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[Axle 96] Axler, S.: Linear Algebra Done Right. Springer, Berlin (1996)[Baru 86] Barut, A., Raczka, R.: Theory of Group Representations and Applications.

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versity Press, Cambridge (1995)[Zeid 95] Zeidler, E.: Applied Functional Analysis. Springer, Berlin (1995)

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