Applied Mathematical Methods in Theoretical Physics

Cover......Page 1
Title page......Page 4
Edition notice......Page 5
Dedication......Page 6
Contents......Page 7
Preface......Page 10
Introduction......Page 13
1.1 Function Spaces......Page 19
1.2 Orthonormal System of Functions......Page 21
1.3 Linear Operators......Page 23
1.4 Eigenvalues and Eigenfunctions......Page 25
1.5 The Fredholm Alternative......Page 27
1.6 Self-Adjoint Operators......Page 30
1.7 Green’s Functions for Differential Equations......Page 32
1.8 Review of Complex Analysis......Page 36
1.9 Review of Fourier Transform......Page 43
2.1 Introduction to Integral Equations......Page 48
2.2 Relationship of Integral Equations with Differential Equations and Green’s Functions......Page 54
2.3 Sturm–Liouville System......Page 60
2.4 Green’s Function for Time-Dependent Scattering Problem......Page 64
2.5 Lippmann–Schwinger Equation......Page 68
2.6 Scalar Field Interacting with Static Source......Page 79
2.7 Problems for Chapter 2......Page 84
3.1 Iterative Solution to Volterra Integral Equation of the Second Kind......Page 122
3.2 Solvable Cases of the Volterra Integral Equation......Page 125
3.3 Problems for Chapter 3......Page 129
4.1 Iterative Solution to the Fredholm Integral Equation of the Second Kind......Page 133
4.2 Resolvent Kernel......Page 136
4.3 Pincherle–Goursat Kernel......Page 139
4.4 Fredholm Theory for a Bounded Kernel......Page 143
4.5 Solvable Example......Page 150
4.6 Fredholm Integral Equation with a Translation Kernel......Page 152
4.8 Problems for Chapter 4......Page 159
5.1 Real and Symmetric Matrix......Page 169
5.2 Real and Symmetric Kernel......Page 171
5.3 Bounds on the Eigenvalues......Page 182
5.4 Rayleigh Quotient......Page 185
5.5 Completeness of Sturm–Liouville Eigenfunctions......Page 188
5.6 Generalization of Hilbert–Schmidt Theory......Page 190
5.7 Generalization of the Sturm–Liouville System......Page 197
5.8 Problems for Chapter 5......Page 203
6.1 Hilbert Problem......Page 209
6.2 Cauchy Integral Equation of the First Kind......Page 213
6.3 Cauchy Integral Equation of the Second Kind......Page 217
6.4 Carleman Integral Equation......Page 221
6.5 Dispersion Relations......Page 227
6.6 Problems for Chapter 6......Page 234
7.1 The Wiener–Hopf Method for Partial Differential Equations......Page 238
7.2 Homogeneous Wiener–Hopf Integral Equation of the Second Kind......Page 252
7.3 General Decomposition Problem......Page 267
7.4 Inhomogeneous Wiener–Hopf Integral Equation of the Second Kind......Page 276
7.5 Toeplitz Matrix and Wiener–Hopf Sum Equation......Page 287
7.6 Wiener–Hopf Integral Equation of the First Kind and Dual Integral Equations......Page 296
7.7 Problems for Chapter 7......Page 300
8.1 Nonlinear Integral Equation of the Volterra Type......Page 310
8.2 Nonlinear Integral Equation of the Fredholm Type......Page 314
8.3 Nonlinear Integral Equation of the Hammerstein Type......Page 318
8.4 Problems for Chapter 8......Page 320
9.1 Historical Background......Page 324
9.2 Examples......Page 328
9.3 Euler Equation......Page 329
9.4 Generalization of the Basic Problems......Page 334
9.5 More Examples......Page 338
9.6 Differential Equations, Integral Equations, and Extremization of Integrals......Page 341
9.7 The Second Variation......Page 345
9.8 Weierstrass–Erdmann Corner Relation......Page 360
9.9 Problems for Chapter 9......Page 364
10.1 Hamilton–Jacobi Equation and Quantum Mechanics......Page 367
10.2 Feynman’s Action Principle in Quantum Theory......Page 375
10.3 Schwinger’s Action Principle in Quantum Theory......Page 382
10.4 Schwinger–Dyson Equation in Quantum Field Theory......Page 385
10.5 Schwinger–Dyson Equation in Quantum Statistical Mechanics......Page 399
10.6 Feynman’s Variational Principle......Page 409
10.7 Poincare Transformation and Spin......Page 421
10.8 Conservation Laws and Noether’s Theorem......Page 425
10.9 Weyl’s Gauge Principle......Page 432
10.10 Path Integral Quantization of Gauge Field I......Page 451
10.11 Path Integral Quantization of Gauge Field II......Page 468
10.12 BRST Invariance and Renormalization......Page 482
10.13 Asymptotic Disaster in QED......Page 489
10.14 Asymptotic Freedom in QCD......Page 493
10.15 Renormalization Group Equations......Page 501
10.16 Standard Model......Page 513
10.17 Lattice Gauge Field Theory and Quark Confinement......Page 532
10.18 WKB Approximation in Path Integral Formalism......Page 537
10.19 Hartree–Fock Equation......Page 540
10.20 Problems for Chapter 10......Page 543
References......Page 581
Index......Page 587