This advanced textbook covers many fundamental, traditional and new branches of electrodynamics, as well as the related fields of special relativity, quantum mechanics and quantum electrodynamics.

The book introduces the material at different levels, oriented towards 3rd-4th year bachelor, master, and PhD students. This is so as to describe the whole complexity of physical phenomena, instead of a mosaic of disconnected data. The required mathematical background is collated in Chapter 1, while the necessary physical background is included in the main text of the corresponding chapters and also given in appendices.

The content is based on teaching material tested on students over many years, and their training to apply general theory for solving scientific and engineering problems. To this aim, the book contains approximately 800 examples and problems, many of which are described in detail. Some of these problems are designed for students to work on their own with only the answers and descriptions of results, and may be solved selectively. The examples are key ingredients to the theoretical course; the user should study all of them while reading the corresponding chapters.

Equally suitable as a reference for researchers specialized in science and engineering.

Igor N.Toptygin is Professor at the Theoretical Physics Department in Saint-Petersburg State Polytechnic University, Russia. He received his academic degrees in the field of physics and mathematics in 1964 (PhD) and 1974 (habilitation). He is an expert in theoretical physics and theoretical astrophysics. He is a member of the Scientific Council on Complex Problem of "Cosmic Rays" of the Russian Academy of Sciences, and a corresponding member of the International Academy of Sciences for High Education. He has been engaged for many years in theoretical studies of quantum paramagnetic amplifiers, acceleration of cosmic rays, radiation of relativistic particles in plasmas, etc.

Preface XI

Fundamental Constants and Frequently Used Numbers XV

Basic Notation XVII

1 The Mathematical Methods of Electrodynamics 1

1.1 Vector and Tensor Algebra 1

1.2 Vector and Tensor Calculus 18

1.3 The Special Functions of Mathematical Physics 41

1.4 Answers and Solutions 71

2 Basic Concepts of Electrodynamics: The Maxwell Equations 91

2.1 Electrostatics 91

2.2 Magnetostatics 112

2.3 Maxwell's Equations. Free Electromagnetic Field 131

2.4 Answers and Solutions 154

3 The Special Theory of Relativity and Relativistic Kinematics 193

3.1 The Principle of Relativity and Lorentz Transformations 193

3.2 Kinematics of Relativistic Particles 214

3.3 Answers and Solutions 233

4 Fundamentals of Relativistic Mechanics and Field Theory 271

4.1 Four-Dimensional Vectors and Tensors 271

4.2 The Motion of Charged Particles in Electromagnetic Fields. Transformation of the Electric Field 280

4.3 The Four-Dimensional Formulation of Electrodynamics. Introduction to Field Theory 313

4.4 Answers and Solutions 332

5 Emission and Scattering of Electromagnetic Waves 395

5.1 Green's Functions and Retarded Potentials 395

5.2 Emission in Nonrelativistic Systems of Charges and Currents 404

5.3 Emission by Relativistic Particles 416

5.4 Interaction of Charged Particles with Radiation 436

5.5 Answers and Solutions 449

6 Quantum Theory of Radiation Processes. Photon Emission and Scattering 513

6.1 Quantum Theory of the Free Electromagnetic Field 513

6.2 Quantum Theory of Photon Emission, Absorption, and Scattering by Atomic Systems 539

6.3 Interaction between Relativistic Particles 560

6.4 Answers and Solutions 581

7 Fundamentals of Quantum Theory of the Electron-Positron Field 631

7.1 Covariant Form of the Dirac Equation. Relativistic Bispinor Transformation 631

7.2 Covariant Quadratic Forms 636

7.3 Charge Conjugation and Wave Functions of Antiparticles 639

7.4 Secondary Quantization of the Dirac Field. Creation and Annihilation Operators for Field Quanta 640

7.5 Energy and Current Density Operators for Dirac Particles 643

7.6 Interaction between Electron-Positron and Electromagnetic Fields 645

7.7 Schrödinger Equation for Interacting Fields and the Evolution Operator 647

7.8 Scattering Matrix and Its Calculation 649

7.9 Calculations of Probabilities and Effective Differential Cross-Sections 652

7.10 Scattering of a Relativistic Particle with a Spin in the Coulomb Field 653

7.11 Green's Functions of Electron-Positron and Electromagnetic Fields 657

7.12 Interaction between Electrons and Muons 662

7.13 Higher-Order Corrections 667

7.14 Answers and Solutions 669

Appendix A Conversion of Electric and Magnetic Quantities between the International System of Units and the Gaussian System 675

Appendix B Variation Principle for Continuous Systems 677

B.1 Vibrations of an Elastic Medium as the Vibration Limit of Discrete Point Masses 677

B.2 The Lagrangian Form of Equations of Motion for a Continuous Medium 680

Appendix C General Outline of Quantum Theory 685

C.1 Spectrum of Physical Values and the Wave Function 685

C.2 State Vector 686

C.3 Indistinguishability of Identical Particles 687

C.4 Operators and Their Properties 688

C.5 Some Useful Formulas of Operator Algebra 698

C.6 Wave Functions of the Hydrogen-Like Atom (the Lowest Levels) 699

C.6.1 Addition of Angular Moments 700

C.6.2 Spin Operators and Wave Functions of Fermions (s D 1/2) 700

References 703

Index 709