals with the familiar linear equations in relativistic field theory and the deduction of certain spin independent theories, which these fields have in common. Other chapter explores the derivation technique of the conservation laws for fields with arbitrary spin directly from the field equations without explicit recourse to Noether's theorem using a configuration space version of the generalized Ward identity. The discussion then shifts to the relativistic quantization method applicable to any field with arbitrary spin; the transformation of various fields under the Lorentz transformation; and a general method for constructing wave functions explicitly, as well as the application of this method to several examples. The concluding chapter focuses on the quantization of interacting fields. This book will prove useful to physicists and researchers.

Preface

I. Introduction

1. Introductory Considerations

2. Creation and Annihilation Operators

II. Non-Relativistic Fields

1. The Schroedinger Field

2. The Radiation Field

3. Harmonic Lattice Vibrations

III. Relativistic Free Fields

1. Notation and Conventions

2. Fields with Spin 0,1, and 2

2.1. The Klein-Gordon Field

2.2. The Proca Field

2.3. The Field with Spin 2

2.4. The Duffin-Kemmer-Petiau Field

3. Fields with Spin 1/2 and 3/2

3.1. The Dirac Field

3.2. The Rarita-Schwinger Field

3.3. The Fierz-Pauli-Gupta Field

4. A General Field Equation of First Order

5. The Bargmann-Wigner Fields

6. Massless Fields

6.1. The Maxwell Field

6.2. The Neutrino

IV. Some Aspects of Linear Field Equations

1. Conservation Laws and the First Identity

2. The Klein-Gordon Divisor and the Second Identity

V. Quantization of Relativistic Free Fields

1. Wave Functions and their Normalization

2. Quantization—A General Theory

2.1. Field Quantization

2.2. Vacuum Expectation Values, Retarded Functions and Reduction Formulae

3. Quantization—Examples

3.1. The Klein-Gordon Field

3.2. The Dirac Field

3.3. The Duffin-Kemmer-Petiau Field

4. Quantization of Massless Fields

4.1. The Electromagnetic Vector Field

4.2. The Neutrino Field

VI. Transformation Properties of Field Operators

1. Lorentz Transformations

2. Uniqueness of Generators of Inhomogeneous Lorentz Transformation

3. Discrete Transformations

3.1. Charge Conjugation

3.2. Space Reflection

3.3. Time Reversal

3.4. CPT theorem

4. Substitution Law

VII. An Explicit Construction of Wave Functions

1. A General Theory

2. Examples

2.1. The Dirac Field

2.2. The Proca Field

2.3. The Duffin-Kemmer-Petiau Field

2.4. The Rarita-Schwinger Field

VIII. Interacting Fields

1. The Quantization of Interacting Fields

2. Examples

2.1. Quantum Electrodynamics of a Dirac Field

2.2. The Duffin-Kemmer-Petiau Field Interacting with the Electromagnetic Field

2.3. A Pseudoscalar Field Interacting with a Dirac Field through a Pseudovector Coupling

3. Selected Topics

3.1. The Normal Dependent Terms

3.2. The Spectral Representation

3.3. The Reduction Formulae

3.4. The Dyson Equations in Quantum Electrodynamics

3.5. Renormalization Constants

Reading Guide

Appendix A. Solution of the Klein-Gordon Equation and Associated Functions

Appendix B. Dirac Matrices

Appendix C. Λ and d for Various Fields

Appendix D. Formulae for 1/2[ε(x0-x'0), F(∂)] Δ(x-x')

Appendix E. Transition Probability, Cross-Section and Lifetime

Problems

References

Index

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Y. Takahashi