A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
Acknowledgements......Page 15
Dedication......Page 17
1 Sets and structures......Page 19
1.1 Sets and logic......Page 20
Unions and intersections......Page 23
Relations......Page 25
Equivalence relations......Page 26
Order relations and posets......Page 27
1.4 Mappings......Page 28
Surjective, injective and bijective maps......Page 29
Countable sets......Page 31
Uncountable sets......Page 32
The continuum hypothesis and axiom of choice......Page 34
1.6 Structures......Page 35
Algebraic structures......Page 36
Geometric structures......Page 37
Dynamical systems......Page 39
1.7 Category theory......Page 41
References......Page 44
2.1 Elements of group theory......Page 45
2.2 Transformation and permutation groups......Page 48
Linear transformations......Page 53
Matrix groups......Page 55
Homomorphisms......Page 58
Automorphisms and conjugacy classes......Page 60
Cosets......Page 63
Factor groups......Page 64
Kernel of a homomorphism......Page 65
2.6 Group actions......Page 67
2.7 Symmetry groups......Page 70
References......Page 76
3.1 Rings and fields......Page 77
3.2 Vector spaces......Page 78
3.3 Vector space homomorphisms......Page 81
3.4 Vector subspaces and quotient spaces......Page 84
Complementary subspaces and quotient spaces......Page 85
Images and kernels of linear maps......Page 88
Subspace spanned by a set......Page 90
Basis of a vector space......Page 91
Matrix of a linear operator......Page 94
Basis extension theorem......Page 97
Summation convention......Page 99
Basis transformations......Page 101
Linear functionals......Page 106
The dual space of a vector space......Page 108
Dual of the dual......Page 109
Transformation law of covector components......Page 111
References......Page 114
4 Linear operators and matrices......Page 116
Invariant subspaces......Page 117
Eigenvectors and eigenvalues......Page 118
Characteristic equation......Page 119
Minimal annihilating polynomial......Page 122
Block diagonal form......Page 125
Nilpotent operators......Page 128
Jordan canonical form......Page 131
4.3 Linear ordinary differential equations......Page 134
Two-dimensional autonomous systems......Page 136
4.4 Introduction to group representation theory......Page 138
Irreducible representations......Page 139
Schur’s lemma......Page 142
References......Page 143
5.1 Real inner product spaces......Page 144
Orthonormal bases......Page 146
5.2 Complex inner product spaces......Page 151
Norm of a vector......Page 153
Orthonormal bases......Page 155
Unitary transformations......Page 157
5.3 Representations of finite groups......Page 159
Orthogonality relations......Page 161
References......Page 166
6.1 Algebras and ideals......Page 167
6.2 Complex numbers and complex structures......Page 170
Complexification of a real vector space......Page 172
Complex structure on a vector space......Page 173
Quaternions......Page 175
Clifford algebras......Page 176
Multivectors......Page 178
Exterior product......Page 181
Properties of exterior product......Page 182
6.5 Lie algebras and Lie groups......Page 184
Lie algebras......Page 185
Matrix Lie groups......Page 187
One-parameter subgroups......Page 189
Complex Lie algebras......Page 190
References......Page 195
Free vector spaces......Page 196
The tensor product V Circled Times W......Page 197
Dual representation of tensor product......Page 199
Free associative algebras......Page 200
Grassmann algebra as a factor algebra of free algebras......Page 201
Multilinear maps and tensor spaces of type (r, s)......Page 204
Covariant tensors of degree 2......Page 205
Contravariant tensors of degree 2......Page 208
Mixed tensors......Page 210
7.3 Basis representation of tensors......Page 211
Tensor product......Page 212
Change of basis......Page 213
Contraction......Page 216
Raising and lowering indices......Page 218
Symmetries......Page 219
References......Page 221
8.1 r-Vectors and r-forms......Page 222
The antisymmetrization operator A......Page 223
8.2 Basis representation of r-vectors......Page 224
8.3 Exterior product......Page 226
Simple Rho-vectors and subspaces......Page 229
8.4 Interior product......Page 231
n-Vectors and n-forms......Page 233
Transformation laws of n-vectors and n-forms......Page 234
Oriented vector spaces......Page 235
Small Element Of-Symbol identities......Page 237
Inner product of Rho-vectors......Page 238
The Hodge star operator......Page 240
References......Page 245
9.1 Minkowski space-time......Page 246
Poincaré and Lorentz transformations......Page 247
Affine geometry......Page 248
Minkowski space and 4-tensors......Page 249
Special Lorentz transformations......Page 253
Relativity of time, length and velocity......Page 254
World-lines and proper time......Page 257
Relativistic particle dynamics......Page 260
4-Tensor fields......Page 262
Electromagnetism......Page 263
Potentials and gauge transformations......Page 265
Conservation of charge......Page 269
Energy–stress tensors......Page 270
References......Page 272
10.1 Euclidean topology......Page 273
10.2 General topological spaces......Page 275
10.3 Metric spaces......Page 282
Induced topologies and topological products......Page 283
Topology by identification......Page 285
10.5 Hausdorff spaces......Page 287
10.6 Compact spaces......Page 289
10.7 Connected spaces......Page 291
10.8 Topological groups......Page 294
Connected component of the identity......Page 296
10.9 Topological vector spaces......Page 297
Banach spaces......Page 299
References......Page 304
Measurable spaces......Page 305
Measurable functions......Page 307
11.2 Measure spaces......Page 310
Lebesgue measure......Page 312
11.3 Lebesgue integration......Page 319
Lebesgue’s dominated convergence theorem......Page 323
References......Page 325
12 Distributions......Page 326
Spaces of test functions......Page 327
Regular distributions......Page 329
12.2 Operations on distributions......Page 332
Differentiation of distributions......Page 333
Change of variable in Delta-functions......Page 336
12.3 Fourier transforms......Page 338
Poisson’s equation......Page 341
Green’s function for the wave equation......Page 344
References......Page 347
13.1 Definitions and examples......Page 348
Orthonormal bases......Page 353
Orthogonal subspaces......Page 359
Riesz representation theorem......Page 360
13.4 Bounded linear operators......Page 362
Adjoint operators......Page 364
Hermitian operators......Page 366
Unitary operators......Page 367
Eigenvectors......Page 369
Spectrum of a bounded operator......Page 371
Spectral theory of hermitian operators......Page 372
13.6 Unbounded operators......Page 375
Self-adjoint and symmetric operators......Page 376
Spectral theory of unbounded operators......Page 380
References......Page 382
Photon polarization experiments......Page 384
The Hilbert space of states......Page 387
Observables......Page 388
Unbounded operators in quantum mechanics......Page 393
14.2 Quantum dynamics......Page 397
The Heisenberg picture......Page 398
Correspondence with classical mechanics and wave mechanics......Page 400
Harmonic oscillator......Page 402
Angular momentum......Page 403
14.3 Symmetry transformations......Page 405
Infinitesimal generators......Page 408
Conserved quantities......Page 410
Discrete symmetries......Page 411
Identical particles......Page 413
14.4 Quantum statistical mechanics......Page 415
Density operator......Page 416
Ensembles......Page 419
Systems of identical particles......Page 422
References......Page 427
15 Differential geometry......Page 428
15.1 Differentiable manifolds......Page 429
15.2 Differentiable maps and curves......Page 433
Tangent vectors......Page 435
Cotangent and tensor spaces......Page 438
Vector and tensor fields......Page 440
Coordinate transformations......Page 441
Tensor bundles......Page 442
The tangent map and pullback of a map......Page 444
Submanifolds......Page 447
Commutators......Page 450
Integral curves and flows......Page 451
Lie derivative......Page 454
15.6 Distributions and Frobenius theorem......Page 458
References......Page 463
16.1 Differential forms and exterior derivative......Page 465
16.2 Properties of exterior derivative......Page 469
16.3 Frobenius theorem: dual form......Page 472
16.4 Thermodynamics......Page 475
Second law of thermodynamics......Page 478
Absolute entropy and temperature......Page 480
Calculus of variations......Page 482
Lagrangian mechanics......Page 485
Hamiltonian mechanics......Page 492
Connection between Lagrangian and Hamiltonian mechanics......Page 495
References......Page 498
17 Integration on manifolds......Page 499
17.1 Partitions of unity......Page 500
17.2 Integration of n-forms......Page 502
17.3 Stokes’ theorem......Page 504
Regular domains......Page 507
Ordered simplices and chains in Euclidean space......Page 511
Simplicial homology on manifolds......Page 514
De Rham cohomology groups and duality......Page 515
17.5 The Poincaré lemma......Page 518
Electrodynamics......Page 520
References......Page 522
18.1 Linear connections and geodesics......Page 524
Coordinate transformations......Page 527
18.2 Covariant derivative of tensor fields......Page 528
Torsion tensor......Page 530
Curvature tensor......Page 531
18.4 Pseudo-Riemannian manifolds......Page 534
Riemannian connection......Page 536
Geodesic coordinates......Page 538
18.5 Equation of geodesic deviation......Page 540
18.6 The Riemann tensor and its symmetries......Page 542
Bianchi identities......Page 544
18.7 Cartan formalism......Page 545
Pseudo-Riemannian spaces in Cartan formalism......Page 547
Locally flat spaces......Page 550
The principle of equivalence......Page 552
Postulates of general relativity......Page 554
Measurement of the curvature tensor......Page 555
The linearized approximation......Page 557
The Schwarzschild solution......Page 560
18.9 Cosmology......Page 566
18.10 Variation principles in space-time......Page 571
Hilbert action......Page 572
Energy–stress tensor of fields......Page 574
References......Page 575
Left-invariant vector fields......Page 577
Lie algebra of a Lie group......Page 578
Maurer–Cartan relations......Page 580
19.2 The exponential map......Page 582
Exponential map......Page 584
19.3 Lie subgroups......Page 587
Matrix Lie groups......Page 588
19.4 Lie groups of transformations......Page 590
Normal subgroups......Page 592
Induced vector fields......Page 593
19.5 Groups of isometries......Page 596
Maximal symmetries and cosmology......Page 597
Spherical symmetry......Page 601
References......Page 603
Bibliography......Page 605
Index......Page 607