Physical mathematics

Unique in its clarity, examples, and range, Physical Mathematics explains simply and succinctly the mathematics that graduate students and professional physicists need to succeed in their courses and research. The book illustrates the mathematics with numerous physical examples drawn from contemporary research. This second edition has new chapters on vector calculus, special relativity and artificial intelligence and many new sections and examples. In addition to basic subjects such as linear algebra, Fourier analysis, complex variables, differential equations, Bessel functions, and spherical harmonics, the book explains topics such as the singular value decomposition, Lie algebras and group theory, tensors and general relativity, the central limit theorem and Kolmogorov's theorems, Monte Carlo methods of experimental and theoretical physics, Feynman's path integrals, and the standard model of cosmology. Contents......Page 7 Preface......Page 20 1.1 Numbers......Page 22 1.2 Arrays......Page 23 1.3 Matrices......Page 25 1.4 Vectors......Page 28 1.5 Linear Operators......Page 31 1.6 Inner Products......Page 33 1.7 Cauchy–Schwarz Inequalities......Page 36 1.8 Linear Independence and Completeness......Page 37 1.9 Dimension of a Vector Space......Page 38 1.10 Orthonormal Vectors......Page 39 1.11 Outer Products......Page 40 1.12 Dirac Notation......Page 41 1.13 Adjoints of Operators......Page 46 1.14 Self-Adjoint or Hermitian Linear Operators......Page 47 1.16 Unitary Operators......Page 48 1.17 Hilbert Spaces......Page 50 1.20 Determinants......Page 51 1.21 Jacobians......Page 59 1.22 Systems of Linear Equations......Page 61 1.24 Lagrange Multipliers......Page 62 1.26 Eigenvectors of a Square Matrix......Page 64 1.27 A Matrix Obeys Its Characteristic Equation......Page 68 1.28 Functions of Matrices......Page 70 1.29 Hermitian Matrices......Page 75 1.30 Normal Matrices......Page 80 1.31 Compatible Normal Matrices......Page 82 1.32 Singular-Value Decompositions......Page 86 1.33 Moore–Penrose Pseudoinverses......Page 91 1.34 Tensor Products and Entanglement......Page 93 1.35 Density Operators......Page 97 1.36 Schmidt Decomposition......Page 98 1.37 Correlation Functions......Page 99 1.39 Software......Page 101 Exercises......Page 102 2.1 Derivatives and Partial Derivatives......Page 105 2.2 Gradient......Page 106 2.3 Divergence......Page 107 2.4 Laplacian......Page 109 2.5 Curl......Page 110 Exercises......Page 113 3.1 Fourier Series......Page 114 3.2 The Interval......Page 117 3.3 Where to Put the 2pi’s......Page 118 3.4 Real Fourier Series for Real Functions......Page 119 3.5 Stretched Intervals......Page 123 3.6 Fourier Series of Functions of Several Variables......Page 124 3.8 How Fourier Series Converge......Page 125 3.9 Measure and Lebesgue Integration......Page 129 3.10 Quantum-Mechanical Examples......Page 131 3.11 Dirac’s Delta Function......Page 138 3.12 Harmonic Oscillators......Page 141 3.13 Nonrelativistic Strings......Page 143 3.14 Periodic Boundary Conditions......Page 144 Exercises......Page 146 4.1 Fourier Transforms......Page 149 4.2 Fourier Transforms of Real Functions......Page 152 4.3 Dirac, Parseval, and Poisson......Page 153 4.4 Derivatives and Integrals of Fourier Transforms......Page 157 4.5 Fourier Transforms of Functions of Several Variables......Page 162 4.6 Convolutions......Page 163 4.7 Fourier Transform of a Convolution......Page 165 4.8 Fourier Transforms and Green’s Functions......Page 166 4.9 Laplace Transforms......Page 167 4.10 Derivatives and Integrals of Laplace Transforms......Page 169 4.11 Laplace Transforms and Differential Equations......Page 170 4.13 Application to Differential Equations......Page 171 Exercises......Page 177 5.1 Convergence......Page 179 5.2 Tests of Convergence......Page 180 5.3 Convergent Series of Functions......Page 182 5.4 Power Series......Page 183 5.5 Factorials and the Gamma Function......Page 184 5.7 Taylor Series......Page 189 5.8 Fourier Series as Power Series......Page 190 5.9 Binomial Series......Page 191 5.11 Dirichlet Series and the Zeta Function......Page 193 5.12 Bernoulli Numbers and Polynomials......Page 195 5.13 Asymptotic Series......Page 196 5.14 Fractional and Complex Derivatives......Page 198 5.15 Some Electrostatic Problems......Page 199 5.16 Infinite Products......Page 202 Exercises......Page 203 6.1 Analytic Functions......Page 206 6.2 Cauchy–Riemann Conditions......Page 207 6.3 Cauchy’s Integral Theorem......Page 208 6.4 Cauchy’s Integral Formula......Page 214 6.5 Harmonic Functions......Page 217 6.6 Taylor Series for Analytic Functions......Page 219 6.8 Liouville’s Theorem......Page 220 6.9 Fundamental Theorem of Algebra......Page 221 6.10 Laurent Series......Page 222 6.11 Singularities......Page 224 6.12 Analytic Continuation......Page 226 6.13 Calculus of Residues......Page 228 6.14 Ghost Contours......Page 230 6.15 Logarithms and Cuts......Page 241 6.16 Powers and Roots......Page 242 6.17 Conformal Mapping......Page 246 6.18 Cauchy’s Principal Value......Page 248 6.19 Dispersion Relations......Page 254 6.20 Kramers–Kronig Relations......Page 256 6.21 Phase and Group Velocities......Page 257 6.22 Method of Steepest Descent......Page 260 6.23 Applications to String Theory......Page 262 Exercises......Page 264 7.1 Ordinary Linear Differential Equations......Page 269 7.2 Linear Partial Differential Equations......Page 271 7.3 Separable Partial Differential Equations......Page 272 7.5 Separable First-Order Differential Equations......Page 278 7.7 Exact First-Order Differential Equations......Page 281 7.8 Meaning of Exactness......Page 282 7.9 Integrating Factors......Page 284 7.10 Homogeneous Functions......Page 285 7.11 Virial Theorem......Page 286 7.12 Legendre’s Transform......Page 288 7.13 Principle of Stationary Action in Mechanics......Page 291 7.14 Symmetries and Conserved Quantities in Mechanics......Page 293 7.15 Homogeneous First-Order Ordinary Differential Equations......Page 294 7.16 Linear First-Order Ordinary Differential Equations......Page 295 7.17 Small Oscillations......Page 298 7.18 Systems of Ordinary Differential Equations......Page 299 7.19 Exact Higher-Order Differential Equations......Page 300 7.20 Constant-Coefficient Equations......Page 301 7.21 Singular Points of Second-Order Ordinary Differential Equations......Page 302 7.22 Frobenius’s Series Solutions......Page 303 7.24 Even and Odd Differential Operators......Page 306 7.25 Wronski’s Determinant......Page 307 7.26 Second Solutions......Page 308 7.27 Why Not Three Solutions?......Page 309 7.28 Boundary Conditions......Page 310 7.29 A Variational Problem......Page 311 7.30 Self-Adjoint Differential Operators......Page 312 7.31 Self-Adjoint Differential Systems......Page 314 7.32 Making Operators Formally Self-Adjoint......Page 316 7.33 Wronskians of Self-Adjoint Operators......Page 317 7.34 First-Order Self-Adjoint Differential Operators......Page 319 7.35 A Constrained Variational Problem......Page 320 7.36 Eigenfunctions and Eigenvalues of Self-Adjoint Systems......Page 325 7.37 Unboundedness of Eigenvalues......Page 328 7.38 Completeness of Eigenfunctions......Page 330 7.39 Inequalities of Bessel and Schwarz......Page 335 7.40 Green’s Functions......Page 336 7.41 Eigenfunctions and Green’s Functions......Page 339 7.42 Green’s Functions in One Dimension......Page 340 7.43 Principle of Stationary Action in Field Theory......Page 342 7.44 Symmetries and Conserved Quantities in Field Theory......Page 343 7.45 Nonlinear Differential Equations......Page 345 7.46 Nonlinear Differential Equations in Cosmology......Page 346 7.47 Nonlinear Differential Equations in Particle Physics......Page 349 Exercises......Page 352 8.1 Differential Equations as Integral Equations......Page 355 8.3 Volterra Integral Equations......Page 356 8.4 Implications of Linearity......Page 357 8.5 Numerical Solutions......Page 358 Exercises......Page 362 9.1 Legendre’s Polynomials......Page 364 9.2 The Rodrigues Formula......Page 365 9.3 Generating Function for Legendre Polynomials......Page 367 9.4 Legendre’s Differential Equation......Page 368 9.5 Recurrence Relations......Page 370 9.7 Schlaefli’s Integral......Page 372 9.8 Orthogonal Polynomials......Page 373 9.9 Azimuthally Symmetric Laplacians......Page 375 9.10 Laplace’s Equation in Two Dimensions......Page 376 9.12 Associated Legendre Polynomials......Page 377 9.13 Spherical Harmonics......Page 379 9.14 Cosmic Microwave Background Radiation......Page 381 Further Reading......Page 383 Exercises......Page 384 10.1 Cylindrical Bessel Functions of the First Kind......Page 386 10.2 Spherical Bessel Functions of the First Kind......Page 397 10.3 Bessel Functions of the Second Kind......Page 403 10.4 Spherical Bessel Functions of the Second Kind......Page 405 Exercises......Page 407 11.1 What Is a Group?......Page 411 11.2 Representations of Groups......Page 414 11.3 Representations Acting in Hilbert Space......Page 415 11.4 Subgroups......Page 416 11.6 Morphisms......Page 418 11.7 Schur’s Lemma......Page 419 11.8 Characters......Page 420 11.9 Direct Products......Page 421 11.10 Finite Groups......Page 422 11.11 Regular Representations......Page 423 11.13 Permutations......Page 424 11.15 Generators......Page 425 11.16 Lie Algebra......Page 426 11.18 Rotation Group......Page 430 11.20 Defining Representation of SU(2)......Page 434 11.21 The Lie Algebra and Representations of SU(2)......Page 435 11.22 How a Field Transforms Under a Rotation......Page 438 11.23 Addition of Two Spin-One-Half Systems......Page 439 11.25 Adjoint Representations......Page 442 11.26 Casimir Operators......Page 443 11.27 Tensor Operators for the Rotation Group......Page 444 11.28 Simple and Semisimple Lie Algebras......Page 445 11.30 SU(3) and Quarks......Page 446 11.31 Fierz Identity for SU(n)......Page 447 11.33 Symplectic Group Sp(2n)......Page 448 11.34 Quaternions......Page 450 11.35 Quaternions and Symplectic Groups......Page 452 11.37 Group Integration......Page 454 11.38 Lorentz Group......Page 456 11.39 Left-Handed Representation of the Lorentz Group......Page 460 11.40 Right-Handed Representation of the Lorentz Group......Page 463 11.41 Dirac’s Representation of the Lorentz Group......Page 465 11.42 Poincaré Group......Page 467 Exercises......Page 468 12.1 Inertial Frames and Lorentz Transformations......Page 472 12.2 Special Relativity......Page 474 12.3 Kinematics......Page 476 12.4 Electrodynamics......Page 477 12.5 Principle of Stationary Action in Special Relativity......Page 480 12.6 Differential Forms......Page 481 Exercises......Page 485 13.1 Points and Their Coordinates......Page 487 13.4 Covariant Vectors......Page 488 13.5 Tensors......Page 489 13.6 Summation Convention and Contractions......Page 490 13.8 Quotient Theorem......Page 491 13.10 Comma Notation for Derivatives......Page 492 13.12 Metric Tensor......Page 493 13.13 Inverse of Metric Tensor......Page 497 13.15 Covariant Derivatives of Contravariant Vectors......Page 498 13.17 Covariant Derivatives of Tensors......Page 500 13.18 The Covariant Derivative of the Metric Tensor Vanishes......Page 502 13.21 What is the Affine Connection?......Page 503 13.22 Parallel Transport......Page 504 13.23 Curvature......Page 505 13.24 Maximally Symmetric Spaces......Page 511 13.25 Principle of Equivalence......Page 513 13.26 Tetrads......Page 514 13.27 Scalar Densities and g = | det(gik)|......Page 515 13.28 Levi-Civita’s Symbol and Tensor......Page 516 13.29 Divergence of a Contravariant Vector......Page 517 13.31 Principle of Stationary Action in General Relativity......Page 519 13.32 Equivalence Principle and Geodesic Equation......Page 522 13.33 Weak Static Gravitational Fields......Page 523 13.35 Einstein’s Equations......Page 524 13.36 Energy–Momentum Tensor......Page 527 13.37 Perfect Fluids......Page 528 13.39 Schwarzschild’s Solution......Page 529 13.40 Black Holes......Page 530 13.41 Rotating Black Holes......Page 531 13.42 Spatially Symmetric Spacetimes......Page 532 13.43 Friedmann–Lemaître–Robinson–Walker Cosmologies......Page 534 13.44 Density and Pressure......Page 536 13.45 How the Scale Factor Evolves with Time......Page 537 13.46 The First Hundred Thousand Years......Page 539 13.47 The Next Ten Billion Years......Page 541 13.49 Before the Big Bang......Page 543 13.50 Yang–Mills Theory......Page 544 13.51 Cartan’s Spin Connection and Structure Equations......Page 546 13.53 Gauge Theory and Vectors......Page 550 Exercises......Page 552 14.1 Exterior Forms......Page 557 14.2 Differential Forms......Page 559 14.3 Exterior Differentiation......Page 564 14.4 Integration of Forms......Page 569 14.5 Are Closed Forms Exact?......Page 574 14.6 Complex Differential Forms......Page 576 14.7 Hodge’s Star......Page 577 14.8 Theorem of Frobenius......Page 581 Exercises......Page 583 15.1 Probability and Thomas Bayes......Page 585 15.2 Mean and Variance......Page 589 15.3 Binomial Distribution......Page 593 15.5 Poisson’s Distribution......Page 595 15.6 Gauss’s Distribution......Page 598 15.7 The Error Function erf......Page 601 15.8 Error Analysis......Page 604 15.9 Maxwell–Boltzmann Distribution......Page 606 15.10 Fermi–Dirac and Bose–Einstein Distributions......Page 607 15.11 Diffusion......Page 608 15.12 Langevin’s Theory of Brownian Motion......Page 609 15.13 Einstein–Nernst Relation......Page 612 15.14 Fluctuation and Dissipation......Page 613 15.15 Fokker–Planck Equation......Page 616 15.16 Characteristic and Moment-Generating Functions......Page 618 15.17 Fat Tails......Page 621 15.18 Central Limit Theorem and Jarl Lindeberg......Page 623 15.19 Random-Number Generators......Page 628 15.20 Illustration of the Central Limit Theorem......Page 630 15.21 Measurements, Estimators, and Friedrich Bessel......Page 632 15.22 Information and Ronald Fisher......Page 635 15.23 Maximum Likelihood......Page 639 15.24 Karl Pearson’s Chi-Squared Statistic......Page 640 15.25 Kolmogorov’s Test......Page 643 Exercises......Page 649 16.2 Numerical Integration......Page 653 16.4 Applications to Experiments......Page 654 16.5 Statistical Mechanics......Page 655 16.8 Evolution......Page 661 Exercises......Page 662 17.2 Slagle’s Symbolic Automatic Integrator......Page 664 17.3 Neural Networks......Page 665 17.4 A Linear Unbiased Neural Network......Page 666 18 Order, Chaos, and Fractals......Page 667 18.1 Hamilton Systems......Page 668 18.2 Autonomous Systems of Ordinary Differential Equations......Page 670 18.3 Attractors......Page 671 18.4 Chaos......Page 672 18.5 Maps......Page 674 18.6 Fractals......Page 677 Further Reading......Page 680 Exercises......Page 681 19.2 Functional Derivatives......Page 682 19.3 Higher-Order Functional Derivatives......Page 684 19.5 Functional Differential Equations......Page 686 Exercises......Page 689 20.2 Gaussian Integrals and Trotter’s Formula......Page 690 20.3 Path Integrals in Quantum Mechanics......Page 691 20.4 Path Integrals for Quadratic Actions......Page 695 20.5 Path Integrals in Statistical Mechanics......Page 700 20.6 Boltzmann Path Integrals for Quadratic Actions......Page 704 20.7 Mean Values of Time-Ordered Products......Page 706 20.8 Quantum Field Theory on a Lattice......Page 709 20.9 Finite-Temperature Field Theory......Page 713 20.10 Perturbation Theory......Page 715 20.11 Application to Quantum Electrodynamics......Page 719 20.12 Fermionic Path Integrals......Page 723 20.14 Faddeev–Popov Trick......Page 730 20.15 Ghosts......Page 733 20.16 Effective Field Theories......Page 734 Exercises......Page 735 21.2 Renormalization Group in Quantum Field Theory......Page 739 21.3 Renormalization Group in Lattice Field Theory......Page 744 21.4 Renormalization Group in Condensed-Matter Physics......Page 745 Exercises......Page 747 22.1 The Nambu–Goto String Action......Page 748 22.2 Static Gauge and Regge Trajectories......Page 750 22.3 Light-Cone Coordinates......Page 752 22.5 Quantized Open Strings......Page 753 22.8 D-branes or P-branes......Page 755 22.10 Riemann Surfaces and Moduli......Page 756 Exercises......Page 757 References......Page 758 Index......Page 765