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Classical Field Theory On Electrodynamics, Non-Abelian Gauge Theories
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Contents 1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Gradient, Curl and Divergence . . . . . . . . . . . . . . 2 1.3 Integral Theorems for the Case of R 3 . . . . . . . . . . . 7 1.4 Maxwell’s Equations in Integral Form . . . . . . . . . . . 11 1.4.1 The Law of Induction . . . . . . . . . . . . . . . 11 1.4.2 Gauss’ Law . . . . . . . . . . . . . . . . . . . 13 1.4.3 The Law of Biot and Savart . . . . . . . . . . . . 15 1.4.4 The Lorentz Force . . . . . . . . . . . . . . . . 17 1.4.5 The Continuity Equation . . . . . . . . . . . . . 18 1.5 Maxwell’s Equations in Local Form . . . . . . . . . . . . 21 1.5.1 Induction Law and Gauss’ Law . . . . . . . . . . . 22 1.5.2 Local Form of the Law of Biot and Savart . . . . . . 23 1.5.3 Local Equations in All Systems of Units . . . . . . . 24 1.5.4 The Question of Physical Units . . . . . . . . . . . 25 1.5.5 Equations of Electromagnetism in SI System . . . . . 28 1.5.6 The Gaussian System of Units . . . . . . . . . . . 29 1.6 Scalar Potentials and Vector Potentials . . . . . . . . . . . 35 1.6.1 A Few Formulae from Vector Analysis . . . . . . . . 35 1.6.2 Construction of a Vector Field from Its Source and Its Curl . . . . . . . . . . . . 40 1.6.3 Scalar Potentials and Vector Potentials . . . . . . . . 42 1.7 Phenomenologyof the Maxwell Equations . . . . . . . . . 46 1.7.1 The Fundamental Equations and Their Interpretation . . 47 1.7.2 Relation Between Displacement Field and Electric Field 50 1.7.3 Relation Between Induction and Magnetic Fields . . . 52 1.8 Static Electric States . . . . . . . . . . . . . . . . . . 55 1.8.1 Poisson and Laplace Equations . . . . . . . . . . . 56 ix x Contents 1.8.2 Surface Charges, Dipoles and Dipole Layers . . . . . 62 1.8.3 Typical Boundary Value Problems . . . . . . . . . . 66 1.8.4 Multipole Expansion of Potentials . . . . . . . . . . 69 1.9 Stationary Currents and Static Magnetic States . . . . . . . . 83 1.9.1 Poisson Equation and Vector Potential . . . . . . . . 84 1.9.2 Magnetic Dipole Density and Magnetic Moment . . . . 84 1.9.3 Fields of Magnetic and Electric Dipoles . . . . . . . 88 1.9.4 Energy and Energy Density . . . . . . . . . . . . 92 1.9.5 Currents and Conductivity . . . . . . . . . . . . . 95 2 Symmetries and Covariance of the Maxwell Equations . . . . . 97 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 97 2.2 The Maxwell Equations in a Fixed Frame of Reference . . . . 97 2.2.1 Rotations and Discrete Spacetime Transformations . . . 98 2.2.2 Maxwell’s Equations and Exterior Forms . . . . . . . 102 2.3 Lorentz Covariance of Maxwell’s Equations . . . . . . . . . 119 2.3.1 Poincaré and Lorentz Groups . . . . . . . . . . . . 120 2.3.2 Relativistic Kinematics and Dynamics . . . . . . . . 123 2.3.3 Lorentz Force and Field Strength . . . . . . . . . . 126 2.3.4 Covariance of Maxwell’s Equations . . . . . . . . . 128 2.3.5 Gauge Invariance and Potentials . . . . . . . . . . 132 2.4 Fields of a Uniformly Moving Point Charge . . . . . . . . . 136 2.5 Lorentz Invariant Exterior Forms and the Maxwell Equations . . 141 2.5.1 Field Strength Tensor and Lorentz Force . . . . . . . 142 2.5.2 Differential Equations for the Two-Forms ! F and ! F . . 145 2.5.3 Potentials and Gauge Transformations . . . . . . . . 148 2.5.4 Behaviour Under the Discrete Transformations . . . . 149 2.5.5 * Covariant Derivative and Structure Equation . . . . . 150 3 Maxwell Theory as a Classical Field Theory . . . . . . . . . . 153 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 153 3.2 Lagrangian Function and Symmetries in Finite Systems . . . . 153 3.2.1 Noether’s Theorem with Strict Invariance . . . . . . . 155 3.2.2 Generalized Theorem of Noether . . . . . . . . . . 156 3.3 Lagrangian Density and Equations of Motion for a Field Theory 157 3.4 Lagrangian Density for Maxwell Fields with Sources . . . . . 163 3.5 Symmetries and Noether Invariants . . . . . . . . . . . . 168 3.5.1 Invariance Under One-Parameter Groups . . . . . . . 169 3.5.2 Gauge Transformations and Lagrangian Density . . . . 171 3.5.3 Invariance Under Translations . . . . . . . . . . . 175 3.5.4 Interpretation of the Conservation Laws . . . . . . . 179 3.6 Wave Equation and Green Functions . . . . . . . . . . . . 183 3.6.1 Solutions in Noncovariant Form . . . . . . . . . . 183 3.6.2 Solutions of the Wave Equation in Covariant Form . . . 188 Contents xi 3.7 Radiation of an Accelerated Charge . . . . . . . . . . . . 193 4 Simple Applications of Maxwell Theory . . . . . . . . . . . . 199 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 199 4.2 Plane Waves in a Vacuum and in Homogeneous Insulating Media . . . . . . . . . . . 199 4.2.1 Dispersion Relation and Harmonic Solutions . . . . . 199 4.2.2 Completely Polarized Electromagnetic Waves . . . . . 205 4.2.3 Description of Polarization . . . . . . . . . . . . . 209 4.3 Simple Radiating Sources . . . . . . . . . . . . . . . . 213 4.3.1 Typical Dimensions of Radiating Sources . . . . . . . 214 4.3.2 Description by Means of Multipole Radiation . . . . . 216 4.3.3 The Hertzian Dipole . . . . . . . . . . . . . . . 220 4.4 Refraction of Harmonic Waves . . . . . . . . . . . . . . 225 4.4.1 Index of Refraction and Angular Relations . . . . . . 225 4.4.2 Dynamics of Refraction and Reflection . . . . . . . . 227 4.5 Geometric Optics, Lenses and Negative Index of Refraction . . 232 4.5.1 Optical Signals in Coordinate and in Momentum Space . 232 4.5.2 Geometric (Ray) Optics and Thin Lenses . . . . . . . 236 4.5.3 Media with Negative Index of Refraction . . . . . . . 240 4.5.4 Metamaterials with Negative Index of Refraction . . . . 247 4.6 The Approximation of Paraxial Beams . . . . . . . . . . . 249 4.6.1 Helmholtz Equation in Paraxial Approximation . . . . 249 4.6.2 The Gaussian Solution . . . . . . . . . . . . . . 250 4.6.3 Analysis of the Gaussian Solution . . . . . . . . . 252 4.6.4 Further Properties of the Gaussian Beam . . . . . . . 256 5 Local Gauge Theories . . . . . . . . . . . . . . . . . . . . 261 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 261 5.2 Klein–Gordon Equation and Massive Photons . . . . . . . . 261 5.3 The Building Blocks of Maxwell Theory . . . . . . . . . . 265 5.4 Non-Abelian Gauge Theories . . . . . . . . . . . . . . . 269 5.4.1 The Structure Group and Its Lie Algebra . . . . . . . 269 5.4.2 Globally Invariant Lagrange Densities . . . . . . . . 276 5.4.3 The Gauge Group . . . . . . . . . . . . . . . . 277 5.4.4 Potential and Covariant Derivative . . . . . . . . . . 278 5.4.5 Field Strength Tensor and Curvature . . . . . . . . . 282 5.4.6 Gauge-InvariantLagrange Densities . . . . . . . . . 284 5.4.7 Physical Interpretation . . . . . . . . . . . . . . 288 5.4.8 *More on the Gauge Group . . . . . . . . . . . . 290 5.5 The U(2) Theory of Electroweak Interactions . . . . . . . . 295 5.5.1 A U(2) Gauge Theory with Massless Gauge Fields . . . 295 5.5.2 Spontaneous Symmetry Breaking . . . . . . . . . . 297 5.5.3 Application to the U(2) Theory . . . . . . . . . . . 303 xii Contents 5.6 Epilogue and Perspectives . . . . . . . . . . . . . . . . 307 6 Classical Field Theory of Gravitation . . . . . . . . . . . . . 309 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 309 6.2 Phenomenologyof Gravitational Interactions . . . . . . . . 310 6.2.1 Parameters and Orders of Magnitude . . . . . . . . . 310 6.2.2 Equivalence Principle and Universality . . . . . . . . 312 6.2.3 Red Shift and Other Effects of Gravitation . . . . . . 316 6.2.4 Some Conjectures and Further Program . . . . . . . 322 6.3 Matter and Nongravitational Fields . . . . . . . . . . . . 322 6.4 Spacetimes as Smooth Manifolds . . . . . . . . . . . . . 325 6.4.1 Manifolds, Curves, and Vector Fields . . . . . . . . 325 6.4.2 One-Forms, Tensors, and Tensor Fields . . . . . . . . 332 6.4.3 Coordinate Expressions and Tensor Calculus . . . . . 335 6.5 Parallel Transport and Connection . . . . . . . . . . . . . 343 6.5.1 Metric, Scalar Product, and Index . . . . . . . . . . 343 6.5.2 Connection and Covariant Derivative . . . . . . . . 345 6.5.3 Torsion and Curvature Tensor Fields . . . . . . . . . 349 6.5.4 The Levi-Civita Connection . . . . . . . . . . . . 351 6.5.5 Properties of the Levi-Civita Connection . . . . . . . 353 6.5.6 Geodesics on Semi-Riemannian Spacetimes . . . . . . 356 6.5.7 More Properties of the Curvature Tensor . . . . . . . 360 6.6 The Einstein Equations . . . . . . . . . . . . . . . . . 363 6.6.1 Energy-MomentumTensor Field in Curved Spacetime . 363 6.6.2 Ricci Tensor, Scalar Curvature, and Einstein Tensor . . 364 6.6.3 The Basic Equations . . . . . . . . . . . . . . . 366 6.7 Gravitational Field of a Spherically Symmetric Mass Distribution 371 6.7.1 The Schwarzschild Metric . . . . . . . . . . . . . 372 6.7.2 Two Observable Effects . . . . . . . . . . . . . . 374 6.7.3 The Schwarzschild Radius is an Event Horizon . . . . 382 6.8 Some Concluding Remarks . . . . . . . . . . . . . . . . 385 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 387 Some Historical Remarks . . . . . . . . . . . . . . . . . . . . 389 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 Selected Solutions of the Exercises . . . . . . . . . . . . . . . . 403 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 About the Author . . . . . . . . . . . . . . . . . . . . . . . . 433 |
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