Mathematical Methods of Theoretical Physics

✍ Scribed by Karl Svozil

• Publisher: WSPC
• Year: 2020
• Tongue: English
• Leaves: 331
• Category: Library

✦ Synopsis

This book contains very explicit proofs and demonstrations through examples for a comprehensive introduction to the mathematical methods of theoretical physics. It also combines and unifies many expositions of this subject, suitable for readers with interest in experimental and applied physics.

✦ Table of Contents

Contents
Why mathematics?
Part I: Linear vector spaces
1 Finite-dimensional vector spaces and linear algebra
1.1 Conventions and basic definitions
1.1.1 Fields of real and complex numbers
1.1.2 Vectors and vector space
1.2 Linear independence
1.3 Subspace
1.3.1 Scalar or inner product
1.3.2 Hilbert space
1.4 Basis
1.5 Dimension
1.6 Vector coordinates or components
1.7 Finding orthogonal bases from nonorthogonal ones
1.8 Dual space
1.8.1 Dual basis
1.8.2 Dual coordinates
1.8.3 Representation of a functional by inner product
1.8.4 Double dual space
1.9 Direct sum
1.10 Tensor product
1.10.1 Sloppy definition
1.10.2 Definition
1.10.3 Representation
1.11 Linear transformation
1.11.1 Definition
1.11.2 Operations
1.11.3 Linear transformations as matrices
1.12 Change of basis
1.12.1 Settlement of change of basis vectors by definition
1.12.2 Scale change of vector components by contra-variation
1.13 Mutually unbiased bases
1.14 Completeness or resolution of the identity operator in terms of base vectors
1.15 Rank
1.16 Determinant
1.16.1 Definition
1.16.2 Properties
1.17 Trace
1.17.1 Definition
1.17.2 Properties
1.17.3 Partial trace
1.18 Adjoint or dual transformation
1.18.1 Definition
1.18.2 Adjoint matrix notation
1.18.3 Properties
1.19 Self-adjoint transformation
1.20 Positive transformation
1.21 Unitary transformation and isometries
1.21.1 Definition
1.21.2 Characterization in terms of orthonormal basis
1.22 Orthonormal (orthogonal) transformation
1.23 Permutation
1.24 Projection or projection operator
1.24.1 Definition
1.24.2 Orthogonal (perpendicular) projections
1.24.3 Construction of orthogonal projections from single unit vectors
1.24.4 Examples of oblique projections which are not orthogonal projections
1.25 Proper value or eigenvalue
1.25.1 Definition
1.25.2 Determination
1.26 Normal transformation
1.27 Spectrum
1.27.1 Spectral theorem
1.27.2 Composition of the spectral form
1.28 Functions of normal transformations
1.29 Decomposition of operators
1.29.1 Standard decomposition
1.29.2 Polar decomposition
1.29.3 Decomposition of isometries
1.29.4 Singular value decomposition
1.29.5 Schmidt decomposition of the tensor product of two vectors
1.30 Purification
1.31 Commutativity
1.32 Measures on closed subspaces
1.32.1 Gleason’s theorem
1.32.2 Kochen-Specker theorem
2 Multilinear algebra and tensors
2.1 Notation
2.2 Change of basis
2.2.1 Transformation of the covariant basis
2.2.2 Transformation of the contravariant coordinates
2.2.3 Transformation of the contravariant (dual) basis
2.2.4 Transformation of the covariant coordinates
2.2.5 Orthonormal bases
2.3 Tensor as multilinear form
2.4 Covariant tensors
2.4.1 Transformation of covariant tensor components
2.5 Contravariant tensors
2.5.1 Definition of contravariant tensors
2.5.2 Transformation of contravariant tensor components
2.6 General tensor
2.7 Metric
2.7.1 Definition
2.7.2 Construction from a scalar product
2.7.3 What can the metric tensor do for you?
2.7.4 Transformation of the metric tensor
2.7.5 Examples
2.8 Decomposition of tensors
2.9 Form invariance of tensors
2.10 The Kronecker symbol δ
2.11 The Levi-Civita symbol ε
2.12 Nabla, Laplace, and D’Alembert operators
2.13 Tensor analysis in orthogonal curvilinear coordinates
2.13.1 Curvilinear coordinates
2.13.2 Curvilinear bases
2.13.3 Infinitesimal increment, line element, and volume
2.13.4 Vector differential operator and gradient
2.13.5 Divergence in three dimensional orthogonal curvilinear coordinates
2.13.6 Curl in three dimensional orthogonal curvilinear coordinates
2.13.7 Laplacian in three dimensional orthogonal curvilinear coordinates
2.14 Index trickery and examples
2.15 Some common misconceptions
2.15.1 Confusion between component representation and “the real thing”
2.15.2 Matrix as a representation of a tensor of type (order, degree, rank) two
3 Groups as permutations
3.1 Basic definition and properties
3.1.1 Group axioms
3.1.2 Discrete and continuous groups
3.1.3 Generators and relations in finite groups
3.1.4 Uniqueness of identity and inverses
3.1.5 Cayley or group composition table
3.1.6 Rearrangement theorem
3.2 Zoology of finite groups up to order 6
3.2.1 Group of order 2
3.2.2 Group of order 3, 4 and 5
3.2.3 Group of order 6
3.2.4 Cayley’s theorem
3.3 Representations by homomorphisms
3.4 Partitioning of finite groups by cosets
3.5 Lie theory
3.5.1 Generators
3.5.2 Exponential map
3.5.3 Lie algebra
3.6 Zoology of some important continuous groups
3.6.1 General linear group GL(n,C)
3.6.2 Orthogonal group over the reals O(n,R) = O(n)
3.6.3 Rotation group SO(n)
3.6.4 Unitary group U(n,C) = U(n)
3.6.5 Special unitary group SU(n)
3.6.6 Symmetric group S(n)
3.6.7 Poincaré group
4 Projective and incidence geometry
4.1 Notation
4.2 Affine transformations map lines into lines as well as parallel lines to parallel lines
4.2.1 One-dimensional case
4.3 Similarity transformations
4.4 Fundamental theorem of affine geometry revised
4.5 Alexandrov’s theorem
Part II: Functional analysis
5 Brief review of complex analysis
5.1 Geometric representations of complex numbers and functions thereof
5.1.1 The complex plane
5.1.2 Multi-valued relationships, branch points, and branch cuts
5.2 Riemann surface
5.3 Differentiable, holomorphic (analytic) function
5.4 Cauchy-Riemann equations
5.5 Definition analytical function
5.6 Cauchy’s integral theorem
5.7 Cauchy’s integral formula
5.8 Series representation of complex differentiable functions
5.9 Laurent and Taylor series
5.10 Residue theorem
5.11 Some special functional classes
5.11.1 Criterion for coincidence
5.11.2 Entire function
5.11.3 Liouville’s theoremfor bounded entire function
5.11.4 Picard’s theorem
5.11.5 Meromorphic function
5.12 Fundamental theorem of algebra
5.13 Asymptotic series
6 Brief review of Fourier transforms
6.0.1 Functional spaces
6.0.2 Fourier series
6.0.3 Exponential Fourier series
6.0.4 Fourier transformation
7 Distributions as generalized functions
7.1 Coping with discontinuities and singularities
7.2 General distribution
7.2.1 Duality
7.2.2 Linearity
7.2.3 Continuity
7.3 Test functions
7.3.1 Desiderata on test functions
7.3.2 Test function class I
7.3.3 Test function class II
7.3.4 Test function class III: Tempered distributions and Fourier transforms
7.3.5 Test function class C∞
7.4 Derivative of distributions
7.5 Fourier transform of distributions
7.6 Dirac delta function
7.6.1 Delta sequence
7.6.2 δ[φ] distribution
7.6.3 Useful formulæ involving δ
7.6.4 Fourier transformof δ
7.6.5 Eigenfunction expansion of δ
7.6.6 Delta function expansion
7.7 Cauchy principal value
7.7.1 Definition
7.7.2 Principle value and pole function 1/x distribution
7.8 Absolute value distribution
7.9 Logarithm distribution
7.9.1 Definition
7.9.2 Connection with pole function
7.10 Pole function 1/xn distribution
7.11 Pole function 1/x±iα distribution
7.12 Heaviside or unit step function
7.12.1 Ambiguities in definition
7.12.2 Unit step function sequence
7.12.3 Useful formulæ involving H
7.12.4 H[φ] distribution
7.12.5 Regularized unit step function
7.12.6 Fourier transform of the unit step function
7.13 The sign function
7.13.1 Definition
7.13.2 Connection to the Heaviside function
7.13.3 Sign sequence
7.13.4 Fourier transform of sgn
7.14 Absolute value function (or modulus)
7.14.1 Definition
7.14.2 Connection of absolute value with the sign and Heaviside functions
7.15 Some examples
Part III: Differential equations
8 Green’s function
8.1 Elegant way to solve linear differential equations
8.2 Nonuniqueness of solution
8.3 Green’s functions of translational invariant differential operators
8.4 Solutions with fixed boundary or initial values
8.5 Finding Green’s functions by spectral decompositions
8.6 Finding Green’s functions by Fourier analysis
9 Sturm-Liouville theory
9.1 Sturm-Liouville form
9.2 Adjoint and self-adjoint operators
9.3 Sturm-Liouville eigenvalue problem
9.4 Sturm-Liouville transformation into Liouville normal form
9.5 Varieties of Sturm-Liouville differential equations
10 Separation of variables.
11 Special functions of mathematical physics
11.1 Gamma function
11.2 Beta function
11.3 Fuchsian differential equations
11.3.1 Regular, regular singular, and irregular singular point
11.3.2 Behavior at infinity
11.3.3 Functional formof the coefficients in Fuchsian differential equations
11.3.4 Frobenius method: Solution by power series
11.3.5 d’Alembert reduction of order
11.3.6 Computation of the characteristic exponent
11.3.7 Examples
11.4 Hypergeometric function
11.4.1 Definition
11.4.2 Properties
11.4.3 Plasticity
11.4.4 Four forms
11.5 Orthogonal polynomials
11.6 Legendre polynomials
11.6.1 Rodrigues formula
11.6.2 Generating function
11.6.3 The three term and other recursion formulæ
11.6.4 Expansion in Legendre polynomials
11.7 Associated Legendre polynomial
11.8 Spherical harmonics
11.9 Solution of the Schrödinger equation for a hydrogen atom
11.9.1 Separation of variables Ansatz
11.9.2 Separation of the radial part from the angular one
11.9.3 Separation of the polar angle θ from the azimuthal angle φ
11.9.4 Solution of the equation for the azimuthal angle factor Φ(φ)
11.9.5 Solution of the equation for the polar angle factor Θ(θ)
11.9.6 Solution of the equation for radial factor R(r)
11.9.7 Composition of the general solution of the Schrödinger equation
12 Divergent series
12.1 Convergence, asymptotic divergence, and divergence: A zoo perspective
12.2 Geometric series
12.3 Abel summation – Assessing paradoxes of infinity
12.4 Riemann zeta function and Ramanujan summation: Taming the beast
12.5 Asymptotic power series
12.6 Borel’s resummation method – “the master forbids it”
12.7 Asymptotic series as solutions of differential equations
12.8 Divergence of perturbation series in quantumfield theory
12.8.1 Expansion at an essential singularity
12.8.2 Forbidden interchange of limits
12.8.3 On the usefulness of asymptotic expansions in quantum field theory
Bibliography
Index