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<br/>A. The Fundamental Equations in Real Space 7<br/>1. The Maxwell-Lorentz Equations 7<br/>2. Some Important Constants of the Motion g<br/>3. Potentials—Gauge Invariance g<br/>B. Electrodynamics in Reciprocal Space U<br/>1. The Fourier Spatial Transformation—Notation 11<br/>2. The Field Equations in Reciprocal Space 12<br/>3. Longitudinal and Transverse Vector Fields I3<br/>4. Longitudinal Electric and Magnetic Fields I5<br/>5. Contribution of the Longitudinal Electric Field to the Total Energy,to<br/>the Total Momentum,and to the Total Angular Momentum—a. The<br/>Total Energy, b. The Total Momentum, c. The Total Angular Momentum 17<br/>6. Equations of Motion for the Transverse Fields 21<br/>C. Normal Variables 23<br/>1. Introduction 23<br/>2. Definition of the Normal Variables 23<br/>3. Evolution of the Normal Variables 24<br/>4. The Expressions for the Physical Observables of the Transverse Field<br/>as a Function of the Normal Variables—a. The Energy i/trans<br/>Transverse Field, b. The Momentum and the Angular Momentum J, trans of the Transverse Field, c. Transverse Electric and Magnetic<br/>Fields in RealSpace, d. The Transverse Vector Potential\ t) ... 26<br/>VVI Contents<br/>5. Similarities and Differences between the Normal Variables and the<br/>Wave Function of a Spin-1 Particle in Reciprocal Space 30<br/>6. Periodic Boundary Conditions.Simplified Notation 31<br/>D. Conclusion: Discussion of Various Possible Quantization Schemes 33<br/>1. Elementary Approach<br/>2. Lagrangian and Hamiltonian Approach 34<br/>Complement A,—The"Transverse" Delta Function<br/>1. Definition in Reciprocal Space—a. Cartesian Coordinates. Transverse and<br/>Longitudinal Components, b. Projection on the Subspace of Transverse<br/>Fields • • •<br/>2. The Expression for the Transverse Delta Function in Real Space—<br/>a. Regularization of 5,^ (p). b. Calculation of g(p). c. Evaluation of the<br/>Derivatives of(p). d. Discussion of the Expression for (p) 38<br/>3. Application to the Evaluation of the Magnetic Field Created by a Magnetization Distribution. Contact Interaction<br/>Complement B,—Angular Momentum of the<br/>Electromagnetic Field. Multipole Waves<br/>Introduction<br/>1. Contribution of the Longitudinal Electric Field to the Total Angular<br/>Momentum o'<br/>2 Angular Momentum of the Transverse Field—a. J, ran Reciprocal Space,<br/>b. Jtr&RS in Terms of Normal Variables, ^ f c. An analogy with the Mean Value of<br/>the total angular Momentum ofa Spin-l Particle<br/>3. Set of Vector Functions of k "Adapted" to the Angular Momentum. General Idea. b. Method for Constructing Vector Eigenfunctions for<br/>J-and J,. c. LongitudinalEigenfunctions. d. Transverse Eigenfunctions.. 51<br/>4. Application: Multiple Waves in Real Space-a. Evaluation of Some<br/>Fourier Transforms, b. Electric Multipole Waves, c. Magnetic Multipole<br/>Waves<br/>complement Exercises<br/>1. //and P as Constants of the Motion 61<br/>2. Transformation from the Coulomb Gauge to the Lorentz Gauge 63<br/>3 Cancellation of the Longitudinal Electric Field by the Instantaneous<br/>Transverse FieldContents * VII<br/>4. Normal Variables and Retarded Potentials 66<br/>5. Field Created by a Charged Particle at Its Own Position. Radiation<br/>Reaction 68<br/>6. Field Produced by an Oscillating Electric Dipole 71<br/>7. Cross-section for Scattering of Radiation by a Classical Elastically Bound<br/>Electron 74<br/><br/>LAGRANGIAN AND HAMILTON APPROACH<br/>TO ELECTRODYNAMICS.THE STANDARD LAGRANGIAN<br/>ANDTHECOULOMB GAUGE<br/>Introduction 79<br/>A. Review of the Lagrangian and Hamiltonian Formalism 81<br/>1. Systems Having a Finite Number of Degrees of Freedom—<br/>a. Dynamical Variables, the Lagrangian,and the Action, b. Lagrange's<br/>Equations, c. Equivalent Lagrangians. d. Conjugate Momenta and<br/>the Hamiltonian. e. Change of Dynamical Variables, f. Use of Complex Generalized Coordinates, g. Coordinates, Momenta, and Hamilton<br/>nian in Quantum Mechanics. 81<br/>2. A System with a Continuous Ensemble of Degrees of Freedom—<br/>a. Dynamical Variables, b. the Lagrangian. c. Lagrange equations<br/>d. Conjugate Momenta and the Hamiltonian. e. Quantization,<br/>f Lagrangian Formalism with Complex Fields, g. Hamiltonian<br/>Formalism and Quantization with Complex Fields 90<br/>B. The Standard Lagrangian of Classical Electrodynamics ICQ<br/>1. The Expression for the Standard Lagrangian—a. The Standard<br/>Lagrangian in Real Space, b. The Standard Lagrangian in Reciprocal<br/>Space 100<br/>2. The Derivation of the Classical Electrodynamic Equations from the<br/>Standard Lagangian—a. Lagrange's Equation for Particles, b. The<br/>Ldgrange Equation Relative to the Scalar Potential, c. The Lagrange<br/>Equation Relative to the Vector Potential IO3<br/>3. General Properties of the Standard Lagrangian—a. Global Symmetries. b. Gauge Invariance. c. Redundancy of the Dynamical Variables 1Q5<br/>C. Electrodynamics in the Coulomb Gauge HI<br/>1. Elimination of the Redundant Dynamical Variables from the Standard<br/>Lagrangian—fl. Elimination of the Scalar Potential, b. The Choice of<br/>the Longitudinal Component of the Vector Potential HI<br/>2. The Lagrangian in the Coulomb Gauge 113Vm Contents<br/>3. Hamiltonian Formalism—a Conjugate Particle Momenta, b. Conjugate Momenta for the Field Variables, c. The Hamiltonian in the<br/>Coulomb Gauge, d. The Physical Variables 115<br/>4. Canonical Quantization in the Coulomb Gauge—a Fundamental<br/>Commutation Relations, b. The Importance of Transversality in<br/>the Case of the Electromagnetic Field, c. Creation and Annihilation<br/>Operators 118<br/>5. Conclusion: Some Important Characteristics of Electrodynamics in<br/>the Coulomb Gauge—a. The Dynamical Variables Are Independent,<br/>b. The Electric Field Is Split into a Coulomb Field and a Transverse<br/>Field, c. The Formalism Is Not Manifestly Covariant. d. The Interaction of the particles with Relativistic ModesIs Not Correctly Described.. 121<br/>ComplementA Functional Derivative.Introduction<br/>AND A Few Applications<br/>1. From a Discrete to a Continuous System. The Limit of Partial<br/>Derivatives 126<br/>2. Functional Derivative 128<br/>3. Functional Derivative of the Action and the Lagrange Equations 128<br/>4. Functional Derivative of the Lagrangian for a Continuous System 130<br/>5. Functional Derivative of the Hamiltonian for a Continuous System 132<br/>Complement Bh—Symmetriesof the Lagrangian in the<br/>Coulomb Gauge and the Constants of the Motion<br/>1. The Variation of the Action between Two Infinitesimally Close Real<br/>Motions 134<br/>2. Constants of the Motion in a Simple Case 136<br/>3. Conservation of Energy for the System Charges + Field 137<br/>4. Conservation of the Total Momentum 138<br/>5. Conservation of the Total Angular Momentum 139<br/>Complement C„—Electrodynamicsin the Presence<br/>of an External Field<br/>1. Separation of the External Field 141<br/>2. The Lagrangian in the Presence of an External Field—a. Introduction of a<br/>Lagrangian. b. The Lagrangian in the Coulomb Gauge 142<br/>3. The Hamiltonian in the Presence of an External Field—a. Conjugate<br/>Momenta, b. The Hamiltonian. c. Quantization 143Contents , IX<br/>Complement Du—Exercises<br/>1. An Example of a Hamiltonian Different from the Energy 146<br/>2. From a Discrete to a Continuous System: Introduction of the Lagrangian<br/>and Hamiltonian Densities 147<br/>3. Lagrange's Equations for the Components of the Electromagnetic Field in<br/>Real Space 150<br/>4. Lagrange's Equations for the Standard Lagrangian in the Coulomb Gauge 151<br/>5. Momentum and Angular Momentum of an Arbitrary Field 152<br/>6. A Lagrangian Using Complex Variables and Linear in Velocity 154<br/>7. Lagrangian and Hamiltonian Descriptions of the Schrodinger Matter Field 157<br/>8. Quantization of the Schrodinger Field 161<br/>9. Schrodinger Equation of a Particle in an Electromagnetic Field: Arbitrariness of Phase and Gauge Invariance 167<br/>111<br/>QUANTUM ELECTRODYNAMICS THE<br/>COULOMB GAUGE<br/>Introduction 169<br/>A. The General Framework 171<br/>1. Fundamental Dynamical Variables. Commutation Relations 171<br/>2. The Operators Associated with the Various Physical Variables of the<br/>System 171<br/>3. State Space 175<br/>B. Time Evolution 176<br/>1. The Schrodinger Picture 176<br/>2. The Heisenberg Picture. The Quantized Maxwell-Lorentz Equations—a. The Heisenberg Equations for Particles, b. The Heisenberg<br/>Equations for Fields, c. The Advantages of the Heisenberg Point of<br/>View 176<br/>C. Observables and States of the Quantized Free Field 183<br/>1. Review of Various Observables of the Free Field—a. Total Energy<br/>and Total Momentum of the Field b. The Fields at a Given Point r of<br/>Space, c. Observables Corresponding to Photoelectric Measurements.. 183<br/>2. Elementary Excitations of the Quantized Free Field. Photons—<br/>a. Eigenstates of the Total Energy and the Total Momentum.<br/>b. The Interpretation in Terms of Photons, c. Single-Photon States.<br/>Propagation<br/>3. Some Properties of the Vacuum—a. Qualitative Discussion, b. Mean<br/>Values and Variances of the Vacuum Field, c. Vacuum Fluctuations .. 189<br/>4. Quasi-classical States— a. Introducing the Quasi-classical States,<br/>b. Characterization of the Quasi-classical States, c. Some Properties of<br/>the Quasi-classical States, d. The Translation Operator 192<br/>D. The Hamiltonian for the Interaction between Particles and Fields. 197<br/>1. Particle Hamiltonian, Radiation Field Hamiltonian, Interaction<br/>197<br/>Hamilton<br/>2. Orders of Magnitude of the Various Interactions Terms for Systems of<br/>Bound Particles<br/>3. Selection Rules<br/>4. Introduction of a Cutoff<br/>Complement Am—The Analysis of Interference Phenomena<br/>in the Quantum Theory of Radiation<br/> |