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Index

Abraham term, 280, 284, 405

Abraham–Lorentz equation, 276, 284

absorption, 262, 361, 404–406

acausality, 221, 232, 247, 393

action, 76, 195

Born–Infeld, 242, 385, 389, 408

Chern–Simons, 382, 407

Fokker, 403, 404, 406

Larmor, 202, 211, 220, 225, 382

Nambu, 235

Poincare–Planck, 85, 202, 210, 378

Polyakov, 292

Yang–Mills, 290, 383

action at a distance, 87, 104, 124, 403

action principle, 197, 236

action–reaction, 51, 52, 104, 113, 120

extended principle, 52, 129, 134, 386

addition of velocities, 7, 10

adjoint representation, 62, 287, 418

advanced solution

to the Maxwell–Lorentz equations,185

to the Yang–Mills equations, 316

aether, 48

affine manifold, 32

Aharonov–Bohm effect, 192

Ampere’s molecular current, 148

law, 138, 139

angular dependence of radiation, 262

angular momentum, 83, 85, 88, 98, 110,207, 263, 335, 342, 345

intrinsic (spin), 98, 120, 207

orbital, 98, 110, 208

ansatz, 142, 147, 189, 284, 340, 343, 350Maxwell–Lorentz theory, 183six-dimensional theory, 186, 377Yang–Mills–Wong theory, 310, 319

anticommutation relations, 200, 424antineutrino, 205antiparticle, 132, 133, 222antiquark, 295, 308, 315, 339, 343antisymmetrization, 26asymptotic condition, 269, 272, 277,

283, 356, 357, 361, 395, 401atomic nucleus, 296automorphism, 14, 19, 38, 49, 222, 315auxiliary variable, 87, 93, 247, 288axial vector, 36, 61, 66, 298, 427

background field, 331, 332, 334, 344,345, 347

Barut regularization, 281–284barycenteric see center of mass, 330,

335baryon, 295, 305, 329, 345, 346, 351basis

matrix, 62, 315, 330, 336, 408, 425orthonormal, 17, 23, 142overcomplete, 320, 421vector, 11

Bianchi identity, 135, 226, 291, 292,383, 387

binary system, 109, 110, 226, 335, 342binding energy, 109, 110Biot–Savart law, 148, 191bivector, 26, 179–183, 186, 187, 190Bondi k-calculus, 6, 49

470 Index

boost, 8, 10, 41, 43, 54, 127, 390, 427Bose–Einstein condensation, 347bound state, 111boundary condition

advanced, 168, 169, 320, 402Dirichlet, 237, 243Neumann, 237, 238, 241, 243periodic, 237retarded, 167, 168, 170–172, 185, 398

brane, 235, 242, 248, 250, 408Dirichlet (D-brane), 237

Brillouin zone, 304

Cabibbo–Ferrari vector potentials, 189,193, 226

canonical equations, 94, 96, 101transformations, 80, 89

Cartan basis, 286, 326, 344, 421, 422differential form, 60, 135, 140, 152subalgebra, 318, 321, 418, 421

Cartan-Weyl basis, 326, 331, 347, 421,422

Cauchy problem, 70, 72, 79, 137, 165for the wave equation, 161, 244

causal cycle, 47causality, 38, 47, 50, 169, 408center of mass, 1, 85, 101, 102, 239, 240,

244, 330frame, 88, 114–117, 119, 123–125

centrifugal term, 106, 107, 384nonrelativistic, 109

characteristic surface, 251, 319charge

chromomagnetic, 339, 350color, 62, 63, 285–287, 293, 308–315,

317–320, 322–331, 333, 342, 347,350, 353, 355

electric, 52, 59, 129, 187, 191, 192elementary, 59magnetic, 52, 61, 110, 119, 133, 150,

187, 188, 190, 191, 226, 292, 340charge density, 130charge quantization, 59charge-coupling, 59, 134charge-source, 128–130, 133, 134chromoelectric flux, 339, 350classical electron radius, 360Clifford algebra, 424, 425clocks, 3, 7

atomic, 4, 6, 44light, 4moving, 44, 51synchronized, 10

closed system, 75, 88, 202, 265collisions, 107, 113, 156, 308, 349

elastic, 114, 117, 118, 245head-on, 107, 113inelastic, 114, 351

color cell, 325, 345neutrality, 307singlet, 289, 314, 329, 344

completeness condition, 23, 29complexification, 316, 344Compton wavelength, 343, 359confinement, 307, 308, 330, 339, 343,

349, 409conformal invariance, 216, 218, 219,

222, 314, 329, 344, 406, 432, 447special, 217, 221, 224, 428, 429, 431,

433conformal transformation, 217, 219,

407, 433conservation

angular momentum, 52, 83, 88, 98,111, 226, 238, 283, 347

charge, 85, 128–130, 132–134, 228,230–233, 241

energy, 82, 114, 118, 246, 363four-momentum, 84, 101, 114, 207momentum, 52, 82, 114

conservation lawcovariant, 291global, 114, 134, 198local, 130, 188, 213, 229, 231, 246,

260, 305, 361conserved quantity, 81, 83, 217, 275constant of motion, 81constraint, 72, 94, 142contraction of indices, 25coordinate variation (local/total), 76coordinates

affine, 15Cartesian, 34, 40, 75, 144, 209, 386curvilinear, 34, 75, 80, 83, 209, 210cylindrical, 139, 337generalized, 75light cone, 159, 239, 257polar, 75, 83, 87, 105, 337

Index 471

rectilinear, 79, 83, 444

spherical, 144, 151, 191, 373, 444

transverse, 239, 241

Coulomb potential, 144, 150

Coulomb singularity, 144, 173, 174, 390

counter-acceleration, 362

coupling constant

electromagnetic, 294

Fermi, 201

gravitational, 248, 408

pseudoscalar, 61

scalar, 59

strong force, 294

vector, 62

weak hypercharge/isospin, 294

Yang–Mills, 285

Yukawa, 201, 203, 218

covariant retarded variables, 171, 174,179, 183, 192, 318

covector, 14, 17, 24, 25, 32, 33, 219, 224

curl, 36, 125, 142, 158, 190

current, 229, 257, 393, 398

axial, 234

Chern–Simons, 293

color charge, 290, 291, 317

electric charge, 130, 135, 187, 232

magnetic charge, 188, 226

Noether, 198, 207, 208, 217, 218, 230

stationary, 138, 146, 148

curvature, 255

principal fiber bundle, 301

world line, 275, 365, 367, 374, 377,382, 404

cutoff, 266, 271, 332, 359

d’Alembert formula, 244

d’Alembertian, 154, 167, 394, 444

damped oscillator, 89, 120

decay, 108, 117, 131

deconfinement, 308

degrees of freedom

color, 286, 305, 347, 349

field, 75, 154, 189, 402, 404, 408

initial, 357, 370, 403

mechanical, 52, 113, 246, 357

rearranged, 249, 362, 401, 406

redundant, 228, 242

delta-function, 129, 131, 150, 167, 168,183, 279, 338, 403, 437, 439, 443,444

derivative

covariant, 230, 232, 255, 288, 289

directional, 411

exterior, 153, 412, 415

normal, 161, 165

partial, 87, 159, 196, 210, 230, 250

total, 88, 199, 290, 292

variational, 210

DeWitt vector potential, 157, 192

diffeomorphism invariance, 386

dilatation, 217–219, 223, 427–429,431–433, 439, 444, 447

dimension

of a vector space , 12

dipole moment, 146, 149, 182

Dirac equation, 119, 140, 200, 298, 424

matrices, 200, 240, 302, 423–425

regularization, 278, 382

spinor, 200, 204, 231, 425, 426

dispersion law, 250, 251

displacement current, 138, 139

distributions, 14, 129, 140, 366, 437,438, 440–442, 444, 447

divergence, 259, 282, 310, 402

cubic, 378

infrared, 375

linear, 276, 378, 385

ultraviolet, 359, 365, 375, 377, 378,402

divergence (differential operator), 36,49, 125, 148, 189

divergence term, 200, 208, 225, 226, 310

Doppler shift, 7

Dothan–Gell-Mann–Ne’eman SL(3, R)

symmetry, 345, 350

double covering, 314, 419, 423

dual vector space, 14

duality

invariance, 225, 226, 228, 230, 389,409

transformation, 225, 226, 228, 388,407

dynamical law, 51, 54, 61, 92, 123, 366

rest frame, 102

dyon, 61, 121, 191, 226, 351

472 Index

effective theory, 359eigenvalue, 42, 155, 192, 345einbein, 93, 95, 97, 102, 368electrodynamics, 131, 319, 349, 372

classical, 130, 283Maxwell–Lorentz, 139, 291, 390nonlinear, 127, 385–388, 408, 433nonlocal, 393–396, 398, 408, 444quantum, 306, 359, 365

electromagnetic field, 59, 119, 123, 342constant, 69–72, 74, 141homogeneous, 70, 72, 74null, 65, 67, 69, 74, 97, 260static, 141uniform, 69, 71

electromagnetic field invariants, 65electromotive force, 138electron, 109, 140, 351, 359

dressed, 246, 360free, 110, 364planetary, 384radiating, 284, 366

electron modelextended, 133, 284, 366point, 131, 140, 284, 358

electron volt, 294embedding, 53, 54, 56, 58, 92, 104, 346energy, 55, 86

indefinite, 274, 275, 363, 364kinetic, 55, 79, 107, 110, 250, 361positive definite, 255, 274, 363potential, 79, 86, 106, 250, 384rest, 109, 335

energy balance, 245, 246, 275, 365energy density, 212, 245energy-momentum, 243energy-momentum balance, 249, 273

global, 288local, 275, 356, 362, 368, 381, 404

equation of continuity, 130, 147, 208equation of motion, 54

linear, 127, 141, 143, 216linearized, 245, 331overdetermined, 142, 147, 152, 158,

184, 189equilibrium

neutral, 330, 331, 342unstable, 1, 107, 131, 370, 385

equivalence class, 43, 154

Euclidean metric, 25, 32, 150, 306quantum field theory, 306space, 16, 49, 125, 416, 417, 422, 430,

435spacetime lattice, 285, 301topology, 135

Euler–Lagrange equations, 79, 198Eulerian, 77, 196, 207event, 1, 6–8, 10, 48, 84, 216extremal, 79

falling to the center, 107, 384Faraday’s law, 138field, 28, 59, 119

acceleration, 261adjunct, 404, 406background, 331, 344, 345, 347boson, 95, 206, 305, 359, 435centrally symmetric, 83, 105charged, 230–232, 246, 247, 249, 258,

297, 298, 313, 338complex-valued, 28, 200, 228, 231,

234, 240, 253, 257, 258, 316, 344,386

Coulomb, 105, 144, 180, 182, 192Dirac, 200, 201, 203, 204, 216, 218,

233, 292, 426electromagnetic, 59, 87, 119, 123fermion, 201, 435free, 141, 158, 199, 218gluon, 291, 306, 344, 347gravitational, 5, 65interacting, 199, 247Klein–Gordon, 150, 165, 185, 201massive, 150, 199, 245, 256–258, 283,

359massless, 239, 257, 265, 294, 304, 340matter, 250, 300Proca, 204, 205, 245, 383Rarita–Schwinger, 204real-valued, 28, 204scalar, 28, 195, 204, 207self-interacting, 200spherically symmetric, 340spinor, 195, 200, 207tachyon, 249, 251, 253–255, 257tensor, 29, 195vector, 28, 195, 207Yang–Mills, 63

Index 473

field configurationof electric type, 67, 70of magnetic type, 187self-dual, 292, 341, 343stable, 146, 257, 308, 334, 398static, 141, 253

field equation, 198elliptic, 158hyperbolic, 124, 158, 232, 408

field four-momentumbound, 261, 274, 283, 375, 380emitted, 262, 459

field strength, 59field variation (local/total), 196flavor, 295, 305, 307flux

angular momentum, 263electric intensity, 137, 247energy, 212, 245Lienard–Wiechert term, 313, 317magnetic induction, 139, 337, 387radiation, 260, 272, 273, 378vortex lines, 336, 337

force, 51, 54attractive, 105–108, 114, 205, 308,

340central, 105, 108, 111, 114centrifugal, 149, 330external, 75, 113, 271, 273, 274, 361Newtonian, 52, 56repulsive, 105, 108, 131, 205, 349

2-form, 27canonical, 27, 31, 65, 67–69decomposable, 28, 67, 179, 182, 192

form factor, 393, 394, 397, 399, 400, 408forward hyperboloid, 72four-acceleration, 44

current, 130, 134, 138, 156, 170divergence, 136, 234momentum, 54vector, 21velocity, 44

Fourier mode, 142, 161, 250, 251, 358series, 141, 239, 244transform, 142, 143, 153, 163, 166,

167, 172, 173, 189frame of reference, 1

fixed (stationary), 56, 66inertial, 1–5

instantaneously comoving, 44–46, 53,54, 56, 172, 263, 270, 381

Lorentz, 1noninertial, 2, 182, 192, 209, 262rest, 9, 53, 54, 56, 57, 98, 99, 102,

175, 240retarded, 176

functionalbilinear, 15, 16linear, 14, 32, 411, 437multilinear, 15

fundamental representation, 292, 417

Galilean particle, 51, 55–57, 64, 85, 94,100, 101, 119

transformation, 10gauge field, 119

Abelian, 63, 296, 313, 318, 325, 329,340, 354, 355, 357

non-Abelian, 291, 305, 313, 316, 323,350, 354, 355, 357

gauge fixing condition, 155, 156, 185,237, 242, 320

Coulomb, 149, 156Fock–Schwinger, 157Lorenz, 155, 160, 185, 372, 404noncovariant, 239, 240orthonormal, 237, 238temporal, 156unitary, 258

gauge group, 237, 298, 305, 309color, 293weak hypercharge/isospin, 294

gauge invariance, 228, 231, 258, 301,305, 340, 399

mode, 154, 160, 228, 331transformation, 147, 154, 190

Gauss’ law, 138, 184, 186Gaussian units, 144, 201Gauss–Ostrogradskiı theorem, 49Gell-Mann matrices, 320, 419general linear group GL(n, R), 14general relativity, 48, 118, 247generator, 41, 94, 319, 331, 417, 421geodesic, 65, 85, 89geometry, 1, 2, 4, 5, 11, 49

affine, 15Euclidean, 16pseudoeuclidean, 1, 16

474 Index

Glashow–Salam–Weinberg model, 254,283, 294

Goldstone mode, 254, 257, 283model, 249, 253

gradient, 49, 147, 191Grassmann algebra, 104, 435Grassmannian variable, 103, 120, 241

even, 103, 131, 434, 435odd, 177, 434, 435

gravitational field, 65gravity, 248, 294, 363Green’s function, 141, 161, 192

advanced, 169, 170, 172, 182retarded, 167–174, 257, 372, 373, 407

ground state, 252, 254, 330, 344, 349,351

group velocity, 165, 199

hadron, 329, 330, 334, 345, 349matter, 308

Hamiltonian, 78, 94, 106, 191harmonic function, 143

mode, 161oscillator, 89, 100, 250, 345

Heaviside step function, 168, 323, 446units, 335, 385

Heisenberg uncertainty principle, 296Helmholtz equation, 150

theorem, 125, 139, 150Hessian matrix, 88, 89Higgs field, 297, 298, 337–339, 344, 350

mechanism, 257, 283, 294, 296, 297,305, 343

model, 254, 258, 307, 316, 335, 349Hodge duality operation, 27homogeneity

space, 5, 82spacetime, 215time, 5, 82

Huygens’ principle, 374hydrogen atom, 105, 109, 284, 384hyperbolic motion, 46, 71, 370, 371hyperplane, 32hypersurface, 34

locally adjusted, 271, 318, 326

identity of mass and energy, 118impact parameter, 112index

color, 62, 63, 285, 286, 289, 307

contravariant, 17, 25, 26, 42

covariant, 17, 26, 44

repeated, 12, 125, 144, 285, 412

spatial, 25, 41, 60

temporal, 126, 156

index summation rule, 144

infinite continuous group, 91

infinitesimal transformation, 81, 84, 88,91, 198, 207, 216, 225, 231, 237,257, 288, 290, 416, 417, 426, 431

instanton, 291, 307

integral of motion, 81

interaction

contact, 51, 113, 408

Coulomb, 105, 108, 328

direct-particle, 401, 408

electromagnetic, 86, 104, 105, 202,213

electroweak, 293, 296

four-fermion, 201, 202

fundamental, 2, 247, 284, 288, 304

Gursey, 200, 202

gauge invariant, 289, 292, 302, 305

instantaneous, 2, 10, 87

local, 131, 134

localizable, 393, 407

nonlocal, 393–400, 407, 439, 444, 447

of a particle with a scalar field, 64,96–97, 204

quartic, 199, 202

Schwarzschild, 202, 211

short-range, 113, 288, 295–296

string, 241

strong, 293–306

van der Waals, 146, 295

weak, 205, 293, 295, 304

Yukawa, 201, 297

interval

closed, 438

finite, 221, 439

null, 104, 164, 216, 280, 283, 401

small, 277, 398

spacelike, 38, 104, 157

timelike, 47–48, 80–81, 85, 278, 283

inverse square law, 182, 339, 408

inversion, 428

irreversibility, 89, 274, 401, 405

Index 475

isomorphism, 12, 14–15, 43, 226, 229,246, 323, 421, 431, 434

Jacobi identity, 62–63, 291, 420Jacobian, 37, 176, 219, 243

k-factor, 5–7Kepler problem, 75, 105–108, 120, 383Killing form, 63, 285, 420

degenerate (singular), 291, 323, 420positive definite, 291, 420

kinematical rest frame, 102kinematics

Newtonian, 6relativistic, 50, 117

Klein–Gordon equation, 150, 199, 250Kronecker delta, 13, 24, 30

Lagrange multiplier, 96, 103Lagrangian, 75, 195

t-dependent, 75acceptable, 91nonsingular, 88standard model, 294–296acceleration-dependent, 366–367coordinate-independent, 82Dirac, 200, 301duality invariant, 225, 388, 407electroweak interaction, 293–296, 304for a colored particle, 286Higgs, 254, 257higher derivative, 366Klein–Gordon, 199, 303rearranged, 252, 256, 282, 353, 401strong interaction, 295, 306time-independent, 82

Laplace equation, 143operator, 164, 175

Larmor formula, 269, 282lattice, 124, 300, 302, 398

spacing, 300Leibnitz’s rule, 162, 414length dimension, 203–204, 217, 235lepton, 294–295, 297Levi-Civita tensor, 25, 30, 60, 61, 126Lienard–Wiechert field, VII, 192

solution, 182vector potential, 172, 179, 185, 192

Lie algebra, 62–63, 119, 415

Abelian, 63, 417–418, 420, 428–429compact, 43, 416, 419, 421complex, 63, 343semisimple, 63, 216, 420, 428simple, 343, 420

Lie group, 14, 416classical/exceptional, 421

light cone, 22backward sheet, 22forward sheet, 22, 169, 374future, 22, 168, 176, 185past, 22, 169, 170, 172, 175

light propagation, 4, 5line element, 38, 64, 95, 209, 210, 217linear

combination, 11connection, 299

locality, VII, 134, 385Lorentz boost

contraction, 9factor, 8, 44, 47force, 61, 69, 119, 131group, 14, 38, 216invariance, 38, 164transformation, 8, 38

proper, 40proper orthochronous, 40, 207

Lorentz–Dirac equation, 274, 277, 284,357

magnetic dipole, 149, 182magnetic monopole, 61, 134, 140, 150

’t Hooft–Polyakov monopole, 292,339, 341, 350

Bogomol’ny–Prasad–Sommerfield,340, 341, 349

Dirac, 140, 150, 187Mandelstam variables, 118manifold, 33, 137, 411mass, 56, 57, 84, 99, 102

bareNewtonian, 52, 56renormalized, 269, 275, 282, 283,

354, 356, 375, 404rest, 56, 57, 71, 99, 102

mass renormalization, 269, 281, 359,402

mass shell, 114maximal electric field, 386

476 Index

maximal parity violation, 296, 298maximal velocity of motion, 2, 3, 10, 48Maxwell stress tensor, 213Maxwell’s equations, 123, 127, 134, 139,

152, 188, 202, 216, 246, 372Maxwell–Lorentz theory, 139, 221, 232,

245, 282, 285, 305, 313, 355, 359,390, 395, 399, 401–403, 406

Meissner effect, 339, 342, 350meron, 308meson, 105, 295, 296, 308, 329, 335, 346metric, 17

Euclidean, 16indefinite, 9induced, 236intrinsic, 238Minkowski, 23, 25pseudo-Riemannian, 64, 65, 85pseudoeuclidean, 64

Millennium Problem, 349minimal coupling, 231, 234, 247, 292Minkowski force, 54, 58

space, 5, 9, 10, 20, 21, 37, 52momentum

conjugate to a coordinate, 78, 86conjugate to a field variable, 197linear, 52, 55, 82transfer, 115, 118

momentum density, 212motion

accelerated, 2, 46finite, 107, 109infinite, 107, 113self-accelerated, 357uniform, 1, 4, 51, 71, 100, 370

multiplication rule, 27, 286, 421multipole moment, 148muon, 201, 295, 351

n-form, 26closed, 414exact, 414, 415

N -quark case, 319, 355natural units, 83, 203, 218, 291, 296Ne’eman–Sijacki SL(4, R) classification

of hadrons, 345, 351neutrino, 95, 206Newton’s first law, 49, 51, 119

second law, 51, 54, 57, 79, 118, 124,274, 381

Newtonian gravitation, 2, 87, 409Nielsen–Olesen vortex, 336, 339, 349,

350Noether first theorem, 76, 82, 196, 198

converse, 84, 88identity, 215, 229, 233, 258, 292, 354second theorem, 92, 118

non-Galilean particle, 51, 56, 58, 119,357, 370

regime of motion, 101, 290, 370nonperturbative solution, 314norm, 16

indefinite, 16nucleon, 204, 205, 309, 346, 351

orthogonal group O(n), 19, 416, 431transformation (rotation), 19, 68

oscillatoranharmonic, 251damped, 75, 89, 277harmonic, 89, 90, 143, 251

pair creation and annihilation, 359Paley–Wiener theorem, 399parallel transport, 299, 301parameter of evolution, 44, 76, 95parametrization, 53, 76, 90, 95parametrized curve, 43, 90particle, 43

bare, 258, 260, 274, 278, 362, 378,380, 395, 404

charged, 59, 69, 95colored, 182, 288, 293, 305, 309dressed, 269, 273, 274, 284, 354, 360,

361, 364, 375, 378, 380, 401, 405free, VI, 55, 58, 89, 95, 98massless, 94, 97, 223, 224, 250, 455neutral, 59point, 52, 54rigid, 79, 367, 370, 378superluminal, 56, 275, 465

path ordering, 294, 299Pauli

blocking principle, 325matrices, 241, 314, 315, 418, 421

Pauli–Lubanski vector, 42Pfaffian, 31

Index 477

phase, 160factor, 298, 301, 306transition, 308, 349, 360

Planck’s constant, 83, 248, 325, 393length, 393, 408mass, 248

Poincare cohesive forces, 131, 202group, 38, 216invariance, 52, 90, 206lemma, 414transformation, 37

point, 10, 15Poisson bracket, 94, 101, 102

equation, 94, 101, 125, 142, 149polarization vector, 160Pontryagin density, 293potential, 106, 142

linearly rising, 109, 308, 313, 323,329, 355

scalar, 63, 142spherically symmetric, 86vector, 86, 141, 147, 152

Poynting vector, 213, 246prepotential, 186principle of least action, 76, 79, 81

of gauge invariance, 231, 457of relativity, 2, 48

productcross, 30exterior, 26, 181inner, 417scalar, 16semidirect, 216tensor, 24

projection operator ⊥, 32, 34, 70, 72,92, 150, 216, 274, 296

propagation of interactions, 2, 5vector, 160

proper length, 9time, 44, 95

pseudoorthogonal group O(m, n), 38puzzle of nucleon spin, 351

quantumchromodynamics, 62, 294, 342, 460electrodynamics, 306, 359, 365field theory, 83, 131, 295, 301, 306,

308, 336, 344, 385, 408, 447mechanics, 140, 216, 348

quark, V, 105, 291, 304bound, 324, 325, 330confined, 343, 394, 399dressed, 354, 356, 360free, 324, 328, 330, 409isolated, 307single, 309, 315, 323, 345, 349, 355

quark–antiquark pair, 295, 308quark-gluon plasma, 308quark/lepton generations, 295, 296quarkonium, 105

radar-location approach, 3radial part of ∇2, 144, 164, 174radiation, 108, 119, 121, 192, 260, 283,

354, 356, 375, 378, 401, 405radiation damping, 284, 361radiation rate, 283, 378, 379radiation reaction, 280, 284, 361, 363radius vector, 16, 22, 144, 175rank

Lie group, 321, 419tensor, 24

rapidity, 10rearranging degrees of freedom, 249,

253, 257, 283, 357Regge sequences as infinite multiplets,

345renormalizability, 297, 305, 359, 366reparametrization invariance, 90, 91

scale dimension, 217, 220invariance, 217, 218, 223, 246

Schott term, 361self-acceleration (runaway), 357

of a dressed particle, 362–364of a rigid particle, 371of a spinning particle, 101

self-energy, 131, 268, 269, 273, 282, 310,384, 390, 396, 398

self-field, 278, 402self-force, 390, 397, 400signal, 38, 167, 171, 183, 319, 320, 374,

388, 408destructive, 47light, 5superluminal, 50

signature, 21, 210, 431signum function, 150, 164, 279, 440

478 Index

similitude transformation, 217, 428simultaneous events, 6singularity, 173, 174

integrable, 260, 284, 390nonintegrable, 260, 284, 402

soliton, 159, 166, 192, 407space, 1

affine, 15, 19, 32, 33color, 62configuration, 75Euclidean, 16, 21event, 76, 80internal, 62isotopic-spin, 309pseudo-Riemannian, 137, 246pseudoeuclidean, 19

space dimension, 123, 135isotropy, 5, 83reflection, 40

spacetime, 1, 3spacetime dimension, 65, 216

reflection, 40special linear group SL(n, R), 181

relativity, 1, 4, 5, 44, 48, 49, 53, 118species doubling, 303, 304speed of light, 2, 51, 56, 83, 248, 393spin originating from isospin, 343, 347spinning particle, 43, 98, 101, 120, 305spinor, 195, 200, 204, 231, 425spontaneous symmetry breaking, 253,

305, 339, 342, 350spontaneous symmetry deformation,

316, 344, 360stability, 142, 149standard synchrony, 6, 48

time scale, 4, 5, 49star-shaped region, 415Stokes theorem, 36, 37, 50, 299stress-energy tensor

canonical, 197, 207, 209, 211, 245improved, 218metric, 210, 211symmetric, 209, 212, 246traceless, 218, 291

stringclosed, 235, 237, 239, 241, 244, 247Dirac, 151, 193, 238free, 241, 250Green–Schwarz, 240

interacting, 241Nambu, 237open, 235, 241, 247Polyakov, 238relativistic, 235, 247tension, 235, 247

strongly/weakly conserved current, 208structure constants, 62, 63, 344, 420,

421superconductivity, 338, 339, 348, 350superstring, 240supersymmetry, 240, 435, 447support, 257, 393, 438

delta-function, 168, 173, 338, 443, 444distribution, 441retarded Green’s function, 170, 173,

373, 374surface element, 34, 36, 176symmetry, 81, 84, 198, 289, 294, 297,

308, 316, 344, 367, 402internal, 198, 225, 229, 305local, 182, 229spacetime, 198, 216, 246, 345

symplectic group Sp(n), 421

tachyon, 56, 249, 275, 360, 365, 406tensor, 24test function space, 437tetrad (vierbein), 23, 67time 1

discrete, 61laboratory, 44, 48proper, 44

time dilation, 44reversal, 40, 61, 89

trajectory, 44, 72, 111, 121planar, 105Regge, 335, 336

translation, 19, 38, 216invariance, 52, 82, 84, 206

transverse mode, 155triangular function, 118two-particle problem, 104, 110, 113,

120, 376, 381, 384two-quark case, 323, 326, 327, 334, 349

uniform acceleration, 46, 58, 362, 363uniformity of light propagation, 4, 5unitary group U(n), 286, 320, 417

Index 479

representation, 292, 345, 419unstable system, 2

vacuum, 3, 48, 308, 342, 344, 351, 387vector

imaginary unit, 20lightlike (null), 22, 45normalized, 20, 171spacelike, 22, 171, 181tangent, 33, 64, 171, 209, 235timelike, 22, 181unit, 17, 64, 144, 171, 181zero, 11

vector bosons (W , Z), 305, 359vector space (complex/real), 11velocity, 2–5

angular, 105, 112, 120, 309, 310constant, 1, 3, 5, 11, 240, 392

velocity field, 261Veneziano model, 336

Ward–Takahashi identity, 192, 233wave

convergent, 169divergent, 168electromagnetic, 52, 192monochromatic, 6, 161plane, 74, 101, 120, 159–161, 165, 307shock, 408

wave equation, 141homogeneous, 141, 156, 158, 160, 167,

173, 270, 279, 319, 403inhomogeneous, 141, 155, 167

wave length, 161zone, 262

Weinberg angle, 294Weyl invariance, 240, 244

rescalings, 220, 238, 429Wilson loop, 300, 302Wong equation, 287, 290, 309, 310, 320

force, 63, 119, 284, 329, 356particle, 290, 305, 308, 342, 347

work, 56, 245, 275, 357, 361, 365world line, 5, 43

Λ- and V -shaped, 81, 133allowable, 47, 133, 284, 401helical, 119, 147, 284lightlike, 45, 56, 80, 95, 97smooth, 47, 80, 137, 175, 185spacelike, 45, 47, 57, 80straight, 5, 55, 147, 180, 184, 187timelike, 45, 47, 57, 80, 132, 134, 170,

175, 179, 185, 187, 190, 203world sheet, 190, 235

volume, 242Wu–Yang ansatz, 350

vector potential, 190, 193

Yang–Mills equations, 141, 183, 289,290, 307, 313, 316, 323, 325, 344,353, 355, 383, 422

Yang–Mills theory, 305, 306, 319, 383pure, 290, 308quantum, 119, 349

Yang–Mills–Wong theory, 183, 283, 285,291, 295, 307–310, 313, 316, 324,344, 347, 349, 350, 355, 357, 359,362, 364

Yukawa model, 204potential, 150, 173, 296

zitterbewegung, 58, 101, 370Zwanziger Lagrangian, 227, 247