References - CERNcds.cern.ch/.../978-3-319-01195-0_BookBackMatter.pdf · References [Abra 85]...
Transcript of References - CERNcds.cern.ch/.../978-3-319-01195-0_BookBackMatter.pdf · References [Abra 85]...
References
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Index
AAbel, 115, 246, 251, 301, 475, 523, 703
biography, 712Abelian group, 704Abelian Lie algebra, 937Abel’s identity, 454Absolute convergence, 319Addition theorem for spherical harmonics,
412, 413Additive identity, 20Adjoint, 147, 171, 177, 434, 454, 495,
551, 564, 614, 618, 624, 675,677, 754, 1092, 1102
classical, 56, 57, 155matrix of, 152–155
differential operators, 433–436formal, 612, 648operator, 56, 57
Adjoint action, 929Adjoint algebra, 944Adjoint boundary conditions, 615, 619Adjoint Green’s function, 618Adjoint map, 926Adjoint of a matrix, 144Adjoint of an operator, 113Adjoint representation, 732
character, 737Affine group, 948Affine motion, 917Affine parameter, 1138Airy’s DE, 456, 490Algebra, 63–72, 101, 119, 191, 398, 515,
740, 784, 797, 829, 891, 921,942
antiderivation, 82, 829, 831, 889, 891associative, 63automorphism, 70center, 64central, 65central simple, 79, 93, 845Clifford, 800, 829–855
anticenter, 839–842canonical element, 838, 839center, 839–842construction, 830–834Dirac equation, 832–834general classification, 843–846
homomorphism with otheralgebras, 837, 838
isomorphisms, 842, 843properties, 834–843
commutative, 63complex numbers, 295decomposition, 83–95definition, 63derivation, 81, 82, 99, 106, 868, 887,
891, 1124derivation of, 80–83derivation of an, 80derived, 66dimension of an, 63epimorphism, 70exterior, 794–801factor, 77, 78generator, 70homomorphism, 70, 71ideal, 73involution, 72, 82, 836, 837, 839, 843,
848, 989, 990isomorphism, 70Lie, 915–936monomorphism, 70operator, 101–107opposite, 66polynomial, 95–97quaternions, 69representation, 125–131semi-simple, 88–91, 92, 92, 94, 130,
764, 799, 844simple, 76
classification, 92–95structure constants, 68symmetric, 791tensor, 784tensor product, 68total matrix, 78–80unital, 63
Algebra direct sum, 67Algebra of endomorphisms, 67Algebra of linear operators, 67Algebra of polynomials, 67Algebra tensor product, 68Algebraic equation, 1010
symmetry group of, 1010
S. Hassani, Mathematical Physics, DOI 10.1007/978-3-319-01195-0,© Springer International Publishing Switzerland 2013
1181
1182 Index
Algebraic multiplicity, 174Analytic continuation, 372–378
SOLDE, 459Analytic function, 297–304
definition, 301derivative, 297derivatives as integrals, 316entire, 301poles of, 342roots (zeros) of, 330
Angular momentum, 25, 398, 933, 976addition theorem, 973commutation relation, 399eigenvalues, 404eigenvector, 406–413intrinsic spin, 1073operator, 398
eigenvalues, 401–405in spherical coordinates, 401
orbital, 1073quantum mechanics, 405
Annihilator, 51, 61, 86left, 73right, 73
Anticommutator, 111Antiderivation, 82, 829, 831, 889, 891Antisymmetric bilinear form, 707Antisymmetric representation, 732Antisymmetrizer, 793Antisymmetry, 793Arc length, 1146Associated bundle, 1084–1086
vector bundle, 1117–1120vector field
horizontal, 1091Associated Legendre functions, 408Associative algebra, 63Asymptotic expansion, 385Atoms, 480–482Automorphism, 43, 74, 836, 841, 945,
1085, 1098, 1102algebra, 70group, 705PFB, 1081
Azimuthal symmetry, 411
BBaker-Campbell-Hausdorff formula, 110Banach space, 218, 1048Basis, 23
dual of, 51dual of a, 782oriented, 800orthonormal, 32standard, 23transformation matrix, 149
Becquerel, 896Bernoulli, 482, 1056Bessel, 246, 671, 1056
biography, 481
Bessel DE, 432, 482Bessel equation, 466
Liouville substitution, 570Bessel function, 482–485, 586
asymptotic behavior, 502–505large argument, 504large order, 503
confluent hypergeometric, 483first kind, 483generating function, 499integral representation of, 498–505modified
first kind, 391, 484second kind, 392, 484
oscillation of, 432recurrence relation, 485second kind, 483spherical, 487, 593third kind, 484
Bessel imaginary (bei), 590Bessel inequality, 219Bessel real (ber), 590Beta function, 378–381
definition, 380Bianchi’s identities, 1123Bianchi’s identity, 1094
abelian case, 1095Bijective map, 6Bilinear inner product
complex, 46Binary operation, 7Binomial theorem, 13Birkhoff’s theorem, 1169Block diagonal, 171, 200Bohr radius, 481Bohr-Sommerfeld quantization, 452Bolzano, 11Bolzano-Weierstrass property, 522Bolzano-Weierstrass theorem, 522Boole, 755Boundary conditions
adjoint, 615Dirichlet, 617, 642general unmixed, 617homogeneous, 611mixed, 616
elliptic PDE, 673Neumann, 617, 643periodic, 571, 617separated, 566unmixed, 616
Boundary functionals, 612Boundary point, 520Boundary value problem, 612, 635
Dirichlet, 642, 665–671Neumann, 643, 671–673
Bounded operator, 513–517continuity, 514
Bra, 20Branch cut, 366
Index 1183
Brouwer, 957Bundle
associated, 1084–1086vector bundle, 1117–1120
Clifford, 1101cotangent, 882spinor, 1101tangent, 877tensor, 883
Bundle of linear frames, 1084, 1120canonical form, 1121
Bundle space, 1080BWHB theorem, 960
CCalculus of variations, 1047–1061
symmetry groups, 1062–1065Canonical 1-form, 928Canonical basis, 802, 902
SOLDE, 463Canonical coordinates, 902Canonical flat connection, 1095Canonical form, 1121Canonical transformation, 902, 907Cantor, 523, 792, 897
biography, 10Cantor set, 12Cardinality, 10–12, 23Cartan, 896, 946, 1015
biography, 799Cartan metric tensor, 945, 970Cartan subalgebra, 948Cartan’s lemma, 798Cartesian product, 2, 7Cartesian vectors, 19Casimir operator, 969, 970, 971, 971,
975–977, 979, 980Cauchy, 154, 340, 366, 475, 533, 581,
640, 702, 1130biography, 301
Cauchy data, 636Cauchy integral formula, 313
for operators, 536Cauchy problem, 636
ill-posed, 642Cauchy sequence, 9, 216, 526Cauchy-Goursat theorem, 310Cauchy-Riemann conditions, 298, 303
differentiability, 300Cauchy’s inequality, 336Cayley, 755, 799Cayley’s theorem, 761Center, 64Center of a group, 708Central algebra, 75, 845Central force field, 25Central simple algebra, 93, 845Centralizer, 708Chain rule, 96Champollion, 267
Characteradjoint representation, 737compound, 736, 737conjugacy class, 736group and its subgroup, 743–746of a representation, 736simple, 736, 737symmetric group
graphical construction, 767–774Character table, 743
for S2, 745for S3, 745
Characteristic hypersurface, 638Characteristic polynomial, 173, 182, 197,
536, 558, 591, 626HNOLDE, 446
Characteristic root, 173, 467Charged scalar field, 1114Chebyshev, 639Chebyshev polynomials, 253Chevalley, biography, 971Chevalley’s theorem, 969Christoffel, biography, 1130Circle of convergence, 319Circuit matrix, 462, 463Circular heat-conducting plate, 600Classical adjoint, 56, 57, 155
matrix of, 152–155Classical field theory
conservation laws, 1069–1073Noether’s theorem, 1069–1073symmetry, 1069–1073
Classical orthogonal polynomial, 241–243classification, 244, 245generating functions, 257, 258recurrence relations, 245–248
Classification of simple algebras, 92–95Clebsch, 246, 755
biography, 753Clebsch-Gordan
coefficients, 756, 757decomposition, 753–756, 774, 973series, 754, 757
Clifford, 799Clifford algebra, 800, 829–855
anticenter, 839–842canonical element, 838, 839center, 839–842conjugation involution, 837construction, 830–834degree involution, 836Dirac equation, 832–834even element, 835general classification, 843–846homomorphism with other algebras,
837, 838isomorphisms, 842, 843odd element, 835properties, 834–843representation, 987–1006
1184 Index
Pauli spin matrices, 997–1001Clifford algebra C1
3(R), 852–855,1004–1006
Clifford algebra Cνμ(R), 846–855
classification, 851, 852Clifford algebra Cν
μ(R)
spin representation, 1003spinor space of, 1003
Clifford algebra Cνμ(R)
standard basis, 1001Clifford algebra Cn
0(R)
classification, 849, 850, 851Clifford algebra C0
n(R)
classification, 849, 850, 851Clifford group, 987–994Clifford product, 830Closed form, 894Closed subset, 520Closure, 520Codifferential, 900
covariant, 1108Codomain, 5Cofactor, 153, 155Commutative algebra, 63Commutative group, 704Commutative Lie algebra, 937Commutator, 106, 107
diagonalizability, 186Commutator subgroup, 707Compact Lie algebra, 945Compact Lie group
representation, 953–963Compact operator, 523–526
spectral theorem, 527–534spectrum, 527
Compact resolvent, 563–569Compact set, 519–523Compact subset, 522Compact support, 234, 898Comparison theorem, 430–432Complement of a set, 2Complete metric space, 10Complete o.n. sequence, 219Completeness relation, 123, 148, 220,
228, 658Complex coordinate space, 21Complex exponential function, 302Complex FOLDEs, 460–462Complex function, 295, 296
analytic, 301analytic continuation, 372branch cut, 366branch point of, 365Cauchy-Riemann conditions, 298continuous, 296derivatives as integrals, 315–319entire, 301essential singularity, 342integration, 309–315isolated singularity, 339
isolated zero, 330meromorphic, 363multivalued
branch, 367Riemann sheet, 367Riemann surface, 367
pole of order m, 342power series, 319
circle of convergence, 319principal part of, 342removable singular point, 342simple pole, 342simple zero, 330zero of order k, 329
Complex GL(V), 922Complex hyperbolic function, 303Complex plane
contour in, 309curve in the, 309multiply connected region, 312path in the, 309simply connected region, 312
Complex potential, 305Complex series, 319–321Complex SOLDE, 463–469Complex structure, 45–48, 139, 163, 202Complex trigonometric function, 303Complex vector space, 20Complexification, 48, 102, 202Composition of maps, 5Compound character, 737Conducting cylindrical can, 586–588Confluent hypergeometric function,
478–485definition, 479integral representation of, 497, 498
Conformal group, 1159in 2 dimensions, 1159
Conformal Killing vector, 1158Conformal map, 304–308, 309
definition, 305translation, 306
Conformal transformation, 1158special, 1159
Conic sections, 196Conjugacy class, 711, 736, 737Conjugate, 711Conjugate subgroup, 707Conjugation
operators, 113, 114Conjunct, 612, 614, 628Connection, 1086–1091
flat, 1095, 1096Levi-Civita, 1145linear, 1120–1140
definition, 1121local expression, 1087–1089matrix structure group, 1096, 1097metric, 1143–1155vector bundle, 1117–1120
Index 1185
Connection 1-form, 1087Connection coefficients, 1133Conservation law, 963, 1065–1069
characteristic, 1067classical field theory, 1069–1073equivalent, 1067trivial of the first kind, 1066trivial of the second kind, 1066
Conserved current density, 1065Constant of the motion, 1066Constrained systems, 905Continuous index, 227–233Contour, 309
simple closed, 309Contractable, 894Contraction, 788Contravariant degree, 784Contravariant tensor, 784Convergence
infinite vector sum, 215–220Convex subset, 528Convolution theorem, 291Coordinate curve, 870Coordinate frame, 870Coordinate functions, 860Coordinate representation of Lg∗, 921Coordinate system
left-handed, 800right-handed, 800
Coordinate transformationorientation preserving, 898orientation reversing, 898
Coset, 708Cosmological constant, 1165Cotangent bundle, 882Coulomb, 1058Coulomb potential, 466Countably infinite set, 11Covariant codifferential, 1108Covariant degree, 784Covariant derivative, 1117, 1123–1125,
1133directional, 1119exterior, 1093Lie derivative, 1135
Covariant differential, 1125Covariant tensor, 784Crystallography, 275Current density, 1065, 1110
energy momentum, 1071formula for, 1111
Curvature, 1125–1132abelian case, 1095and gravity, 1161as relative acceleration, 1160matrix structure group, 1096, 1097
Curvature form, 1093principal fiber bundle, 1091–1097structure equation, 1093
Curvature scalar, 1163
Curvature tensor field, 1126Curvature transformation, 1126Curve
coordinate, 870development of, 1137differentiable, 866
Curvilinear coordinates, 1150Cyclic permutation, 717Cyclic subgroup, 707
DD’Alembert
biography, 397D’Alembert, 1057Damping factor, 447Darboux, 799, 1015Darboux inequality, 313Darboux theorem, 902De Broglie, 907Decomposition
algebra, 83–95Clebsch-Gordan, 753–756
Dedekind, 11, 764, 792, 1130Degeneracy
energy, 656lifting of, 748
Degenerate eigenvectors, 402Degenerate kernel, 556–559Delta function, 229, 512, 624
derivative of, 233expansion
Fourier, 273general, 257
Fourier transform, 279Green’s function, 644, 685integral representation of, 281Legendre polynomials, 256limit of sequence, 229, 231potential, 653spherical harmonics, 692step function, 231, 232Sturm-Liouville eigenfunctions, 691variational problem, 1054
Dense subset, 520Density function, 936Density of states, 584Derivation, 81, 82, 99, 106, 887, 891, 944,
1124of an algebra, 80–83tangent vector, 868
Derivation algebra, 82, 944Derivative
complex function, 315–319covariant, 1117function of operator, 108functional, 1050–1053Hilbert spaces, 1047–1050of operators, 107–112total, 1027
1186 Index
Derivative operator, 40unboundedness of, 515
Derived algebra, 66, 98Descartes, 791Determinant, 7, 55, 56, 118, 153–155,
158, 160–162, 173, 175, 201,205, 557, 558, 567, 610, 641,644, 661, 706, 719, 788, 800,801, 806, 816–818, 839, 897,898, 916, 924
analytic definition of, 205connection with trace, 161derivative of, 161exponential of trace, 162minor, 153relation to trace, 160
Determinant function, 54–56, 152, 158,159, 162, 167, 799, 805, 815,838, 848
dual, 158–160normed, 815
Determinant of a matrix, 151–160Development, 1137Diagonalization
simultaneous, 185–188Diffeomorphism, 865Differentiable curve, 866
tangent vector, 868Differentiable manifold, 859–866
dimension of a, 860Differentiable map, 864
coordinate expression of, 864Differential
of a constant map, 873of a map, 872real-valued maps, 874
Differential equationanalytic, 460analytic properties, 460–463associated Legendre, 411Bessel, 466completely homogeneous problem,
612Euler, 471Fuchsian, 469–473
definition, 470homogeneous, 418hypergeometric, 466
definition, 473inhomogeneous, 418Legendre, 411linear, 418
superposition principle, 423multiparameter symmetry group,
1040–1043Riemann, 471second order linear
behavior at infinity, 469second-order linear
Frobenius method, 440
regular, 422symmetry group, 1014–1024
Differential form, 888closed, 894exact, 894Lorentz force law, 892Maxwell’s equations, 890pullback of, 888
Differential geometry, 1117–1140Differential one-form, 882Differential operator, 418, 970
adjoint, 433–436linear, 605
Diffusion equation, 643, 673one-dimensional
parabolic, 642time-dependent, 581, 582
Dilation, 306Dilitation, 1159Dimension theorem, 42, 44, 45, 61, 99,
193, 518, 803, 810, 857, 876Dirac, 957
biography, 235Dirac delta function, 229, 512, 624
derivative of, 233expansion
Fourier, 273general, 257
Fourier transform, 279Green’s function, 644, 685integral representation of, 281Legendre polynomials, 256limit of sequence, 229spherical harmonics, 692step function, 231Sturm-Liouville eigenfunctions, 691variational problem, 1054
Dirac equation, 832–834, 997Dirac gamma matrices, 834
Majorana representation, 855, 1003Direct product
group, 712, 713Direct sum, 25–28, 75, 92, 119, 169, 201,
528, 558, 567, 712, 731, 797,803, 810, 834–836, 840, 852,947, 959, 999, 1001, 1087
algebra, 67definition, 25inner product, 32
Directional covariant derivative, 1119Directional derivative, 884Dirichlet, 246, 791, 1130, 1144
biography, 666Dirichlet boundary condition, 642Dirichlet BVP, 642, 665–671
in two dimensions, 690Discrete Fourier transform, 286, 287Dispersion relation, 376–378
with one subtraction, 377Distribution, 234, 418, 686, 688
Index 1187
Distribution (cont.)density, 234derivative of, 236Fourier transform, 287, 288Fourier transform of a, 288Green’s function as, 606limit of functions, 235
Divergencenull Lagrangians, 1060, 1061of tensors, 1135total, 1060
Divergence theorem, 648Division algebra, 69
of a Clifford algebra, 999DOLDE
hypergeometricKummer’s solutions, 477
Domain, 5Dot product, 7, 29Dual
basis, 51, 782of an operator, 51space, 48
Dual determinant function, 158–160Dual space, 49
EEffective action, 713Eigenfunction expansion technique
2D Laplacian, 689Eigenspace, 173, 180
compact operator, 527compact resolvent operator, 565involution, 836normal operator, 179perturbation theory, 655Weyl operator, 955
Eigenvalue, 172–175angular momentum, 401–405, 970Casimir operator, 969characteristic polynomial, 173circuit matrix, 462compact operators, 527definition, 172discrete, 630extrema of functions, 197Green’s functions, 630, 688harmonic oscillator, 444hermitian operator, 178integral equation, 544invertible operator, 611involution, 836largest, 181orthogonal operator, 201perturbation theory, 656positive operator, 181projection operator, 174simple, 173smallest, 181Sturm-Liouville, 691
Sturm-Liouville system, 568, 578unitary operator, 178upper-triangular matrix, 175Weyl operator, 959
Eigenvector, 172–175, 178, 199angular momentum, 402, 406–413Casimir operator, 969compact normal operator, 532compact operators, 527definition, 172harmonic oscillator, 444hermitian operator, 433infinite dimensions, 518integral equation, 554normalized, 190perturbation theory, 656simultaneous, 185SOLDE, 463Sturm-Liouville system, 567Weyl operator, 959
Einstein, 897, 956, 1070, 1131, 1145,1146, 1164, 1166
Einstein tensor, 1163Einstein’s equation, 1163–1166
Schwarzschild solution, 1169spherically symmetric solutions,
1167–1169Einstein’s summation convention, 781Electromagnetic field tensor, 145, 826,
889, 892, 893, 895Elementary column operation, 156Elementary row operation, 156Elliptic PDE, 641, 665–673Elsewhere, 941Empty set, 2Endomorphism, 39, 40, 42, 80, 81, 101,
102, 125, 164, 705, 806, 807,1126
involution, 72Energy function, 905Energy levels, 11Energy quantum number, 727Entire function, 301, 343
Bessel functions, 572bounded, 317confluent HGF, 479inverse of gamma function, 379with simple zeros, 364
Epimorphismalgebra, 70
Equivalence class, 3representative, 3
Equivalence relation, 3, 4, 24Equivalent representations, 726Error function, 436
as solution of a DE, 437Essential singularity, 342Essentially idempotent, 741, 772η-orthogonal matrices, 940Euclid, 220, 907
1188 Index
Euclidean metric, 1149Euler, 301, 474, 482, 570, 1057, 1144
biography, 1055Euler angles, 146, 172, 934, 972Euler equation, 457Euler kernel, 494Euler operator, 1054Euler theorem, 973Euler transform, 493Euler-Lagrange equation, 1055, 1069
classical, 1053field, 1053
Euler-Mascheroni constant, 380Evaluation function, 1051Event, 941Evolution operator, 109, 678Exact form, 894Expectation value, 115Exponential function
complex, 302Exponential map, 925Exterior algebra, 794–801Exterior calculus, 888–897Exterior covariant derivative, 1093Exterior derivative, 889
covariant, 1093Exterior product, 794
inner product, 819, 820
FF -related vector fields, 877Factor algebra, 77, 78, 92, 709Factor group, 710Factor map, 6Factor set, 4, 24Factor space, 24, 25, 77Factorial function, 378
Stirling approximation of, 386Faithful representation, 126, 726Fast Fourier transform, 287Fermi energy, 584Feynman diagram, 654Feynman propagator, 688Fiber, 1080Fiber bundle, 1079–1097
abelian case, 1095principal, 1079–1086
Fiber metric, 1143Field, 20
gauge, 1099–1105magnetic, 3particle, 1101tensor
manifold, 876–888vector
manifold, 877–882Fine-structure constant, 481, 655Finite-rank operator, 524First integral, 1066First variation, 1057
Flat connection, 1095, 1096Flat manifold, 1153, 1154Flat map, 801, 902Flow, 881FODE
existence, 419–421existence and uniqueness
local, 421linear, 420normal form, 420Peano existence theorem, 420uniqueness, 419–421uniqueness theorem, 420
FOLDE, 433complex, 460–462irregular singular point, 461regular singular point, 461removable singularity, 461
Forminvariant
Lie group, 927, 928pseudotensorial, 1092tensorial, 1092torsion, 1122
Form factor, 284Formal adjoint, 612, 633, 648Four-potential, 895Four-vector, 808
energy momentum, 1071Fourier, 666, 703, 1056
biography, 267Fourier integral transforms, 278Fourier series, 265–276, 563
angular variable, 266fundamental cell, 267general variable, 268group theory, 960higher dimensions, 275, 276main theorem, 272Peter-Weyl, 960sawtooth, 270square wave, 269to Fourier transform, 276–278two-dimensional, 581
Fourier transform, 276–288, 493Coulomb potential
charge distribution, 283point charge, 282
definition, 278derivatives, 284, 285discrete, 286, 287distribution, 287, 288Gaussian, 280Green’s functions, 680–688higher dimensions, 281quark model, 284scattering experiments, 282
Fourier-Bessel series, 587Fredholm, 220Fredholm, biography, 551
Index 1189
Fredholm alternative, 551Fredholm equation, 543, 652
second kindcharacteristic values, 544
Fredholm integral equation, 549–559Free action, 713Friedmann, biography, 1166Friedmann metric, 1149Frobenius, 734, 957, 981
biography, 764Frobenius method, 439–444Frobenius Theorem, 93Fuchsian DE, 469–473
definition, 470Function, 5
analytic, 297–304complex, 295, 296
derivatives as integrals, 315–319integration, 309–315
determinant, 54generalized, 233–237inner product, 32meromorphic, 363–365multivalued, 365–371of operators, 104–106operator, 188–191p-linear, 53piecewise continuous, 266square-integrable, 221–227
Function algebra, 67Function of operator
derivative, 108Functional, 1054
linear, 48–53Functional derivative, 1050–1053Fundamental theorem of algebra, 318Fundamental vector field, 1086Future light cone, 941
GG-invariance, 1009G-invariant Lagrangian, 1106g-orthogonal, 808g-orthonormal, 813g-transpose, 806Galois, 154, 764, 946, 1015
biography, 702Gamma function, 250, 378–381
definition, 378Gamma matrices, 834
Majorana representation, 855Gauge
choice of, 1099Gauge field, 1099–1105Gauge invariance, 895Gauge Lagrangian, 1105Gauge Lagrangian density, 1109Gauge potential, 1099–1105Gauge theories, 1099–1114Gauge theory
local equation, 1112–1114Gauge transformation, 1102Gauss, 154, 251, 301, 482, 523, 533, 666,
791, 895, 1055, 1130, 1144biography, 474
Gay-Lussac, 581Gegenbauer function, 478Gegenbauer polynomials, 253General linear group, 705
representation, 963–966General relativity, 1163–1174Generalized Fourier coefficients, 220Generalized function, 233–237, 418, 606,
688Generalized Green’s identity, 613, 626,
648Generating function, 257Generator
Clifford algebra, 997, 1000conformal group, 1036, 1159coordinate transformation, 934cyclic group, 710group, 933, 1113group action, 962infinitesimal, 929, 932, 970, 976,
1010–1013, 1023, 1030, 1037,1040, 1041, 1062, 1064, 1068,1069
Lorentz, 1073of an algebra, 70rotation, 106, 750, 933, 998, 1157translation, 112, 948
Geodesic, 1137–1140relative acceleration, 1160
Geodesic deviation, 1159–1163equation of, 1161
Geodesic equation, 1138massive particles, 1170, 1171massless particles, 1170, 1172
Geometric multiplicity, 173Geometry
Riemannian, 1143–1174symplectic, 51, 901–909
Gibbs, 523, 907Gibbs phenomenon, 273–275GL(n,R) as a Lie group, 916GL(V)
as a Lie group, 915representation of, 963
Gödel, 897Gordan, 1070
biography, 755Gradient
for Hilbert spaces, 1050Gradient operator, 1133Gram, 34Gram-Schmidt process, 33–35, 164, 210,
241, 532Graph, 5Grassmann, 799, 1070
1190 Index
Grassmann product, 794Gravitational red-shift, 1173Gravity
and curvature, 1161Newtonian, 1161–1163
Green, biography, 613Green’s function, 358
adjoint, 618advanced, 686as a distribution, 606Dirichlet BC
circle, 670eigenfunction expansion, 630–632for d/dx, 606for d2/dx2, 607formal considerations, 610–617Helmholtz operator
in 2D, 694in one dimension, 605indefinite, 606–610multidimensional
delta function, 643–648diffusion operator, 684, 685Dirichlet BVP, 665–671eigenfunction expansion, 688–693Fourier transform, 680–688fundamental solution, 649–651general properties, 648, 649Helmholtz operator, 682–684integral equations, 652–655Laplacian, 647, 648, 681, 682Neumann BVP, 671–673perturbation theory, 655–661wave equation, 685–688
Neumann BVP, 673exterior, 673interior, 673
physical interpretation, 629properties, 619regular part of, 651resolvent, 630retarded, 686second order DO, 614–616self-adjoint SOLDOs, 616, 617singular part of, 651SOLDO, 617–629
construction, 621–626inhomogeneous BCs, 626–629properties, 619–621uniqueness, 621–626
symmetry, 619Green’s identity, 619, 648, 675, 679
generalized, 648Group, 8, 702–705
1st isomorphism theorem, 710abelian, 704affine, 948algebra
symmetric group, 771automorphism, 705
center of, 708commutative, 704commutator of, 707direct product, 712, 713
external, 712internal, 712
external direct product, 712finite
Lagrange’s theorem, 721homomorphism, 705
kernel of, 708internal direct product, 712isomorphism, 705left action, 713Lie, 915–936multiplication, 702multiplication table, 705of affine motions, 917order of, 703orthogonal, 706realization, 715representation, 725–732
character table, 743criterion for irreducibility, 738crystallography, 727irreducible, projection operator,
749irreducible basis function, 746–750matrix, 727particles and fields, 751quantum state parity, 727tensor product, 750–758
right action, 713rigid rotations, 706simply reducible, 753special orthogonal, 706special unitary, 706subset
left invariant, 713right invariant, 713word on, 720
symmetric, 715–720symmetry of Hamiltonian, 725symplectic, 707, 803unitary, 706
Group action, 713–715effective, 713, 918free, 713, 918infinitesimal, 928–935infinitesimal generator, 929Lie groups, 917–920orbit, 713stabilizer, 713transitive, 713, 918
Group algebra, 740representations, 740–743
Guided wavesTE, 585TEM, 585TM, 585
Index 1191
HHaar measure, 935Halley, 481Hamilton, 246, 545, 1070
biography, 906Hamiltonian
group of symmetry of, 725Hamiltonian mechanics, 801, 904Hamiltonian system, 905Hamiltonian vector field, 905
energy function, 905Hankel function, 484
first kindasymptotic expansion of, 386
second kind, 391Hankel transform, 494Harmonic functions, 304Harmonic oscillator, 443, 444–446
critically damped, 447ground state, 444Hamiltonian, 444overdamped, 447underdamped, 447
Heat equation, 395, 643, 673symmetry group, 1030–1034
Heat transfertime-dependent, 581, 582
Heat-conducting plate, 597Hegel, 791Heisenberg, 115, 236Helicity, 982Helmholtz, 246, 639, 957Helmholtz equation, 593Hermite, 251, 896
biography, 115Hermite polynomials, 245, 248, 249, 442,
573Hermitian, 31, 48, 116, 117, 120, 144,
147, 172, 177, 178, 181, 186,189, 205, 402, 525, 533, 555,558, 564, 613, 924, 945, 955,968, 982
Hermitian conjugate, 113–116, 144, 146,162, 171, 202, 404, 513, 661
Hermitian inner product, 31Hermitian kernel, 552–556Hermitian operator, 114–119Hilbert, 11, 34, 268, 523, 755, 897, 956,
1070, 1164biography, 220
Hilbert space, 215–227, 435basis of, 219bounded operators in, 513compact hermitian operator in, 530compact normal operator in, 532compact operator in, 524compact resolvent, 564convex subset, 528countable basis, 228definition, 218
derivative, 1047–1050differential of functions, 1049directional derivative, 1050functions on, 1052
derivative of, 1049invertible operator in, 611operator norm, 513perturbation theory, 658representation theory, 726, 953square-integrable functions, 222
Hilbert transform, 377Hilbert-Schmidt kernel, 525, 549Hilbert-Schmidt operator, 525, 955Hilbert-Schmidt theorem, 552HNOLDE, 446, 448
characteristic polynomial, 446Hodge star operator, 820–823, 893Hölder, 523Homographic transformations, 307Homomorphism
algebra, 70, 71, 77, 82, 98, 125Clifford algebra, 837Clifford group, 991group, 705, 710, 726, 731, 732, 987,
992Lie algebra, 922, 944, 953, 1101Lie group, 915, 922, 928, 953, 967PFB, 1081symmetric, 705trivial, 705
Horizontal lift, 1089Horizontal vector field, 1087HSOLDE
basis of solutions, 425comparison theorem, 431exact, 433integrating factor, 433second solution, 426–428separation theorem, 430
Hydrogen, 11Hydrogen-like atoms, 480–482Hyperbolic PDE, 641, 678–680Hypergeometric DE, 466Hypergeometric function, 473–478
confluent, 478–485integral representation of, 497, 498
contiguous functions, 476Euler formula, 496integral representation of, 494–498
Hypergeometric series, 473Hypersurface, 635
IIdeal, 73–78Idempotent, 83, 86–89, 119–125, 175,
741, 844, 852, 999, 1002essentially, 741, 772primitive, 88, 94, 999, 1001, 1002principal, 87–89rank, 94
1192 Index
Identityadditive, 20multiplicative, 20
Identity map, 5Identity operator, 101Identity representation, 726Ignorable coordinate, 645Image
map, 5Image of a subset, 5Implicit function theorem, 419Index
continuous, 227–233Indicial equation, 465
SOLDE, 465Indicial polynomial, 465Induced representations, 978Induction principle, 12Inductive definition, 14Inequality
Bessel, 219Cauchy, 336Darboux, 313Parseval, 219Schwarz, 35triangle, 36
Infinitesimal actionadjoint, 929
Infinitesimal generator, 929, 932Initial conditions, 418Initial value problem, 611, 635Injective map, 5Inner automorphism, 926Inner product, 29–38, 804–820
bra and ket notation, 31complex bilibear, 46definition of, 30direct sum, 32Euclidean, 31exterior product, 819, 820G-orthogonal, 1107hermitian, 31indefinite
orthonormal basis, 812–819subspaces, 809–812
isotropic vector, 808norm and, 37null vector, 808positive definite, 30pseudo-Euclidean, 31sesquilinear, 31signature, 813
Inner product space, 31INOLDE
particular solution, 448Integral
principal value, 354–358Integral curve, 879Integral equation, 543–548
characteristic value, 544
first kind, 543Fredholm, 549–559Green’s functions, 652–655kernel of, 543second kind, 543Volterra, 543Volterra, of second kind
solution, 545Integral operator, 512Integral transform, 493
Bessel function, 494Integration
complex functions, 309–315Lie group, 935, 936manifolds, 897–901
Integration operator, 40Interior product, 829, 891Intersection, 2Intrinsic spin, 1073Invariant, 1010
map, 1010operator
matrix representation, 171subspace, 169–172
definition, 170Invariant subspace, 728, 729Inverse
image, 5of a map, 6of a matrix, 155–158
Inverse mapping theorem, 873Inversion, 154, 306, 1159Involution, 72, 82, 836, 837, 839, 843,
848, 989, 990Irreducible basis function, 746–750Irreducible representation, 729
i-th rowfunctions, 747
norm of functions, 747Irreducible set of operators, 757Irreducible tensor operators, 756–758Irreducible tensorial set, 757Isolated singularity, 342–344Isolated zero, 330ISOLDE
general solution, 428–430Isometric map, 39, 1155Isometry, 40, 42, 43, 125, 205, 539, 806,
807, 811, 826, 992, 1143,1155–1159, 1168
time translation, 1167Isomorphism, 43–45, 52, 68, 74, 78, 127,
139, 140, 158, 222, 228, 661,704, 719, 721, 726, 789, 796,801, 838, 845, 847, 851, 871,872, 884, 905, 921, 922, 926,930, 945, 972, 998, 1085–1087,1089, 1103, 1118, 1128, 1143
algebra, 70Clifford algebras, 842, 843
Index 1193
Isomorphism (cont.)group, 705Lie algebra, 922Lie group, 915linear, 43–45natural, 785PFB, 1081
Isotropic vector, 808
JJacobi, 251, 475, 545, 666, 713, 753, 755,
791, 907, 1144biography, 246
Jacobi functionfirst kind, 477second kind, 478
Jacobi identity, 879, 887, 927Jacobi polynomials, 245, 250, 252, 478
special cases, 245Jacobian matrix, 873Jordan arc, 309Jordan canonical form, 539Jordan’s lemma, 345
KKant, 791Kelvin, 613Kelvin equation, 589Kelvin function, 589Kepler problem, 1074Kernel, 41, 42, 51, 130, 158, 173, 192,
198, 498, 529, 546, 558, 560,635, 678, 708, 826, 937, 944,995, 999
degenerate, 556–559hermitian, 552–556Hilbert-Schmidt, 525, 544, 555integral operator, 512integral transforms, 493separable, 556
Ket, 20Killing, 799, 1015
biography, 946Killing equation, 1156Killing form, 945, 948
of gl(n,R), 947Killing parameter, 1167Killing vector field, 1155–1159, 1167,
1170, 1173conformal, 1158
Kirchhoff, 639Klein, 799, 896, 956, 1015, 1070, 1131,
1164Klein-Gordon equation, 396Korteweg-de Vries equation, 1044Kovalevskaya, 523
biography, 639Kramers-Kronig relation, 378Kronecker, 11, 154
biography, 791Kronecker delta, 32, 50, 161, 782, 939Kronecker product, 751Kummer, 36, 755, 791, 946, 1130
LLagrange, 154, 246, 251, 267, 474, 482,
581, 755biography, 1057
Lagrange identity, 435, 494, 570, 578,613, 805
Lagrange multiplier, 1064Lagrange’s equation, 1111Lagrangian, 904, 1054
G-invariant, 1106gauge, 1109gauge-invariant, 1105–1107
construction, 1107–1111null, 1060, 1061
Lagrangian density, 1105Laguerre polynomials, 245, 249, 250Laplace, 34, 267, 666, 906, 1056, 1058,
1164biography, 581
Laplace transform, 493Laplace’s equation, 395
Cartesian coordinates, 579cylindrical coordinates, 586elliptic, 642
LaplacianGreen’s function for, 647separated
angle radial, 399spherical coordinates
separation of angular part, 398–401Laurent, biography, 340Laurent series, 321–330, 657
construction, 322uniqueness, 325
Lavoisier, 153, 1058Least square fit, 225–227Lebesgue, 221Left annihilator, 73Left coset, 708Left ideal, 73, 74, 84, 740, 773, 1000,
1002minimal, 74, 76, 79, 94, 128, 129, 772,
999, 1001, 1003Left translation
as action, 929Left-invariant 1-form, 921Left-invariant vector field, 920Legendre, 246, 267, 545, 666, 1144
biography, 251Legendre equation, 436, 441, 572Legendre function, 478Legendre polynomial, 225, 250–252, 256,
408, 411, 428, 555and Laplacian, 256asymptotic formula, 576
1194 Index
delta function, 256Legendre transformation, 904Leibniz, 154, 791Leibniz formula, 81Leibniz rule, 16Length
vector, 36–38Levi-Civita, 1131
biography, 1146Levi-Civita connection, 1145Levi-Civita tensor, 799, 976Lie, 764, 799, 896, 946
biography, 1014Lie algebra, 915–936
abelian, 937adjoint map, 926Cartan metric tensor, 945Cartan theorem, 948center, 937commutative, 937compact, 945decomposition, 947derivation, 944ideal, 937Killing form of, 945of a Lie group, 920–927of SL(V), 924of unitary group, 924of vector fields, 879representation, 966–983
definition, 953semisimple, 948simple, 948structure constants, 937theory, 936–948
Lie bracket, 879Lie derivative, 885
covariant derivative, 1135of a 1-form, 886of p-forms, 890of vectors, 886
Lie group, 405, 915–936canonical 1-form on, 928compact
characters, 960matrix representation, 959representation, 953–963unitary representation, 954Weyl operator, 955
group action, 917–920homomorphism, 915infinitesimal action, 928–935integration, 935, 936
density function, 936invariant forms, 927, 928left translation, 920local, 917representation, 953
Lie multiplication, 937Lie subalgebra, 937
Lie’s first theorem, 932Lie’s second theorem, 927Lie’s third theorem, 927Light cone, 941Linear combination, 21Linear connection, 1120–1140
definition, 1121Linear frame, 1083Linear functional, 48–52, 53, 53, 61, 233,
234, 287, 515, 617, 783, 787,796, 809, 829, 883
Linear independence, 21Linear isomorphism, 43–45, 49Linear map, 38–45, 51, 70, 78, 95, 116,
563, 789, 801, 814, 837, 838,840, 856, 1048, 1049, 1073
invertible, 43Linear operator, 39–41, 47, 55, 56, 66,
113, 115, 116, 119, 139, 140,151, 170, 171, 174, 422, 513,515, 517, 522, 529, 531, 564,785, 793, 799, 810, 944
determinant, 55, 56null space of a, 41
Linear PDE, 636Linear transformation, 53
bounded, 514definition, 39pullback of a, 51
Liouville, 568, 703biography, 570
Liouville substitution, 569, 573, 576, 577Liouville’s theorem, 908Lipschitz condition, 420Little algebra, 978Little group, 714, 978–981Local diffeomorphism, 865Local group of transformations, 917Local Lie group, 917Local operator, 512Local trivialization, 1080Logarithmic function, 365Lorentz, 897Lorentz algebra, 972Lorentz force law, 892Lorentz group, 707, 940Lorentz metric, 1149Lorentz transformation, 940
orthochronous, 941proper orthochronous, 941
Lowering indices, 805Lowering operator, 403
MMaclaurin series, 321Magnetic field, 3Manifold, 859–866
atlas, 860chart, 860coordinate functions, 860
Index 1195
Manifold (cont.)differentiable, 859–866differential of a map, 872–876flat, 1153integration, 897–901orientable, 898product, 863pseudo-Riemannian, 1144Riemannian, 1144semi-Riemannian, 1144subset
contractable to a point, 894symplectic, 902tangent vectors, 866–872tensor fields, 876–888vector fields, 877–882with boundary, 899
Map, 4–8bijective, 6codomain, 5conformal, 304–309differentiable, 864differential
Jacobian matrix of, 873domain, 5equality of, 5functions and, 5graph of a, 5identity, 5image of a subset, 5injective, 5inverse of a, 6isometric, 39linear, 38–45
invertible, 43manifold, 872–876multilinear, 53–57, 782–789
skew-symmetric, 53one-to-one, 5onto, 6p-linear, 53range of a, 5surjective, 5target space, 5
Maschke’s Theorem, 759Mathematical induction, 12–14Matrix, 137–142
antisymmetric, 144basis transformation, 149block diagonal, 171, 200circuit, 462, 463complex conjugate of, 144determinant of, 151–160diagonal, 144diagonalizable, 162hermitian, 144hermitian conjugate of, 144inverse of, 155–158irreducible, 171
operations on a, 142–146orthogonal, 144rank of, 158reducible, 171representation
orthonormal basis, 146–148row-echelon, 156strictly upper triangular, 66symmetric, 144symplectic, 804transpose of, 142triangular, 156unitary, 144upper triangular, 66upper-triangular, 175, 176
Matrix algebra, 66, 78–80Matrix of the classical adjoint, 152–155Maurer-Cartan equation, 928, 1095Maximally symmetric spaces, 1157Maxwell’s equations, 894Mellin transform, 493Mendelssohn, 666, 792Meromorphic functions, 363–365Method of images, 668
sphere, 669Method of steepest descent, 383, 577Metric, 37
Friedmann, 1149Schwarzschild, 1149
Metric connection, 1143–1155Metric space, 8–10
complete, 10convergence, 9definition, 8
Minimal ideal, 963Minimal left ideal, 74, 76, 79, 94, 128,
129, 772, 999, 1001, 1003Minkowski, 1164Minkowski metric, 1149Mittag-Leffler, 523, 640Mittag-Leffler expansion, 364Modified Bessel function, 484
first kindasymptotic expansion of, 391
second kindasymptotic expansion of, 392
Moment of inertia, 145, 195matrix, 145
Momentum operator, 398Monge, 153, 267Monomorphism
algebra, 70Morera’s theorem, 319Multidimensional diffusion operator
Green’s function, 684, 685Multidimensional Helmholtz operator
Green’s function, 682–684Multidimensional Laplacian
Green’s function, 681, 682Multidimensional wave equation
1196 Index
Green’s function, 685–688Multilinear, 152, 783, 789, 883, 1092,
1124Multilinear map, 53–57, 782–789
tensor-valued, 787Multiplicative identity, 20Multivalued functions, 365–371
Nn-equivalent functions, 1018n-sphere, 860, 865n-th jet space, 1018n-tuple, 3
complex, 21real, 21
Napoleon, 267, 581Natural isomorphism, 785, 820Natural numbers, 2, 9Natural pairing, 783Neighborhood
open round, 519Neumann, 246, 753
biography, 671Neumann BC, 643Neumann BVP, 643, 671–673Neumann function, 483Neumann series, 548, 653, 654Newton, 397, 474, 581, 896, 906, 1056Newtonian gravity, 1161–1163Nilpotent, 83–85, 88, 91, 539Noether, 755
biography, 1069Noether’s theorem, 1065–1069
classical field theory, 1069–1073NOLDE
circuit matrix, 462constant coefficients, 446–449existence and uniqueness, 611integrating factor, 632simple branch point, 463
Non-local potential, 683Nondegenerate subspace, 810Norm, 215, 217, 291, 513–515, 529, 544,
812of a vector, 36operator, 514product of operators, 516
Normal coordinates, 1138–1140Normal operator, 177Normal subgroup, 709Normal vectors, 32Normed determinant function, 815Normed linear space, 36Null divergence, 1066Null Lagrangian, 1060, 1061Null space, 41, 551, 554Null vector, 808, 941Nullity, 41Number
complex, 2
integer, 2natural, 2, 9rational, 4, 9, 10real, 2
OODE, 417–419
first ordersymmetry group, 1037–1039
higer ordersymmetry group, 1039, 1040
Ohm, 666Olbers, 482One-form, 882One-parameter group, 881One-to-one correspondence, 6Open ball, 519Open subset, 520Operation
binary, 7Operations on matrices, 142–146Operator, 39
adjoint, 113existence of, 517
adjoint of, 46angular momentum, 398
eigenvalues, 401–405annihilation, 444anti-hermitian, 115bounded, 513–517Casimir, 969–971closed, 564
bounded, 564compact, 523–526
spectral theorem, 527–534compact Hermitian
spectral theorem, 530compact normal
spectral theorem, 532compact resolvent, 564conjugation, 113, 114creation, 444derivative, 40, 107–112determinant, 55, 56diagonalizable, 174differential, 511, 512domain of, 563evolution, 109expectation value of, 115extension of, 564finite rank, 524formally self-adjoint, 649functions of, 104–106, 188–191hermitian, 114–119, 564
eigenvalue, 178hermitian conjugate of, 113Hilbert-Schmidt, 525, 551, 567Hodge star, 820–823idempotent, 119–125integral, 511, 512
Index 1197
Operator (cont.)integration, 40inverse, 101involution, 72kernel of an, 41local, 512negative powers of, 103norm of, 514normal, 177
diagonalizable, 181eigenspace of, 179
null space of an, 41polar decomposition, 205–208polarization identity, 41polynomials, 102–104positive, 117positive definite, 117projection, 120–125
orthogonal, 121pullback of an, 51raising and lowering, 403regular point, 517representation of, 138resolvent of, 534right-shift, 513scalar, 757self-adjoint, 46, 115, 564skew, 46spectrum, 517, 518spectrum of, 173square root, 189square root of, 189strictly positive, 117Sturm-Liouville, 564, 566symmetric, 193tensor
irreducible, 756–758trace of, 161unbounded
compact resolvent, 563–569unitary, 114–119, 189
eigenvalue, 178Operator algebra, 101–107Lie algebra o(p,n − p), 940–943Opposite algebra, 66Optical theorem, 378Orbit, 728, 918Orbital angular momentum, 1073Ordered pairs, 2Orientable manifolds, 898Orientation, 800, 801, 898
positive, 801Oriented basis, 800Orthogonal, 40Orthogonal basis
Riemannian geometry, 1148–1155Orthogonal complement, 169, 528–530,
551, 729, 747, 802, 812, 841Orthogonal group, 706, 925
Lie algebra of, 925Orthogonal polynomial, 222–225, 579
classical, 241, 241–243classification, 245differential equation, 243generating functions, 257recurrence relations, 245
expansion in terms of, 254–257least square fit, 225
Orthogonal transformation, 154Orthogonal vectors, 32Orthogonality, 32, 33
group representation, 732–737Orthonormal basis, 32
indefinite inner product, 812–819matrix representation, 146–148
Pp-form, 796
vector-valued, 800Pairing
natural, 783Parabolic PDE, 641, 673–678Parallel displacement, 1090Parallel section, 1091, 1119Parallelism, 1089–1091Parallelogram law, 37Parameter
affine, 1138Parity, 718
Hermite polynomials, 262Legendre polynomials, 262
Parseval equality, 220Parseval inequality, 219, 958Parseval’s relation, 291Particle field, 1101Particle in a box, 582–584Particle in a cylindrical can, 601Particle in a hard sphere, 593Partition, 4, 720Past light cone, 941Pauli spin matrices, 146, 938, 944
Clifford algebra representations,997–1001
PDE, 635–643Cauchy data, 636Cauchy problem, 636characteristic hypersurface, 636–640characteristic system of, 1012elliptic, 665–673
mixed BCs, 673homogeneous, 397hyperbolic, 678–680inhomogeneous, 397order of, 636parabolic, 673–678principal part, 636second order, 640–643second-order
elliptic, 641
1198 Index
PDE (cont.)hyperbolic, 641parabolic, 641ultrahyperbolic, 641
PDEs of mathematical physics, 395–398Peano, 897Peirce decomposition, 87, 89, 90, 100Periodic BC, 571Permutation, 53
cyclic, 717even, 719odd, 719parity of, 718
Permutation group, 715Permutation tensor, 816Perturbation theory, 655, 748
degenerate, 660, 661first-order, 660nondegenerate, 659, 660second-order, 660
Peter-Weyl theorem, 960Fourier series, 960
PFBlocal section, 1083
Phase space, 801Photon capture
cross section, 1173Piecewise continuous, 266Pin(μ, ν), 995Planck, 523, 1164Poincaré, 115, 533, 552, 672, 799, 1164
biography, 895Poincaré algebra, 943, 948
representation, 975–983Poincaré group, 707, 917, 943, 979Poincaré lemma, 894
converse of, 895Poisson, 246, 568, 581, 666, 703Poisson bracket, 908Poisson integral formula, 671Poisson’s equation, 395, 648, 1162Polar decomposition, 205–208Polarization identity, 41, 812Pole, 342Polynomial, 20
inner product, 32operators, 102–104orthogonal, 222–225
Polynomial algebra, 95–97Positive definite operator, 117Positive operator, 117Positive orientation, 801Potential
gauge, 1099–1105non-local, 683separable, 683
Power series, 319differentiation of, 320integration of, 320
SOLDE solutions, 436–446uniform convergence, 320
Lie algebra p(p,n − p), 940–943Preimage, 5Primitive idempotent, 88, 94, 999, 1001,
1002Principal fiber bubdle
curvature form, 1091Principal fiber bundle, 1079–1086
associated bundle, 1084–1086base space, 1080connection, 1086–1091
matrix structure group, 1096, 1097curvature
matrix structure group, 1096, 1097curvature form, 1097curve
horizontal lift, 1089fundamental vector field, 1086global section, 1083lift of curve, 1089parallelism, 1089–1091reducible, 1082structure group, 1080
matrix, 1096, 1097trivial, 1080vector field
horizontal lift, 1089Principal idempotent, 87–89Principal part
PDE, 636Principal value, 354–358, 685Product
Cartesian, 2, 7dot, 7inner, 29–38tensor, 28, 29
Product manifold, 863Projectable symmetry, 1017Projection, 6Projection operator, 120–125, 169, 174,
180, 527, 529, 532, 536, 552,655–657, 688, 748, 809
completeness relation, 123orthogonal, 121
Projective groupdensity function, 936one-dimensional, 920
Projective space, 4Prolongation, 1017–1024
functions, 1017–1021groups, 1021, 1022of a function, 1019vector fields, 1022–1024
Propagator, 654, 678Feynman, 688
Proper subset, 2Prüfer substitution, 574Pseudo-Riemannian manifold, 1144Pseudotensorial form, 1092
Index 1199
Puiseux, biography, 365Pullback, 789, 883, 888, 898, 1094, 1112
linear transformation, 51of p-forms, 796
QQuadratic form, 843Quantization
harmonic oscillatoralgebraic, 445analytic, 443
hydrogen atom, 481Quantum electrodynamics, 654Quantum harmonic oscillator, 444–446Quantum mechanics
angular momentum, 405Quantum particle in a box, 582–584Quantum state
even, odd, 727Quark, 753, 754, 980Quaternion, 69, 98, 831, 846, 847, 856,
907, 989, 990, 993, 996, 1070absolute value, 70conjugate, 69pure part, 69real part, 69
Quotient group, 710Quotient map, 6Quotient set, 4, 24Quotient space, 24, 25
Rr-cycle, 716Radical, 84–88Radon-Hurwitz number, 1002Raising indices, 805Raising operator, 403Range of a map, 5Rank of a matrix, 158Rational function, 343
integration of, 345–348Rational numbers, 4, 9, 10
dense subset of reals, 520Rational trig function
integration of, 348–350Real coordinate space, 21Real normal operator
spectral decomposition, 198–205Real vector space, 20Realization, 715Reciprocal lattice vectors, 276Recurrence relations, 222Redshift, 1173Reduced matrix elements, 758Reducible bundle, 1082Reducible representation, 729Reflection, 808Reflection operator, 121Reflection principle, 374–376Reflexivity, 3
Regular point, 301, 460operator, 517, 551
Regular representation, 128, 739Regular singular point
SOLDE, 464Relation, 3, 24
equivalence, 3, 4Relative acceleration, 1160Relativistic electromagnetism, 889Relativity
general, 1163–1174Removable singularity
FOLDE, 461Representation
abelian group, 733action on Hilbert space, 726adjoint, 732, 755, 1092, 1102algebra, 125–131angular momentum, 402carrier space, 726character of, 736classical adjoint, 152Clifford algebras, 987–1006compact Lie group, 945, 953–963complex conjugate, 732dimension of, 726direct sum, 128, 731equivalent, 127, 726faithful, 126, 726general linear group, 715, 963–966
Representation ofgl(n,R), 968
Representationgroup, 725–732
adjoint, 755analysis, 737–739antisymmetric, 745, 771identity, 754, 758, 771irreducible, 734, 737irreducible basis function, 746–750irreducible in regular, 739orthogonality, 732–737tensor product, 750–758trivial, 769
group algebra, 740–743hermitian operator, 182identity, 726, 1092irreducible, 127, 729
compact Lie group, 957finite group, 730general linear group, 964Lie group, 1072semi-simple algebra, 130
Kronecker product, 751Lie algebra, 948, 966–983
Casimir operator, 969Lie group, 937, 953
unitary, 953matrix
orthonormal basis, 146–148
1200 Index
Representation (cont.)operator, 161, 169, 199, 923operators, 138orthogonal operator, 201quantum mechanics, 734, 748quaternions, 126reducible, 729regular, 128, 739semi-simple algebra, 130simple algebra, 129
Representation ofsl(n,C), 968
Representationso(3), 972so(3,1), 974structure group, 1092, 1101, 1117,
1143, 1144subgroup, 743subgroups of GL(V), 967–969
Representation ofsu(n), 969
Representationsymmetric group, 761–776
analytic construction, 761–763graphical construction, 764–767products, 774–776Young tableaux, 766
tensor product, 128, 751antisymmetrized, 752character, 751symmetrized, 752
trivial, 732, 1092twisted adjoint, 987
Representation ofu(n), 968
Representationunitary, 730
compact Lie group, 954upper-triangular, 175vectors, 137
Residue, 339–341definite integrals, 344–358definition, 340integration
rational function, 345–348rational trig function, 348–350trig function, 350–352
Residue theorem, 340Resolution of identity, 536, 740, 774Resolvent, 534–539
compact, 564unbounded operator, 563–569
Green’s functions, 630Laurent expansion, 535perturbation theory, 655
Resolvent set, 517openness of, 521
Resonant cavity, 585, 597Riccati equation, 455, 1040
Ricci, 1131, 1146Ricci tensor, 1162, 1163, 1165Riemann, 36, 268, 366, 755, 896, 956,
1055, 1130biography, 1144
Riemann identity, 472Riemann normal coordinates, 1138–1140Riemann sheet, 365, 367Riemann surface, 366–371Riemann-Christoffel symbols, 1130Riemannian geometry, 1143–1174
gravityNewtonian, 1161–1163
isometry, 1155–1159Killing vector field, 1155–1159Newtonian gravity, 1161–1163orthogonal bases, 1148–1155
Riemannian manifold, 1144Riesz-Fischer theorem, 222Right annihilator, 73Right coset, 708Right ideal, 73Right translation, 921Right-invariant 1-form, 921Right-invariant vector field, 921Right-shift operator, 513
eigenvalues of, 518Rigid rotations, 706Rodriguez formula, 243, 245, 446Rosetta stone, 267Rotation algebra, 972Rotation group, 727, 970
character, 973Rotation matrix, 972
Wigner formula, 972Russell, 11, 897
SSaddle point approximation, 382Sawtooth voltage, 270Scalar, 20Scalar operator, 757, 758Scalar product, 29Scale transformations, 920Scattering theory, 595Schelling, 791Schmidt, biography, 34Schopenhauer, 791Schrödinger, 115, 907Schrödinger equation, 109, 396, 442, 469,
480, 582, 593, 683, 727classical limit, 452, 453one dimensional, 451
Schur, 764, 957, 981biography, 734
Schur’s lemma, 732, 733, 758, 953, 969Schwarz, 523, 792
biography, 36Schwarz inequality, 35, 59, 211, 218, 222,
515, 540, 950, 956
Index 1201
Schwarz reflection principle, 374–376Schwarzschild, biography, 1164Schwarzschild geodesic, 1169–1174Schwarzschild metric, 1149Schwarzschild radius, 1169Second order PDE, 640–643Second-order PDE
classification, 641Section
global, 1083local, 1083parallel, 1091, 1119
Selection rules, 753Self-adjoint, 115, 193, 194, 198, 201, 206,
433, 435, 533, 566, 569, 613,616, 619, 628, 633, 649, 663,665, 673, 679, 692, 694, 956
formally, 613Semi-Riemannian manifold, 1144Semi-simple algebra, 88–91, 92, 92, 94,
130, 764, 799, 844Semi-simple Lie algebra, 948Separable kernel, 556Separable potential, 683Separated boundary conditions, 566Separation of variables, 396
Cartesian, 579–585conducting box, 579–581conducting plate, 581, 582quantum particle in a box, 582–584wave guides, 584, 585
cylindrical, 586–590conducting cylindrical can,
586–588current distribution, 589, 590cylindrical wave guide, 588, 589
spherical, 590–595Helmholtz equation, 593particle in a sphere, 593, 594plane wave expansion, 594, 595radial part, 591, 592
Separation theorem, 430–432Sequence, 9
Cauchy, 9complete orthonormal, 219
SeriesClebsch-Gordan, 754complex, 319–321Fourier, 265–276Fourier-Bessel, 587Laurent, 321–330Neumann, 653, 654SOLDE solutions, 436–446Taylor, 321–330vector, 215–220
Sesquilinear inner product, 31Set, 1–4
Cantor, 12compact, 519–523complement of, 2
countably infinite, 11element of, 1empty, 2intersection, 2matrices, 7natural numbers, 2partition of a, 4uncountable, 12union, 2universal, 2
Sharp map, 801, 902Signature of g, 813Similarity transformation, 148–151
orthonormal basis, 149Simple algebra, 76, 88, 90–92, 94, 126,
129, 852, 948, 999classification, 92–95
Simple arc, 309Simple character, 737Simple Lie algebra, 948Simple pole, 342Simple zero, 330Simultaneous diagonalizability, 185Simultaneous diagonalization, 185–188Singleton, 2Singular point, 301, 339, 354, 355
differential equation, 422irregular, 461isolated, 463regular, 461, 470removable, 342Sturm-Liouville equation, 572transformation, 644
Singularity, 301, 302, 324, 439, 637confluent HGDE, 479essential, 342Green’s function, 651isolated, 339, 342–344
classification, 342rational function, 343removable, 343, 355Schwarzschild solution, 1169
Skew-symmetry, 53, 793Skin depth, 589SL(V) as a Lie group, 916SL(V)
Lie algebra of, 924normal subgroup of GL(V), 711
Smooth arc, 309SOLDE, 421–425
adjoint, 434branch point, 464canonical basis, 463characteristic exponents, 465complex, 463–469confluent hypergeometric, 479constant coefficients, 446–449existence theorem, 440Frobenius method, 439–444homogeneous, 422
1202 Index
SOLDE (cont.)hypergeometric
Jacobi functions, 477hypergeometric function, 473indicial equation, 465integral equation of, 545Lagrange identity, 435normal form, 422power-series solutions, 436–446regular singular point, 464singular point, 422Sturm-Liouville systems, 569–573uniqueness theorem, 424variation of constants, 429WKB method, 450–453Wronskian, 425
SOLDO, 614Solid angle
m-dimensional, 646Solid-state physics, 275Space
Banach, 218complex coordinate, 21dual, 48factor, 24, 25, 77inner product, 31metric, 8–10
complete, 10projective, 4quotient, 24, 25real coordinate, 21square-integrable functions, 221target, 5vector, 19–29
Spacelike vector, 941Spacetime
spherically symmetric, 1168static, 1167stationary, 1167
Spacetime translation, 1070Span, 22Special linear group, 706Special orthogonal group, 706, 925
Lie algebra of, 925Special relativity, 808, 940, 975, 979,
1059Special unitary group, 706
Lie algebra of, 924Spectral decomposition
complex, 177–188orthogonal operator, 201real, 191–205real normal operator, 198–205symmetric operator, 193–198
Spectral decomposition theorem, 688Spectral theorem
compact hermitian, 530compact normal, 532compact operators, 527–534
Spectrumbounded operator, 522closure of, 521compact operator, 527Hilbert space operator, 517integral operator, 545linear operator, 517, 518permutation operator, 208
Spherical Bessel functions, 487, 593expansion of plane wave, 594
Spherical coordinatesmultidimensional, 645, 646
Spherical harmonics, 406–413, 970addition theorem, 412, 413, 974definition, 408expansion in terms of, 411, 412expansion of plane wave, 595, 698first few, 410
Spin representation, 1003faithful, 1003
Spin(μ, ν), 996Spinor, 995–1006
algebra Cνμ(R), 1001–1003
Spinor bundles, 1101Spinor space, 1003Spinoza, 791Split complex numbers, 847Square wave voltage, 269Square-integrable functions, 221–227Stabilizer, 918Standard basis, 23Standard horizontal vector field, 1121Standard model, 1079Static spacetime, 1167Stationary spacetime, 1167Steepest descent method, 382–388Step function, 231, 357, 684Stereographic projection
n-sphere, 865two-sphere, 862
Stirling approximation, 385Stokes’ Theorem, 899Stone-Weierstrass theorem, 222
generalized, 265Stress energy tensor, 1165Strictly positive operator, 117Strictly upper triangular matrices, 66Structure
complex, 45–48Structure constant, 78, 937, 939, 976, 984,
1093, 1095, 1113Lie algebra, 927
Structure equation, 1093Structure group
matrix, 1096, 1097Sturm, biography, 568Sturm-Liouville
operator, 566problem, 243, 674system, 411, 567, 569–573, 689
Index 1203
Sturm-Liouville (cont.)asymptotic behavior, 573–577completeness, 577eigensolutions, 567eigenvalues, 568expansion in eigenfunctions,
577–579large argument, 577large eigenvalues, 573–576regular, 567singular, 572
Subalgebra, 64, 73–78Subgroup, 705–713
conjugate, 707generated by a subset, 707normal, 709trivial, 706
Submanifold, 863open, 863
Subset, 2bounded, 520closed, 520convex, 528dense, 520open, 520proper, 2
Subspace, 22–24invariant, 44, 127, 169–172, 175, 177,
192, 193, 198, 402, 530, 728,731, 733, 734, 738, 740, 749,758, 840, 955, 959, 967, 969,977, 989
nondegenerate, 810stable, 99, 127, 989
Sumdirect, 25–28
Superposition principlelinear DEs, 422
Surjective map, 5Symmetric algebra, 791Symmetric bilinear form, 804
classification, 807definite, 807indefinite, 807index of, 807inner product, 805negative definite, 807negative semidefinite, 807nondegenerate, 805positive definite, 807positive semidefinite, 807semidefinite, 807
Symmetric group, 704, 715–720characters
graphical construction, 767–771cycle, 716identical particles, 774irreducible representation of, 772permutation
parity of, 718representation, 761–776
analytic construction, 761–763antisymmetric, 732graphical construction, 764–767products, 774–776Young operators, 771–774
transposition, 717Symmetric homomorphism, 705Symmetric operator
extremum problem, 197spectral decomposition, 193–198
Symmetric product, 791Symmetrizer, 790Symmetry, 3, 8
algebraic equations, 1009–1014calculus of variations, 1062–1065conservation laws, 1065–1069
classical field theory, 1069–1073differential equations, 1014–1024first-order ODEs, 1037–1039heat equation, 1030–1034higher-order ODEs, 1039, 1040multiparameter, 1040–1043tensors, 789–794wave equation, 1034–1036
Symmetry groupdefining equations, 1030of a subset, 1009of a system of DEs, 1017projectable, 1017transform of a function, 1016variational, 1062
Symplectic algebra, 939Symplectic charts, 902Symplectic form, 801, 902
rank of, 801Symplectic geometry, 51, 901–909, 1079
conservation of energy, 906Symplectic group, 707, 803, 939Symplectic manifold, 902Symplectic map, 801, 902Symplectic matrix, 804Symplectic structure, 902Symplectic transformation, 801Symplectic vector space, 801–804
canonical basis of, 802Hamiltonian dynamics, 803
TTangent bundle, 877Tangent space, 869Tangent vector, 868
manifold, 866–872Tangential coordinates, 637Tangents to a curve
components, 874Target space, 5Taylor expansion, 104Taylor formula, 96
1204 Index
Taylor series, 321–330construction, 321
Tensor, 784classical definition, 787components of, 785contravariant, 784contravariant-antisymmetric, 793contravariant-symmetric, 789covariant, 784covariant-antisymmetric, 793covariant-symmetric, 789dual space, 782Levi-Civita, 799multilinear map, 782–789symmetric, 789symmetric product, 791symmetries, 789–794transformation law, 786types of, 784
Tensor algebra, 784Tensor bundle, 883Tensor field, 883, 887
crucial property of, 883curvature, 1125–1132manifold, 876–888torsion, 1125–1132
Tensor operatorirreducible, 756–758
Tensor product, 28, 29, 783, 784algebra, 68group representation
Clebsch-Gordan decomposition,753–756
of vector spaces, 751Tensorial form, 1092Test function, 233Theta function, 357Timelike vector, 941Topology, 8Torsion, 1125–1132Torsion form, 1122Torsion tensor field, 1125Total derivative, 1027Total divergence, 1060Total matrix algebra, 78–80, 92, 846, 850,
852, 997, 999Total space, 1080Trace, 160–162
and determinant, 161definition, 160log of determinant, 162relation to determinant, 160
Transformationsimilarity, 148–151
Transformation group, 704Transition function, 1081Transivity, 3Translation, 919Translation operator, 209Transpose of a matrix, 142
Traveling waves, 584Triangle inequality, 8, 36, 38, 133, 216,
301Trigonometric function
integration of, 350–352Trivial bundle, 1080Trivial homomorphism, 705Trivial representation, 732Trivial subgroup, 706Twin paradox
as a variational problem, 1060Twisted adjoint representation, 987
UUnbounded operator, 563–569Uncertainty principle, 133Uncertainty relation, 279Uncountable set, 12Union, 2Unit circle, 7Unital algebra, 63, 72Unital homomorphism, 72Unitary, 40Unitary group, 706
Lie algebra of, 924Unitary operator, 114–119Unitary representation, 730Universal set, 2Upper-triangular matrix, 66, 83, 175, 176
VVandermonde, biography, 153Variational derivative, 1051Variational problem, 1053–1060
twin paradox, 1060Variational symmetry group, 1062Vector, 19
Cartesian, 19component, 23dual of, 51infinite sum, 215–220isotropic, 808length, 36–38norm of, 36normal, 32null, 808orthogonal, 32tangent
manifold, 866–872Vector bundle, 1117Vector field, 877
as streamlines, 879complete, 881curl of, 889flow of a, 881fundamental, 1086gauge transformation of, 1104Hamiltonian, 905horizontal, 1087
Index 1205
Vector field (cont.)integral curve of, 879Killing, 1155–1159left-invariant, 920Lie algebra of, 879manifold, 877–882standard horizontal, 1121vertical, 1087
Vector potential, 3, 1099Vector space, 8, 19–29
automorphism, 43basis
components in a, 23basis of a, 23complete, 216complex, 20definition, 19dual, 48endomorphism of a, 39finite-dimension
criterion for, 522finite-dimensional, 23indefinite inner product
orthonormal basis, 812–819subspaces, 809–812
isomorphism, 43linear operator on a, 39Minkowski, 815normed, 36
compact subset of, 522operator on a, 39orientation, 800, 801oriented, 800real, 20self-dual, 805semi-Euclidean, 815symplectic, 801–804
Vertical vector field, 1087Volterra, biography, 545Volterra equation, 543Volume element, 801
relative to an inner product, 816Von Humboldt, 246, 666, 792Von Neumann, 981
biography, 532
WWave equation, 395, 584
hyperbolic, 642symmetry group, 1034–1036
Wave guide, 584cylindrical, 588, 589rectangular, 584, 585, 600
Weber-Hermite equation, 487Wedderburn decomposition, 92Wedge product, 794Weierstrass, 10, 36, 366, 640, 792, 946
biography, 523Weight function, 32Weyl, 799, 946, 1015, 1070
biography, 956Weyl basis, 938, 947Weyl operator, 955Wigner, 236, 1015
biography, 981Wigner formula, 972Wigner-Eckart theorem, 758Wigner-Seitz cell, 276WKB method, 450–453
connection formulas, 451Wordsworth, 907Wronski, biography, 425Wronskian, 425–432, 567
YYoung, 957Young antisymmetrizer, 772Young frame, 765, 772
negative application, 768positive application, 768regular application, 767
Young operator, 771–774, 963Young pattern, 765Young symmetrizer, 772Young tableaux, 766, 964
horizontal permutation, 772regular graphs, 766vertical permutation, 772
Yukawa potential, 282
ZZero of order k, 329