In this post, we will see the book Quantum Electrodynamics by A. I. Akhiezer; V. B. Berestetskii.

# About the book

At the present time a number of particles is known which correspond to various quantum fields interacting with each other. However, of the many types of physical interactions existing in nature the only one, apart from gravitation, that has been studied in sufficient detail is the electromagnetic interaction. The theory of the latter interaction is the subject of quantum electrodynamics, to which the systematic exposition of this book is devoted.

Since the electromagnetic interaction is the fundamental one in the case of electrons and photons, quantum electrodynamics enables us to explain and to predict a wide range of phenomena related to the behavior of these particles.

In preparing the second edition we have subjected the book to extensive revision. The principal aim and the contents of the book have not been altered; the book is devoted to the systematic presentation of electromagnetic processes only. Only some general theorems and methods go beyond the framework of electrodynamics proper. In the second edition the space devoted to these has been increased

(reflection properties, Green’s functions, functional methods, etc.).

In the presentation of the principles of quantum electrodynamics the theory of renormalizations has been subjected to the most extensive revision. Without claiming complete mathematical rigor we have attempted to present the concept of renormalization from a single simple physical point of view, avoiding purely prescription-like methods of eliminating divergences, and making maximum use of the general properties of quantum mechanical systems.

The book was translated from Russian by G. M. Volkoff was published in 1965.

Credits to original uploader.

You can get the book here.

Follow us on The Internet Archive: https://archive.org/details/@mirtitles

Follow Us On Twitter: https://twitter.com/MirTitles

Write to us: mirtitles@gmail.com

Fork us at GitLab: https://gitlab.com/mirtitles/

Add new entries to the detailed book catalog here.

# Contents

## CHAPTER I

QUANTUM MECHANICS OF THE PHOTON

§ 1. The Photon Wave Function 1

1. Introduction.

2. The Photon Wave Function in k-Space.

3. Energy.

4. Normalization of the Photon Wave Function.

§ 2. Photon States of Definite Momentum 9

1. Photon Momentum Operator.

2. Impossibility of Introducing a Photon Wave Function in the Coordinate

Representation.

3. Plane Waves.

4. Polarization Density Matrix for the Photon.

§ 3. Angular Momentum. Photon Spin 17

1. Angular Momentum Operator.

2. Photon Spin Operator.

3. Photon Spin Wave Functions.

§ 4. Photon States of Definite Angular Momentum and Parity

1. Eigenfunctions of the Photon Angular Momentum Operator.

2. Longitudinal and Transverse Vector Spherical Harmonics.

3. Parity of Photon States.

4. Expansion in Spherical Waves.

5. Expressions for the Electric and Magnetic Fields.

§ 5. Scattering of Photons by a System of Charges 36

1. Incoming and Outgoing Waves.

2. Effective Scattering Cross Section.

3. The Optical Theorem. 4. Dispersion Relations.

§ 6. The Photon Field Potentials 46

1. Transverse, Longitudinal, and Scalar Potentials.

2. Longitudinally Polarized “Photon.”

3. Potentials for Plane and Spherical Waves.

§ 7. System of Photons 52

1. Wave Function for a System of Two Photons.

2. Even and Odd States of Two Photons.

3. Classification of the States of Two Photons of Definite Angular Momentum.

4. Wave Function for a System of an Arbitrary Number of Photons.

§ 8. L-Vectors and Spherical Harmonics 62

1. Irreducible Tensors.

2. The Algebra of L-Vectors.

3. Spherical Harmonics.

## CHAPTER II

RELATIVISTIC QUANTUM MECHANICS OF THE ELECTRON

§ 9. The Dirac Equation 73

1. Spinors. Pauli Matrices.

2. Dirac Equations. Dirac Matrices.

3. Unitary Transformations of Bispinors.

4. The Necessity for Four-Component Electron Wave Functions.

5. Symmetric Form of the Dirac Equation. Equation of Continuity.

6. Invariance of the Dirac Equation.

7. Bilinear Combinations of the Components of the Wave Function.

§ 10. Electron and Positron States. States of Definite

Momentum and Polarization 86

1. Solutions with Positive and Negative Frequencies.

2. The Charge Conjugation Transformation.

3. The Positron Wave Function.

4. Plane Waves.

5. Polarization of a Plane Wave.

6. Polarization Density Matrix for the Electron.

7. Averaging over Polarization States.

§ 11. Electron States of Definite Angular Momentum and Parity 105

1. Orbital and Spin Functions. Spherical Spinors.

2. Wave Function of a State of Definite Angular Momentum.

3. Parity of a State.

4. Expansion in Spherical Waves.

§ 12. Electron in an External Field 122

1. The Dirac Equation with an External Field.

2. Separation of Variables in a Central Field.

3. Asymptotic Behavior of the Radial Functions.

4. Behavior of Energy Levels as Functions of the Potential Well Depth.

5. Electron in a Constant Homogeneous Magnetic Field.

§ 13. Motion of an Electron in the Field of a Nucleus 131

1. Solution of the Radial Equations for the Coulomb Field.

2. Wave Functions for the Continuous Spectrum.

3. Isotopic Level Shift.

4. General Investigation of the Effect of Finite Nuclear Size.

§ 14. Electron Scattering 144

1. Spinor Scattering Amplitude.

2. Expression for the Cross Section in Terms of Phases.

3. Polarization and Azimuthal Asymmetry.

4. Scattering by a Coulomb Field.

5. Small Angle Scattering.

§ 15. Nonrelativistie Approximation 153

1. Transition to the Pauli Equation.

2. Second Approximation.

3. Application of the Dirac Equation to Nucleons.

## CHAPTER III

QUANTIZED ELECTROMAGNETIC AND ELECTRON-POSITRON FIELDS

§ 16. Quantization of the Electromagnetic Field 153

1. Four-Dimensional Form of the Field Equations.

2. Variational Principle. Energy-Momentum Tensor of the Electromagnetic Field.

3. Expansion of the Potentials into Plane Waves.

4. Quantization of the Electromagnetic Field.

5. Use of the Indefinite Metric.

§ 17. Commutators of the Electromagnetic Field 174

1. Commutation Relations for the Potentials and the Field Components.

2. Chronological and Normal Products of Components of the Potential. 3. Singular Functions Associated with the Operators ▢ and (▢^2 — m^2).

§ 18. Quantization of the Electron-Positron Field 195

1. Variational Principle for the Dirac Equation. Energy-Momentum Tensor of the Electron-Positron Field.

2. Quantization Rules for the Electron-Positron Field.

§ 19. Anticommutators of the Electron-Positron Field. Chronological and Normal Products of Field Components. Current Density 205

1. Commutation Relations for Field Components.

2. Chronological and Normal Products of Operators of

the Electron-Positron Field. 3. Electric Current Density.

§ 20. General Properties of Wave Fields 214

1. Wave Functions of a Field and the Lorentz Group.

2. Irreducible Finite-Dimensional Representations of the Lorentz Group.

3. Energy-Momentum Tensor and Angular Momentum Tensor.

4. Current Density Vector.

5. Relativistically Invariant Field Equations.

6. Wave Equations for Particles of Spin Zero and Unity.

§ 21. Quantization of Fields. Connection between Spin and

Statistics 237

1. Nondefiniteness of the Charge in the Case of Integral Spin and of the Energy in the Case of Half-Integral Spin.

2. Quantization of Fields for Integral and Half-Integral Spin. Pauli’s Theorem.

3. Inversion of Coordinates and Time Reversal.

## CHAPTER IV

FUNDAMENTAL EQUATIONS OF QUANTUM ELECTRODYNAMICS

§ 22. Interacting Electromagnetic and Electron-Positron

Fields 253

1. System of Equations for Interacting ) Fields.

2. Lagrangian. Energy-Momentum Tensor.

3. Field Equations in Poisson Bracket Form.

4. Invariance Properties of the Equations of Quantum Electrodynamics.

§ 23. Equations of Quantum Electrodynamics in the Interaction Picture. Invariant Perturbation Theory 268

1. Heisenberg and Schrödinger Pictures. Interaction Picture.

2. Transition to the Interaction Picture in Quantum Electrodynamics. 3. Charge Conjugation Operator.

4. Perturbation Theory.

§ 24. The Scattering Matrix 290

1. The Scattering Problem and the Definition of the Scattering Matrix.

2. Matrix Elements of Field Operators.

3. Representation of the Scattering Matrix as a Sum of Normal Products.

4. General Relation between T- and N-Orderings.

5. Symmetry of the Scattering Matrix under Time Reversal.

§ 25. Graphical Representation of the Elements of the Scattering Matrix. The Scattering Matrix in Momentum Space 307

1. Graphical Representation of Normal Products.

2. Various Interaction Processes between Fields.

3. Transition to Momentum Space.

4. Closed Electron Loops with an Odd Number of Vertices.

5. Rules for Writing Down Matrix Elements.

§ 26. Probabilities of Various Processes 327

1. General Formula for the Probability.

2. Effective Cross Section.

3. Summation and Averaging over Polarization States of Electrons and Photons.

4. Probabilities of Processes Involving Polarized Particles.

5. Probabilities of Processes in the Presence of an External Field.

6. Feynman’s Notation.

## CHAPTER V

INTERACTION OF ELECTRONS WITH PHOTONS

§ 27. Emission and Absorption of a Photon 345

1. General Expression for the Matrix Element.

2. Electric Multipole Radiation.

3. Magnetic Multipole Radiation.

4. Selection Rules.

5. Angular Distribution and Polarization of the Radiation.

§ 28. Scattering of a Photon by a Free Electron 363

1. Scattering Matrix Element.

2. Application of Conservation Laws.

3. Differential Cross Section for Unpolarized Particles.

4. Angular Distribution and Total Cross Section.

5. Distribution of Recoil Electrons.

6. Scattering of Polarized Photons.

7. Scattering of Photons by Polarized Electrons.

§ 29. Bremsstrahlung. 378

1. Perturbation Theory for an Electron Wave Function in the Continuum. Incoming and Outgoing Waves.

2. Effective Cross Section for Bremsstrahlung.

3. Angular Distribution of the Radiation in a Coulomb Field.

4. Polarization of the Radiation.

5. Spectrum of the Radiation.

6. Screening.

7. Radiative Energy Losses.

8. Exact Theory of Bremsstrahlung in the Nonrelativistic Domain.

9. Exact Theory of Bremsstrahlung in the Extreme Relativistic Domain.

10. Radiation Emitted in Electron-Electron and Electron-Positron Collisions.

§ 30. Emission of Photons of Long Wavelength 413

1. “The Infrared Catastrophe.”

2. Investigation of the Divergence in the Low Frequency Domain by Means of the Scattering Matrix.

3. Relation between the Photon “Mass” and the Minimum Frequency.

§ 31. Photoeffect 429

1. Photoeffect in the Nonrelativistic Domain.

2. Photoeffect in the Relativistic Domain.

§ 32. Production of Electron-Positron Pairs 438

1. Production of an Electron-Positron Pair by a Photon in the Field of a Nucleus.

2. Exact Theory of Pair Production by a Photon in the Field of a Nucleus in the Nonrelativistic and Extreme Relativistic Cases.

3. Pair Production by Two Photons.

4. Pair Production in a Photon-Electron Collision.

5. Pair Production in a Collision of Two Fast Charged Particles.

§ 33. Annihilation of Electron-Positron Pairs into Photons 457

1. Annihilation of a Pair into Two Photons.

2. Polarization Effects in the Two-Photon Annihilation of a Pair.

3. Annihilation of a Pair into One Photon.

4. Positronium Decay.

5. Three-Photon Decay of Orthopositronium.

6. Multiple Photon Production Accompanying the Annihilation of a Pair.

§ 34. The Method of Equivalent Photons 473

1. The Number of Equivalent Photons.

2. Bremsstrahlung from a Fast Electron in the Field of a Nucleus.

3. Radiation Emitted in an Electron-Electron Collision.

4. Pair Production by a Photon in the Field of a Nucleus.

5. Pair Production in a Collision of Two Fast Particles.

§ 35. Scattering of a Photon by a Bound Electron. Emission of Two Photons 484

1, The Dispersion Formula.

2. Resonance Scattering.

3. Compton Scattering by Bound Electrons.

4. Emission of Two Photons. The Metastable 2s;, State of the Hy-

drogen Atom.

§ 36. Electron-Electron and Positron-Electron Scattering 499

1. Electron-Electron Scattering.

2. Positron-Electron Scattering.

3. Scattering of Polarized Electrons and Positrons.

4. Annihilation of an Electron-Positron Pair into a 𝜋-Meson Pair.

## CHAPTER VI

RETARDED INTERACTION BETWEEN TWO CHARGES

§ 37. Retarded Potentials 509

1. Interaction Function for Two Charges.

2. General Form of the Matrix Element.

3. Retarded Potentials and Transition Currents.

§ 38. Interaction Energy of Two Electrons to Terms of Order v2/c? 517

1. The Breit Formula.

2. Schrödinger Equation for a Two-Electron System.

3. Interaction between an Electron and a Positron.

4. Exchange Interaction between an Electron and a Positron.

§ 39. Positronium. 527

1. Hamiltonian Operator and the Unperturbed Equation.

2. Perturbation Operator.

3. Fine Structure.

4. Zeeman Effect.

§ 40. Internal Conversion of Gamma-Rays 537

1. Expansion of Retarded Potentials in Spherical Waves.

2. Conversion Coefficient.

3. Conversion in the K-Shell.

4. Effect of Finite Nuclear Size.

5. Effect of Electron Shells on Radiation from the Nucleus.

§ 41. Conversion Accompanied by Pair Production. Excitation of Nuclei by Electrons 554

1. Conversion of Magnetic Multipole Radiation.

2. Conversion of Electric Multipole Radiation.

3. Excitation of Nuclei by Electrons.

4. Monoenergetic Positrons.

§ 42. Coulomb (Monopole) Transitions 565

1. Reduction to the Static Interaction.

2. Conversion and Nuclear Excitation in the Case of an E0-Transition.

## CHAPTER VII

INVESTIGATION OF THE SCATTERING MATRIX

§ 43. Properties of Exact Solutions of the Equations of Quantum Electrodynamics. Propagators 571

1. Stationary States of a System of Interacting Fields.

2. Propagators and Their Spectral Representation.

3. Connection between Propagators and the Scattering

Matrix. Integral Equations for Propagators.

4. Electromagnetic Mass of the Electron.

§ 44. Structure of the Scattering Matrix 593

1. Self-Energy Parts of Diagrams.

2. Vertex Parts of Diagrams.

3. Renormalization of Electron Mass.

§ 45. Renormalization of Electron Charge 605

1. Physical Charge of the Electron.

2. Renormalization of Propagators and Vertex Parts.

3. Three-Photon Vertex Parts.

4. Renormalization of Matrix Elements.

5. Formulation of Perturbation Theory as an Expansion of Powers of e_c.

§ 46. Divergences in the Scattering Matrix and their Removal 619

1. Divergences in Irreducible Diagrams.

2. Introduction of a Cut-Off Momentum.

3. Convergence of Regularized Expressions for Irreducible Vertex Parts and Self-Energy Parts.

4. Convergence of Regularized Quantities in the Case of Reducible Diagrams.

§ 47. Evaluation of Self-Energy and Vertex Parts 631

1. Evaluation of Integrals over Four-Dimensional Regions.

2. Second Order Electron Self-Energy Part.

3. Second Order Photon Self-Energy Part.

4. Third Order Vertex Part in the Case of External Electron Lines.

5. Third Order Vertex Part in the Case of One External Electron Line.

§ 48. Functional Properties of Green’s Functions. Limits of

Applicability of Quantum Electrodynamics 657

1. Expansion Parameters of Perturbation Theory.

2. Zero Order Approximation in the Expansion in Powers of e_c.

3. Integral Equations for the Zero Order Approximation.

4. The Renormalization Group. 5. Derivation of Asymptotic Expressions for the Green’s Functions with the Aid of Differential Equations of the Renormalization Group.

6. The Problem of Closure of Quantum Electrodynamics.

§ 49. Generalized Green’s Functions 676

1. Green’s Functions in the Presence of External Fields.

2. Green’s Function for Two Electrons. Equation for Bound States of the Electron-Positron System.

3. Equations for Green’s Functions in Terms of Variational Derivatives.

4. Expressions for Green’s Functions in Terms of Functional Integrals.

## CHAPTER VIII

RADIATION CORRECTIONS TO ELECTROMAGNETIC PROCESSES

§ 50. Effective Potential Energy of the Electron. Radiation Corrections to the Electron Magnetic Moment and to Coulomb’s Law 693

1. Energy of Interaction of the Electron with the Electromagnetic Field Taking into Account Corrections of Order 𝛼.

2. Radiation Corrections to the Electron Magnetic Moment.

3. Radiation Corrections to Coulomb’s Law.

§ 51. Radiation Corrections to Electron Scattering 705

1. Electron Scattering by the Coulomb Field of a Nucleus in the Second Born Approximation.

2. Differential Cross Section for the Scattering of an Electron by the Coulomb Field of a Nucleus taking into Account Radiation Corrections of Order 𝛼.

3. Elimination of the Photon “Mass” from the Scattering Cross Section.

4. Removal of the Infrared Divergence for an Arbitrary Scattering Process.

5. Scattering of High Energy Electrons by an External Field.

6. Radiation Corrections to Electron-Electron and Electron-Positron Scattering.

§ 52. Radiation Corrections to Photon-Electron Scattering, to Pair Creation and Annihilation, and to Bremsstrahlung. 731

1. Radiation Corrections to the Compton Effect.

2. Limiting Cases of Low and High Energies.

3. Radiation Corrections to Two-Photon Pair Annihilations.

4. Radiation Corrections to Bremsstrahlung.

5. Radiation Corrections to Photon Production and Single Photon Annihilation of Pairs.

§ 53. Radiation Corrections to Atomic Levels 751

1. Radiation Shift of Atomic Levels.

2. Radiation Shift of the Levels of 𝜇-Mesohydrogen.

3. Natural Line Widths.

4. Photon Scattering near Resonance.

§ 54. Photon-Photon Scattering and the Lagrangian for the

Electromagnetic Field 764

1. Photon-Photon Scattering Tensor of the Fourth Rank.

2. Photon-Photon Scattering.

3. Connection between the Photon-Photon Scattering Cross Section and the Radiation Corrections to the Lagrangian of the Electromagnetic Field.

4. Exact Expressions for the Lagrangian of the Electromagnetic Field.

§ 55. Photon Scattering by the Coulomb Field of a Nucleus 792

1. General Expression for the Cross Section for Photon Scattering by a Constant Electromagnetic Field.

2. Relation between the Forward Scattering Amplitude for a Photon and Pair-Production by a Photon in the Field of a Nucleus.

3. Momentum Distribution of Recoil Nuclei Accompanying Pair Production by a Photon in the Field of a Nucleus.

4. Angular Distribution of Recoil Nuclei and Total Cross Section for Pair Production by a Photon in the Coulomb Field of a Nucleus.

5. Small Angle Coherent Scattering of Photons by the Field of a Nucleus.

## CHAPTER IX

ELECTRODYNAMICS OF PARTICLES OF SPIN ZERO

§ 56. Field Equations for Scalar Particles 819

1. First Order Equations.

2. Quantization of the Free Scalar Field.

3. Commutators of the Field. Vacuum Expectation Values of Products of Field Components.

§ 57. The Scattering Matrix in Scalar Electrodynamics 827

1. The Interaction Picture.

2. Rules for Calculating Elements of the Scattering Matrix.

3. Divergences of the Scattering Matrix.

§ 58. Scattering of Scalar Particles 835

1. Scattering of Scalar Particles by the Coulomb Field of a Nucleus. 2. Scattering of a Charged Scalar Particle by a Scalar Particle.

§ 59. Scattering of a Photon by a Scalar Particle. Bremsstrahlung Photons from a Scalar Particle 838

1. Scattering of a Photon by a Scalar Particle.

2. Bremsstrahlung from Scalar Particles.

§ 60. Production and Annihilation of Pairs of Scalar Particles. 842

1. Production of Pairs of Scalar Particles by a Photon in the Coulomb Field of a Nucleus.

2. Production of a Pair of Scalar Particles by Two Photons.

3. Two-Photon Annihilation of a Pair of Scalar Particles.

4. Annihilation of Pairs of Scalar Particles into Electron-Positron Pairs and the Inverse Process.

§ 61. Polarization of the Vacuum in the Case of Charged Scalar Particles 847

1. Vacuum Polarization Tensor for Scalar Particles.

2. Correction to Coulomb’s Law.

3. Photon-Photon Scattering. Radiation Corrections to the Lagrangian of the Electromagnetic Field.

Concluding Remarks 852

References 855

Subject Index 863