Sadri HASSANI - Mathematical Physics: A Modern Introduction to Its Foundations, Second Edition


CONTENTS


1 Mathematical Preliminaries ................... 1

1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Equivalence Relations ................ 3

1.2 Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Metric Spaces ........................ 8

1.4 Cardinality .......................... 10

1.5 Mathematical Induction ................... 12

1.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 14



Part I Finite-Dimensional Vector Spaces


2 Vectors and Linear Maps .................... 19

2.1 Vector Spaces ........................ 19

2.1.1 Subspaces . . . ................... 22

2.1.2 Factor Space . . ................... 24

2.1.3 Direct Sums . . . . . . . . . . . . . . . . . . . . . 25

2.1.4 Tensor Product of Vector Spaces . ......... 28

2.2 Inner Product ......................... 29

2.2.1 Orthogonality . ................... 32

2.2.2 The Gram-Schmidt Process ............. 33

2.2.3 The Schwarz Inequality ............... 35

2.2.4 Length of a Vector .................. 36

2.3 Linear Maps ......................... 38

2.3.1 Kernel of a Linear Map ............... 41

2.3.2 Linear Isomorphism ................. 43

2.4 Complex Structures . . . . . . . . . . . . . . . . . . . . . 45

2.5 Linear Functionals . . . ................... 48

2.6 Multilinear Maps ....................... 53

2.6.1 Determinant of a Linear Operator . ......... 55

2.6.2 Classical Adjoint . . . . . . . . . . . . . . . . . . . 56

2.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 57


3 Algebras .............................. 63

3.1 From Vector Space to Algebra ................ 63

3.1.1 General Properties .................. 64

3.1.2 Homomorphisms . . . . . . . . . . . . . . . . . . . 70

3.2 Ideals . ............................ 73

3.2.1 Factor Algebras ................... 77

3.3 Total Matrix Algebra ..................... 78

3.4 Derivation of an Algebra ................... 80

3.5 Decomposition of Algebras ................. 83

3.5.1 The Radical . . . . . . . . . . . . . . . . . . . . . 84

3.5.2 Semi-simple Algebras ................ 88

3.5.3 Classification of Simple Algebras .......... 92

3.6 Polynomial Algebra ..................... 95

3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 97


4 Operator Algebra ......................... 101

4.1 Algebra of End(V) ...................... 101

4.1.1 Polynomials of Operators . . ............ 102

4.1.2 Functions of Operators . . . ............ 104

4.1.3 Commutators . . . . . . . . . . . . . . . . . . . . . 106

4.2 Derivatives of Operators . . . . . . . . . . . . . . . . . . . 107

4.3 Conjugation of Operators . . . . . . . . . . . . . . . . . . 113

4.3.1 Hermitian Operators ................. 114

4.3.2 Unitary Operators . . . . . . . . . . . . . . . . . . 118

4.4 Idempotents . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.4.1 Projection Operators . . . . . . . . . . . . . . . . . 120

4.5 Representation of Algebras ................. 125

4.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 131


5 Matrices .............................. 137

5.1 Representing Vectors and Operators . . . . . . . . . . . . . 137

5.2 Operations on Matrices . . . . . . . . . . . . . . . . . . . 142

5.3 Orthonormal Bases ...................... 146

5.4 Change of Basis . ...................... 148

5.5 Determinant of a Matrix ................... 151

5.5.1 Matrix of the Classical Adjoint . . . . . . . . . . . 152

5.5.2 Inverse of a Matrix . . . . . . . . . . . . . . . . . . 155

5.5.3 Dual Determinant Function . ............ 158

5.6 The Trace . . . . . . . . . . . . . . . . . . . . . . . . . . 160

5.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 163


6 Spectral Decomposition ..................... 169

6.1 Invariant Subspaces ..................... 169

6.2 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . 172

6.3 Upper-Triangular Representations . . ............ 175

6.4 Complex Spectral Decomposition . . ............ 177

6.4.1 Simultaneous Diagonalization ............ 185

6.5 Functions of Operators .................... 188

6.6 Real Spectral Decomposition ................ 191

6.6.1 The Case of Symmetric Operators . . . . . . . . . . 193

6.6.2 The Case of Real Normal Operators . . . . . . . . . 198

6.7 Polar Decomposition ..................... 205

6.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 208



Part II Infinite-Dimensional Vector Spaces


7 Hilbert Spaces .......................... 215

7.1 The Question of Convergence ................ 215

7.2 The Space of Square-Integrable Functions ......... 221

7.2.1 Orthogonal Polynomials ............... 222

7.2.2 Orthogonal Polynomials and Least Squares ..... 225

7.3 Continuous Index . . . ................... 227

7.4 Generalized Functions . ................... 233

7.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 237


8 Classical Orthogonal Polynomials ................ 241

8.1 General Properties . . . ................... 241

8.2 Classification . . . . . . . . . . . . . . . . . . . . . . . . . 244

8.3 Recurrence Relations . ................... 245

8.4 Details of Specific Examples ................ 248

8.4.1 Hermite Polynomials ................ 248

8.4.2 Laguerre Polynomials ................ 249

8.4.3 Legendre Polynomials ................ 250

8.4.4 Other Classical Orthogonal Polynomials ...... 252

8.5 Expansion in Terms of Orthogonal Polynomials ...... 254

8.6 Generating Functions . ................... 257

8.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 258


9 Fourier Analysis ......................... 265

9.1 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . 265

9.1.1 The Gibbs Phenomenon ............... 273

9.1.2 Fourier Series in Higher Dimensions ........ 275

9.2 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 276

9.2.1 Fourier Transforms and Derivatives . . . . . . . . . 284

9.2.2 The Discrete Fourier Transform . . . . . . . . . . . 286

9.2.3 Fourier Transform of a Distribution . . . . . . . . . 287

9.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 288



Part III Complex Analysis


10 Complex Calculus ........................ 295

10.1 Complex Functions . . ................... 295

10.2 Analytic Functions . . . ................... 297

10.3 Conformal Maps . . . . . . . . . . . . . . . . . . . . . . . 304

10.4 Integration of Complex Functions .............. 309

10.5 Derivatives as Integrals . . . . . . . . . . . . . . . . . . . 315

10.6 Infinite Complex Series . . . . . . . . . . . . . . . . . . . 319

10.6.1 Properties of Series ................. 319

10.6.2 Taylor and Laurent Series . . . . . . . . . . . . . . 321

10.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 330


11 Calculus of Residues ....................... 339

11.1 Residues ........................... 339

11.2 Classification of Isolated Singularities . . ......... 342

11.3 Evaluation of Definite Integrals . . . . . . . . . . . . . . . 344

11.3.1 Integrals of Rational Functions ........... 345

11.3.2 Products of Rational and Trigonometric Functions . 348

11.3.3 Functions of Trigonometric Functions ....... 350

11.3.4 Some Other Integrals . . . . . . . . . . . . . . . . 352

11.3.5 Principal Value of an Integral . . . . . . . . . . . . 354

11.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 359


12 Advanced Topics ......................... 363

12.1 Meromorphic Functions ................... 363

12.2 Multivalued Functions .................... 365

12.2.1 Riemann Surfaces .................. 366

12.3 Analytic Continuation . . . . . . . . . . . . . . . . . . . . 372

12.3.1 The Schwarz Reflection Principle .......... 374

12.3.2 Dispersion Relations . . . . . . . . . . . . . . . . . 376

12.4 The Gamma and Beta Functions . . . ............ 378

12.5 Method of Steepest Descent ................. 382

12.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 388



Part IV Differential Equations


13 Separation of Variables in Spherical Coordinates ....... 395

13.1 PDEs of Mathematical Physics . . . . . . . . . . . . . . . 395

13.2 Separation of the Angular Part ................ 398

13.3 Construction of Eigenvalues of L2 .............. 401

13.4 Eigenvectors of L2: Spherical Harmonics .......... 406

13.4.1 Expansion of Angular Functions .......... 411

13.4.2 Addition Theorem for Spherical Harmonics . . . . 412

13.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 413


14 Second-Order Linear Differential Equations .......... 417

14.1 General Properties of ODEs ................. 417

14.2 Existence/Uniqueness for First-Order DEs ......... 419

14.3 General Properties of SOLDEs . . . ............ 421

14.4 The Wronskian . . . . . . . . . . . . . . . . . . . . . . . . 425

14.4.1 A Second Solution to the HSOLDE ......... 426

14.4.2 The General Solution to an ISOLDE ........ 428

14.4.3 Separation and Comparison Theorems ....... 430

14.5 Adjoint Differential Operators . . . . . . . . . . . . . . . . 433

14.6 Power-Series Solutions of SOLDEs . . . . . . . . . . . . . 436

14.6.1 Frobenius Method of Undetermined Coefficients . . 439

14.6.2 Quantum Harmonic Oscillator ............ 444

14.7 SOLDEs with Constant Coefficients . . . . . . . . . . . . 446

14.8 The WKB Method . . . . . . . . . . . . . . . . . . . . . . 450

14.8.1 Classical Limit of the Schrödinger Equation . . . . 452

14.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 453


15 Complex Analysis of SOLDEs .................. 459

15.1 Analytic Properties of Complex DEs ............ 460

15.1.1 Complex FOLDEs . . . . . . . . . . . . . . . . . . 460

15.1.2 The Circuit Matrix . . . . . . . . . . . . . . . . . . 462

15.2 Complex SOLDEs . . . . . . . . . . . . . . . . . . . . . . 463

15.3 Fuchsian Differential Equations ............... 469

15.4 The Hypergeometric Function ................ 473

15.5 Confluent Hypergeometric Functions . . . ......... 478

15.5.1 Hydrogen-Like Atoms ................ 480

15.5.2 Bessel Functions ................... 482

15.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 485


16 Integral Transforms and Differential Equations ........ 493

16.1 Integral Representation of the Hypergeometric Function . . 494

16.1.1 Integral Representation of the Confluent Hypergeometric Function .............. 497

16.2 Integral Representation of Bessel Functions ......... 498

16.2.1 Asymptotic Behavior of Bessel Functions ..... 502

16.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 505



Part V Operators on Hilbert Spaces


17 Introductory Operator Theory ................. 511

17.1 From Abstract to Integral and Differential Operators . . . . 511

17.2 Bounded Operators in Hilbert Spaces . . . ......... 513

17.2.1 Adjoints of Bounded Operators . . ......... 517

17.3 Spectra of Linear Operators ................. 517

17.4 Compact Sets ........................ 519

17.4.1 Compactness and Infinite Sequences ........ 521

17.5 Compact Operators . . ................... 523

17.5.1 Spectrum of Compact Operators . ......... 527

17.6 Spectral Theorem for Compact Operators . ......... 527

17.6.1 Compact Hermitian Operator . . . ......... 529

17.6.2 Compact Normal Operator ............. 531

17.7 Resolvents . . . . . . . . . . . . . . . . . . . . . . . . . . 534

17.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 539


18 Integral Equations ........................ 543

18.1 Classification . . . . . . . . . . . . . . . . . . . . . . . . . 543

18.2 Fredholm Integral Equations ................. 549

18.2.1 Hermitian Kernel .................. 552

18.2.2 Degenerate Kernels ................. 556

18.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 560


19 Sturm-Liouville Systems ..................... 563

19.1 Compact-Resolvent Unbounded Operators ......... 563

19.2 Sturm-Liouville Systems and SOLDEs . . ......... 569

19.3 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . 573

19.3.1 Large Eigenvalues . . . . . . . . . . . . . . . . . . 573

19.3.2 Large Argument . . . . . . . . . . . . . . . . . . . 577

19.4 Expansions in Terms of Eigenfunctions . . ......... 577

19.5 Separation in Cartesian Coordinates ............. 579

19.5.1 Rectangular Conducting Box . . . ......... 579

19.5.2 Heat Conduction in a Rectangular Plate ....... 581

19.5.3 Quantum Particle in a Box . . ............ 582

19.5.4 Wave Guides . . . . . . . . . . . . . . . . . . . . . 584

19.6 Separation in Cylindrical Coordinates ............ 586

19.6.1 Conducting Cylindrical Can . ............ 586

19.6.2 Cylindrical Wave Guide . . . . . . . . . . . . . . . 588

19.6.3 Current Distribution in a Circular Wire . . . . . . . 589

19.7 Separation in Spherical Coordinates . ............ 590

19.7.1 Radial Part of Laplace’s Equation .......... 591

19.7.2 Helmholtz Equation in Spherical Coordinates . . . 593

19.7.3 Quantum Particle in a Hard Sphere ......... 593

19.7.4 Plane Wave Expansion ................ 594

19.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 595



Part VI Green’s Functions


20 Green’s Functions in One Dimension .............. 605

20.1 Calculation of Some Green’s Functions ........... 606

20.2 Formal Considerations . . . . . . . . . . . . . . . . . . . . 610

20.2.1 Second-Order Linear DOs . . ............ 614

20.2.2 Self-adjoint SOLDOs . . . . . . . . . . . . . . . . 616

20.3 Green’s Functions for SOLDOs . . . ............ 617

20.3.1 Properties of Green’s Functions ........... 619

20.3.2 Construction and Uniqueness of Green’s Functions . 62

 20.3.3 Inhomogeneous BCs ................. 626

20.4 Eigenfunction Expansion .................. 630

20.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 632


21 Multidimensional Green’s Functions: Formalism ....... 635

21.1 Properties of Partial Differential Equations ......... 635

21.1.1 Characteristic Hypersurfaces ............ 636

21.1.2 Second-Order PDEs in m Dimensions . . . . . . . 640

21.2 Multidimensional GFs and Delta Functions ......... 643

21.2.1 Spherical Coordinates in m Dimensions . . . . . . 645

21.2.2 Green’s Function for the Laplacian ......... 647

21.3 Formal Development . . . . . . . . . . . . . . . . . . . . . 648

21.3.1 General Properties .................. 648

21.3.2 Fundamental (Singular) Solutions .......... 649

21.4 Integral Equations and GFs . . . . . . . . . . . . . . . . . 652

21.5 Perturbation Theory ..................... 655

21.5.1 The Nondegenerate Case . . ............ 659

21.5.2 The Degenerate Case ................ 660

21.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 661


22 Multidimensional Green’s Functions: Applications ...... 665

22.1 Elliptic Equations ...................... 665

22.1.1 The Dirichlet Boundary Value Problem ....... 665

22.1.2 The Neumann Boundary Value Problem ...... 671

22.2 Parabolic Equations . . . . . . . . . . . . . . . . . . . . . 673

22.3 Hyperbolic Equations .................... 678

22.4 The Fourier Transform Technique . . ............ 680

22.4.1 GF for the m-Dimensional Laplacian ........ 681

22.4.2 GF for the m-Dimensional Helmholtz Operator . . . 682

22.4.3 GF for the m-Dimensional Diffusion Operator . . . 684

22.4.4 GF for the m-Dimensional Wave Equation ..... 685

22.5 The Eigenfunction Expansion Technique . ......... 688

22.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 693



Part VII Groups and Their Representations


23 Group Theory ........................... 701

23.1 Groups ............................ 702

23.2 Subgroups .......................... 705

23.2.1 Direct Products ................... 712

23.3 Group Action . . . . . . . . . . . . . . . . . . . . . . . . 713

23.4 The Symmetric Group Sn .................. 715

23.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 720


24 Representation of Groups .................... 725

24.1 Definitions and Examples . . . . . . . . . . . . . . . . . . 725

24.2 Irreducible Representations ................. 728

24.3 Orthogonality Properties ................... 732

24.4 Analysis of Representations . . . . . . . . . . . . . . . . . 737

24.5 Group Algebra ........................ 740

24.5.1 Group Algebra and Representations ......... 740

24.6 Relationship of Characters to Those of a Subgroup ..... 743

24.7 Irreducible Basis Functions ................. 746

24.8 Tensor Product of Representations ............. 750

24.8.1 Clebsch-Gordan Decomposition . . ......... 753

24.8.2 Irreducible Tensor Operators . . . ......... 756

24.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 758


25 Representations of the Symmetric Group ........... 761

25.1 Analytic Construction . . . . . . . . . . . . . . . . . . . . 761

25.2 Graphical Construction ................... 764

25.3 Graphical Construction of Characters . . . ......... 767

25.4 Young Operators ....................... 771

25.5 Products of Representations of Sn .............. 774

25.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 776



Part VIII Tensors and Manifolds


26 Tensors .............................. 781

26.1 Tensors as Multilinear Maps ................. 782

26.2 Symmetries of Tensors . . . . . . . . . . . . . . . . . . . . 789

26.3 Exterior Algebra ....................... 794

26.3.1 Orientation . . . . . . . . . . . . . . . . . . . . . . 800

26.4 Symplectic Vector Spaces .................. 801

26.5 Inner Product Revisited ................... 804

26.5.1 Subspaces . . . ................... 809

26.5.2 Orthonormal Basis .................. 812

26.5.3 Inner Product on Λp(V,U) ............. 819

26.6 The Hodge Star Operator .................. 820

26.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 823


27 Clifford Algebras ......................... 829

27.1 Construction of Clifford Algebras . . ............ 830

27.1.1 The Dirac Equation . . . . . . . . . . . . . . . . . 832

27.2 General Properties of the Clifford Algebra ......... 834

27.2.1 Homomorphism with Other Algebras ........ 837

27.2.2 The Canonical Element . . . ............ 838

27.2.3 Center and Anticenter ................ 839

27.2.4 Isomorphisms . . . . . . . . . . . . . . . . . . . . 842

27.3 General Classification of Clifford Algebras ......... 843

27.4 The Clifford Algebras Cνμ(ℝ) ................ 846

27.4.1 Classification of C0n(ℝ) and Cn0(ℝ) ......... 849

27.4.2 Classification of Cνμ(ℝ) ............... 851

27.4.3 The Algebra C13(ℝ) ................. 852

27.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 856


28 Analysis of Tensors ........................ 859

28.1 Differentiable Manifolds . . . . . . . . . . . . . . . . . . . 859

28.2 Curves and Tangent Vectors ................. 866

28.3 Differential of a Map . . . . . . . . . . . . . . . . . . . . 872

28.4 Tensor Fields on Manifolds . . . . . . . . . . . . . . . . . 876

28.4.1 Vector Fields . . . . . . . . . . . . . . . . . . . . . 877

28.4.2 Tensor Fields . . . . . . . . . . . . . . . . . . . . . 882

28.5 Exterior Calculus . . . . . . . . . . . . . . . . . . . . . . 888

28.6 Integration on Manifolds . . . . . . . . . . . . . . . . . . . 897

28.7 Symplectic Geometry . . . . . . . . . . . . . . . . . . . . 901

28.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 909



Part IX Lie Groups and Their Applications


29 Lie Groups and Lie Algebras .................. 915

29.1 Lie Groups and Their Algebras . . . ............ 915

29.1.1 Group Action . . . . . . . . . . . . . . . . . . . . 917

29.1.2 Lie Algebra of a Lie Group . ............ 920

29.1.3 Invariant Forms . . . . . . . . . . . . . . . . . . . 927

29.1.4 Infinitesimal Action . . . . . . . . . . . . . . . . . 928

29.1.5 Integration on Lie Groups . . ............ 935

29.2 An Outline of Lie Algebra Theory . . ............ 936

29.2.1 The Lie Algebras o(p,n − p) and p(p,n − p) ... 940

29.2.2 Operations on Lie Algebras . ............ 944

29.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 948


30 Representation of Lie Groups and Lie Algebras ........ 953

30.1 Representation of Compact Lie Groups ........... 953

30.2 Representation of the General Linear Group ........ 963

30.3 Representation of Lie Algebras . . . ............ 966

30.3.1 Representation of Subgroups of GL(V) ....... 967

30.3.2 Casimir Operators . . . . . . . . . . . . . . . . . . 969

30.3.3 Representation of so(3) and so(3, 1) ........ 972

30.3.4 Representation of the Poincaré Algebra ....... 975

30.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 983


31 Representation of Clifford Algebras .............. 987

31.1 The Clifford Group . . . . . . . . . . . . . . . . . . . . . 987

31.2 Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995

31.2.1 Pauli Spin Matrices and Spinors . . . . . . . . . . . 997

31.2.2 Spinors for Cνμ(ℝ) .................. 1001

31.2.3 C13(ℝ) Revisited . . . . . . . . . . . . . . . . . . . 1004

31.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006


32 Lie Groups and Differential Equations ............. 1009

32.1 Symmetries of Algebraic Equations ............. 1009

32.2 Symmetry Groups of Differential Equations ........ 1014

32.2.1 Prolongation of Functions .............. 1017

32.2.2 Prolongation of Groups ............... 1021

32.2.3 Prolongation of Vector Fields . . . . . . . . . . . . 1022

32.3 The Central Theorems . ................... 1024

32.4 Application to Some Known PDEs . . . . . . . . . . . . . 1029

32.4.1 The Heat Equation . . . . . . . . . . . . . . . . . . 1030

32.4.2 The Wave Equation . . . . . . . . . . . . . . . . . 1034

32.5 Application to ODEs . . . . . . . . . . . . . . . . . . . . . 1037

32.5.1 First-Order ODEs . . . . . . . . . . . . . . . . . . 1037

32.5.2 Higher-Order ODEs ................. 1039

32.5.3 DEs with Multiparameter Symmetries . . . . . . . 1040

32.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043


33 Calculus of Variations, Symmetries, and Conservation Laws . 1047

33.1 The Calculus of Variations . . . . . . . . . . . . . . . . . . 1047

33.1.1 Derivative for Hilbert Spaces . . . ......... 1047

33.1.2 Functional Derivative ................ 1050

33.1.3 Variational Problems ................. 1053

33.1.4 Divergence and Null Lagrangians . ......... 1060

33.2 Symmetry Groups of Variational Problems ......... 1062

33.3 Conservation Laws and Noether’s Theorem ......... 1065

33.4 Application to Classical Field Theory . . . ......... 1069

33.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073



Part X Fiber Bundles


34 Fiber Bundles and Connections ................. 1079

34.1 Principal Fiber Bundles ................... 1079

34.1.1 Associated Bundles ................. 1084

34.2 Connections in a PFB . ................... 1086

34.2.1 Local Expression for a Connection ......... 1087

34.2.2 Parallelism . . . . . . . . . . . . . . . . . . . . . . 1089

34.3 Curvature Form . . . . . . . . . . . . . . . . . . . . . . . 1091

34.3.1 Flat Connections ................... 1095

34.3.2 Matrix Structure Group . . . . . . . . . . . . . . . 1096

34.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097


35 Gauge Theories .......................... 1099

35.1 Gauge Potentials and Fields ................. 1099

35.1.1 Particle Fields . . . . . . . . . . . . . . . . . . . . 1101

35.1.2 Gauge Transformation ................ 1102

35.2 Gauge-Invariant Lagrangians ................ 1105

35.3 Construction of Gauge-Invariant Lagrangians ........ 1107

35.4 Local Equations . ...................... 1112

35.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115


36 Differential Geometry ...................... 1117

36.1 Connections in a Vector Bundle . . . ............ 1117

36.2 Linear Connections ..................... 1120

36.2.1 Covariant Derivative of Tensor Fields . . . . . . . . 1123

36.2.2 From Forms on P to Tensor Fields on M ...... 1125

36.2.3 Component Expressions . . . ............ 1128

36.2.4 General Basis .................... 1132

36.3 Geodesics .......................... 1137

36.3.1 Riemann Normal Coordinates . . . . . . . . . . . . 1138

36.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1140


37 Riemannian Geometry ...................... 1143

37.1 The Metric Connection ................... 1143

37.1.1 Orthogonal Bases .................. 1148

37.2 Isometries and Killing Vector Fields . ............ 1155

37.3 Geodesic Deviation and Curvature . ............ 1159

37.3.1 Newtonian Gravity . . . . . . . . . . . . . . . . . . 1161

37.4 General Theory of Relativity ................ 1163

37.4.1 Einstein’s Equation . . . . . . . . . . . . . . . . . 1163

37.4.2 Static Spherically Symmetric Solutions ....... 1167

37.4.3 Schwarzschild Geodesics . . ............ 1169

37.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174


References ............................... 1179


Index .................................. 1181


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