Physical mathematics

Contents
Preface
1 Linear Algebra
	1.1 Numbers
	1.2 Arrays
	1.3 Matrices
	1.4 Vectors
	1.5 Linear Operators
	1.6 Inner Products
	1.7 Cauchy–Schwarz Inequalities
	1.8 Linear Independence and Completeness
	1.9 Dimension of a Vector Space
	1.10 Orthonormal Vectors
	1.11 Outer Products
	1.12 Dirac Notation
	1.13 Adjoints of Operators
	1.14 Self-Adjoint or Hermitian Linear Operators
	1.15 Real, Symmetric Linear Operators
	1.16 Unitary Operators
	1.17 Hilbert Spaces
	1.18 Antiunitary, Antilinear Operators
	1.19 Symmetry in Quantum Mechanics
	1.20 Determinants
	1.21 Jacobians
	1.22 Systems of Linear Equations
	1.23 Linear Least Squares
	1.24 Lagrange Multipliers
	1.25 Eigenvectors and Eigenvalues
	1.26 Eigenvectors of a Square Matrix
	1.27 A Matrix Obeys Its Characteristic Equation
	1.28 Functions of Matrices
	1.29 Hermitian Matrices
	1.30 Normal Matrices
	1.31 Compatible Normal Matrices
	1.32 Singular-Value Decompositions
	1.33 Moore–Penrose Pseudoinverses
	1.34 Tensor Products and Entanglement
	1.35 Density Operators
	1.36 Schmidt Decomposition
	1.37 Correlation Functions
	1.38 Rank of a Matrix
	1.39 Software
	Exercises
2 Vector Calculus
	2.1 Derivatives and Partial Derivatives
	2.2 Gradient
	2.3 Divergence
	2.4 Laplacian
	2.5 Curl
	Exercises
3 Fourier Series
	3.1 Fourier Series
	3.2 The Interval
	3.3 Where to Put the 2pi’s
	3.4 Real Fourier Series for Real Functions
	3.5 Stretched Intervals
	3.6 Fourier Series of Functions of Several Variables
	3.7 Integration and Differentiation of Fourier Series
	3.8 How Fourier Series Converge
	3.9 Measure and Lebesgue Integration
	3.10 Quantum-Mechanical Examples
	3.11 Dirac’s Delta Function
	3.12 Harmonic Oscillators
	3.13 Nonrelativistic Strings
	3.14 Periodic Boundary Conditions
	Exercises
4 Fourier and Laplace Transforms
	4.1 Fourier Transforms
	4.2 Fourier Transforms of Real Functions
	4.3 Dirac, Parseval, and Poisson
	4.4 Derivatives and Integrals of Fourier Transforms
	4.5 Fourier Transforms of Functions of Several Variables
	4.6 Convolutions
	4.7 Fourier Transform of a Convolution
	4.8 Fourier Transforms and Green’s Functions
	4.9 Laplace Transforms
	4.10 Derivatives and Integrals of Laplace Transforms
	4.11 Laplace Transforms and Differential Equations
	4.12 Inversion of Laplace Transforms
	4.13 Application to Differential Equations
	Exercises
5 Infinite Series
	5.1 Convergence
	5.2 Tests of Convergence
	5.3 Convergent Series of Functions
	5.4 Power Series
	5.5 Factorials and the Gamma Function
	5.6 Euler’s Beta Function
	5.7 Taylor Series
	5.8 Fourier Series as Power Series
	5.9 Binomial Series
	5.10 Logarithmic Series
	5.11 Dirichlet Series and the Zeta Function
	5.12 Bernoulli Numbers and Polynomials
	5.13 Asymptotic Series
	5.14 Fractional and Complex Derivatives
	5.15 Some Electrostatic Problems
	5.16 Infinite Products
	Exercises
6 Complex-Variable Theory
	6.1 Analytic Functions
	6.2 Cauchy–Riemann Conditions
	6.3 Cauchy’s Integral Theorem
	6.4 Cauchy’s Integral Formula
	6.5 Harmonic Functions
	6.6 Taylor Series for Analytic Functions
	6.7 Cauchy’s Inequality
	6.8 Liouville’s Theorem
	6.9 Fundamental Theorem of Algebra
	6.10 Laurent Series
	6.11 Singularities
	6.12 Analytic Continuation
	6.13 Calculus of Residues
	6.14 Ghost Contours
	6.15 Logarithms and Cuts
	6.16 Powers and Roots
	6.17 Conformal Mapping
	6.18 Cauchy’s Principal Value
	6.19 Dispersion Relations
	6.20 Kramers–Kronig Relations
	6.21 Phase and Group Velocities
	6.22 Method of Steepest Descent
	6.23 Applications to String Theory
	Further Reading
	Exercises
7 Differential Equations
	7.1 Ordinary Linear Differential Equations
	7.2 Linear Partial Differential Equations
	7.3 Separable Partial Differential Equations
	7.4 First-Order Differential Equations
	7.5 Separable First-Order Differential Equations
	7.6 Hidden Separability
	7.7 Exact First-Order Differential Equations
	7.8 Meaning of Exactness
	7.9 Integrating Factors
	7.10 Homogeneous Functions
	7.11 Virial Theorem
	7.12 Legendre’s Transform
	7.13 Principle of Stationary Action in Mechanics
	7.14 Symmetries and Conserved Quantities in Mechanics
	7.15 Homogeneous First-Order Ordinary Differential Equations
	7.16 Linear First-Order Ordinary Differential Equations
	7.17 Small Oscillations
	7.18 Systems of Ordinary Differential Equations
	7.19 Exact Higher-Order Differential Equations
	7.20 Constant-Coefficient Equations
	7.21 Singular Points of Second-Order Ordinary Differential Equations
	7.22 Frobenius’s Series Solutions
	7.23 Fuch’s Theorem
	7.24 Even and Odd Differential Operators
	7.25 Wronski’s Determinant
	7.26 Second Solutions
	7.27 Why Not Three Solutions?
	7.28 Boundary Conditions
	7.29 A Variational Problem
	7.30 Self-Adjoint Differential Operators
	7.31 Self-Adjoint Differential Systems
	7.32 Making Operators Formally Self-Adjoint
	7.33 Wronskians of Self-Adjoint Operators
	7.34 First-Order Self-Adjoint Differential Operators
	7.35 A Constrained Variational Problem
	7.36 Eigenfunctions and Eigenvalues of Self-Adjoint Systems
	7.37 Unboundedness of Eigenvalues
	7.38 Completeness of Eigenfunctions
	7.39 Inequalities of Bessel and Schwarz
	7.40 Green’s Functions
	7.41 Eigenfunctions and Green’s Functions
	7.42 Green’s Functions in One Dimension
	7.43 Principle of Stationary Action in Field Theory
	7.44 Symmetries and Conserved Quantities in Field Theory
	7.45 Nonlinear Differential Equations
	7.46 Nonlinear Differential Equations in Cosmology
	7.47 Nonlinear Differential Equations in Particle Physics
	Further Reading
	Exercises
8 Integral Equations
	8.1 Differential Equations as Integral Equations
	8.2 Fredholm Integral Equations
	8.3 Volterra Integral Equations
	8.4 Implications of Linearity
	8.5 Numerical Solutions
	Exercises
9 Legendre Polynomials and Spherical Harmonics
	9.1 Legendre’s Polynomials
	9.2 The Rodrigues Formula
	9.3 Generating Function for Legendre Polynomials
	9.4 Legendre’s Differential Equation
	9.5 Recurrence Relations
	9.6 Special Values of Legendre Polynomials
	9.7 Schlaefli’s Integral
	9.8 Orthogonal Polynomials
	9.9 Azimuthally Symmetric Laplacians
	9.10 Laplace’s Equation in Two Dimensions
	9.11 Helmholtz’s Equation in Spherical Coordinates
	9.12 Associated Legendre Polynomials
	9.13 Spherical Harmonics
	9.14 Cosmic Microwave Background Radiation
	Further Reading
	Exercises
10 Bessel Functions
	10.1 Cylindrical Bessel Functions of the First Kind
	10.2 Spherical Bessel Functions of the First Kind
	10.3 Bessel Functions of the Second Kind
	10.4 Spherical Bessel Functions of the Second Kind
	Further Reading
	Exercises
11 Group Theory
	11.1 What Is a Group?
	11.2 Representations of Groups
	11.3 Representations Acting in Hilbert Space
	11.4 Subgroups
	11.5 Cosets
	11.6 Morphisms
	11.7 Schur’s Lemma
	11.8 Characters
	11.9 Direct Products
	11.10 Finite Groups
	11.11 Regular Representations
	11.12 Properties of Finite Groups
	11.13 Permutations
	11.14 Compact and Noncompact Lie Groups
	11.15 Generators
	11.16 Lie Algebra
	11.17 Yang and Mills Invent Local Nonabelian Symmetry
	11.18 Rotation Group
	11.19 Rotations and Reflections in 2n Dimensions
	11.20 Defining Representation of SU(2)
	11.21 The Lie Algebra and Representations of SU(2)
	11.22 How a Field Transforms Under a Rotation
	11.23 Addition of Two Spin-One-Half Systems
	11.24 Jacobi Identity
	11.25 Adjoint Representations
	11.26 Casimir Operators
	11.27 Tensor Operators for the Rotation Group
	11.28 Simple and Semisimple Lie Algebras
	11.29 SU(3)
	11.30 SU(3) and Quarks
	11.31 Fierz Identity for SU(n)
	11.32 Cartan Subalgebra
	11.33 Symplectic Group Sp(2n)
	11.34 Quaternions
	11.35 Quaternions and Symplectic Groups
	11.36 Compact Simple Lie Groups
	11.37 Group Integration
	11.38 Lorentz Group
	11.39 Left-Handed Representation of the Lorentz Group
	11.40 Right-Handed Representation of the Lorentz Group
	11.41 Dirac’s Representation of the Lorentz Group
	11.42 Poincaré Group
	11.43 Homotopy Groups
	Further Reading
	Exercises
12 Special Relativity
	12.1 Inertial Frames and Lorentz Transformations
	12.2 Special Relativity
	12.3 Kinematics
	12.4 Electrodynamics
	12.5 Principle of Stationary Action in Special Relativity
	12.6 Differential Forms
	Exercises
13 General Relativity
	13.1 Points and Their Coordinates
	13.2 Scalars
	13.3 Contravariant Vectors
	13.4 Covariant Vectors
	13.5 Tensors
	13.6 Summation Convention and Contractions
	13.7 Symmetric and Antisymmetric Tensors
	13.8 Quotient Theorem
	13.9 Tensor Equations
	13.10 Comma Notation for Derivatives
	13.11 Basis Vectors and Tangent Vectors
	13.12 Metric Tensor
	13.13 Inverse of Metric Tensor
	13.14 Dual Vectors, Cotangent Vectors
	13.15 Covariant Derivatives of Contravariant Vectors
	13.16 Covariant Derivatives of Covariant Vectors
	13.17 Covariant Derivatives of Tensors
	13.18 The Covariant Derivative of the Metric Tensor Vanishes
	13.19 Covariant Curls
	13.20 Covariant Derivatives and Antisymmetry
	13.21 What is the Affine Connection?
	13.22 Parallel Transport
	13.23 Curvature
	13.24 Maximally Symmetric Spaces
	13.25 Principle of Equivalence
	13.26 Tetrads
	13.27 Scalar Densities and g = | det(gik)|
	13.28 Levi-Civita’s Symbol and Tensor
	13.29 Divergence of a Contravariant Vector
	13.30 Covariant Laplacian
	13.31 Principle of Stationary Action in General Relativity
	13.32 Equivalence Principle and Geodesic Equation
	13.33 Weak Static Gravitational Fields
	13.34 Gravitational Time Dilation
	13.35 Einstein’s Equations
	13.36 Energy–Momentum Tensor
	13.37 Perfect Fluids
	13.38 Gravitational Waves
	13.39 Schwarzschild’s Solution
	13.40 Black Holes
	13.41 Rotating Black Holes
	13.42 Spatially Symmetric Spacetimes
	13.43 Friedmann–Lemaître–Robinson–Walker Cosmologies
	13.44 Density and Pressure
	13.45 How the Scale Factor Evolves with Time
	13.46 The First Hundred Thousand Years
	13.47 The Next Ten Billion Years
	13.48 Era of Dark Energy
	13.49 Before the Big Bang
	13.50 Yang–Mills Theory
	13.51 Cartan’s Spin Connection and Structure Equations
	13.52 Spin-One-Half Fields in General Relativity
	13.53 Gauge Theory and Vectors
	Further Reading
	Exercises
14 Forms
	14.1 Exterior Forms
	14.2 Differential Forms
	14.3 Exterior Differentiation
	14.4 Integration of Forms
	14.5 Are Closed Forms Exact?
	14.6 Complex Differential Forms
	14.7 Hodge’s Star
	14.8 Theorem of Frobenius
	Further Reading
	Exercises
15 Probability and Statistics
	15.1 Probability and Thomas Bayes
	15.2 Mean and Variance
	15.3 Binomial Distribution
	15.4 Coping with Big Factorials
	15.5 Poisson’s Distribution
	15.6 Gauss’s Distribution
	15.7 The Error Function erf
	15.8 Error Analysis
	15.9 Maxwell–Boltzmann Distribution
	15.10 Fermi–Dirac and Bose–Einstein Distributions
	15.11 Diffusion
	15.12 Langevin’s Theory of Brownian Motion
	15.13 Einstein–Nernst Relation
	15.14 Fluctuation and Dissipation
	15.15 Fokker–Planck Equation
	15.16 Characteristic and Moment-Generating Functions
	15.17 Fat Tails
	15.18 Central Limit Theorem and Jarl Lindeberg
	15.19 Random-Number Generators
	15.20 Illustration of the Central Limit Theorem
	15.21 Measurements, Estimators, and Friedrich Bessel
	15.22 Information and Ronald Fisher
	15.23 Maximum Likelihood
	15.24 Karl Pearson’s Chi-Squared Statistic
	15.25 Kolmogorov’s Test
	Further Reading
	Exercises
16 Monte Carlo Methods
	16.1 The Monte Carlo Method
	16.2 Numerical Integration
	16.3 Quasirandom Numbers
	16.4 Applications to Experiments
	16.5 Statistical Mechanics
	16.6 Simulated Annealing
	16.7 Solving Arbitrary Problems
	16.8 Evolution
	Further Reading
	Exercises
17 Artificial Intelligence
	17.1 Steps Toward Artificial Intelligence
	17.2 Slagle’s Symbolic Automatic Integrator
	17.3 Neural Networks
	17.4 A Linear Unbiased Neural Network
	Further Reading
18 Order, Chaos, and Fractals
	18.1 Hamilton Systems
	18.2 Autonomous Systems of Ordinary Differential Equations
	18.3 Attractors
	18.4 Chaos
	18.5 Maps
	18.6 Fractals
	Further Reading
	Exercises
19 Functional Derivatives
	19.1 Functionals
	19.2 Functional Derivatives
	19.3 Higher-Order Functional Derivatives
	19.4 Functional Taylor Series
	19.5 Functional Differential Equations
	Exercises
20 Path Integrals
	20.1 Path Integrals and Richard Feynman
	20.2 Gaussian Integrals and Trotter’s Formula
	20.3 Path Integrals in Quantum Mechanics
	20.4 Path Integrals for Quadratic Actions
	20.5 Path Integrals in Statistical Mechanics
	20.6 Boltzmann Path Integrals for Quadratic Actions
	20.7 Mean Values of Time-Ordered Products
	20.8 Quantum Field Theory on a Lattice
	20.9 Finite-Temperature Field Theory
	20.10 Perturbation Theory
	20.11 Application to Quantum Electrodynamics
	20.12 Fermionic Path Integrals
	20.13 Application to Nonabelian Gauge Theories
	20.14 Faddeev–Popov Trick
	20.15 Ghosts
	20.16 Effective Field Theories
	20.17 Complex Path Integrals
	Further Reading
	Exercises
21 Renormalization Group
	21.1 Renormalization and Interpolation
	21.2 Renormalization Group in Quantum Field Theory
	21.3 Renormalization Group in Lattice Field Theory
	21.4 Renormalization Group in Condensed-Matter Physics
	Further Reading
	Exercises
22 Strings
	22.1 The Nambu–Goto String Action
	22.2 Static Gauge and Regge Trajectories
	22.3 Light-Cone Coordinates
	22.4 Light-Cone Gauge
	22.5 Quantized Open Strings
	22.6 Superstrings
	22.7 Covariant and Polyakov Actions
	22.8 D-branes or P-branes
	22.9 String–String Scattering
	22.10 Riemann Surfaces and Moduli
	Further Reading
	Exercises
References
Index