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Mathematical Methods of Theoretical Physics V KARLSVOZIL MATHEMATICALMETH- ODSOFTHEORETICAL PHYSICS arXiv:1203.4558v8 [math-ph] 1 Feb 2019 EDITIONFUNZL Copyright © 2019 Karl Svozil Published by Edition Funzl For academic use only. You may not reproduce or distribute without permission of the author. First Edition, October 2011 Second Edition, October 2013 Third Edition, October 2014 Fourth Edition, October 2016 Fifth Edition, October 2018 Sixth Edition, February 2019 Contents Unreasonable effectiveness of mathematics in the natural sciences xi Part I: Linear vector spaces 1 1 Finite-dimensional vector spaces and linear algebra 3 1.1 Conventions and basic definitions3 1.1.1 Fields of real and complex numbers,5.—1.1.2 Vectors and vector space, 5. 1.2 Linear independence6 1.3 Subspace7 1.3.1 Scalar or inner product,7.—1.3.2 Hilbert space,9. 1.4 Basis9 1.5 Dimension 10 1.6 Vector coordinates or components 11 1.7 Finding orthogonal bases from nonorthogonal ones 13 1.8 Dual space 15 1.8.1 Dual basis,16.—1.8.2 Dual coordinates,19.—1.8.3 Representation of a functional by inner product,20.—1.8.4 Double dual space, 23. 1.9 Direct sum 23 1.10 Tensor product 24 1.10.1 Sloppy definition, 24.—1.10.2 Definition,24.—1.10.3 Representation, 25. 1.11 Linear transformation 26 1.11.1 Definition, 26.—1.11.2 Operations, 26.—1.11.3 Linear transforma- tions as matrices, 28. 1.12 Change of basis 29 1.12.1 Settlement of change of basis vectors by definition, 29.—1.12.2 Scale change of vector components by contra-variation,31. 1.13 Mutually unbiased bases 33 iv Karl Svozil 1.14 Completeness or resolution of the identity operator in terms of base vectors 34 1.15 Rank 35 1.16 Determinant 36 1.16.1 Definition, 36.—1.16.2 Properties, 37. 1.17 Trace 39 1.17.1 Definition, 39.—1.17.2 Properties, 40.—1.17.3 Partial trace,40. 1.18 Adjoint or dual transformation 42 1.18.1 Definition, 42.—1.18.2 Adjoint matrix notation, 42.— 1.18.3 Properties, 43. 1.19 Self-adjoint transformation 43 1.20 Positive transformation 44 1.21 Unitary transformation and isometries 44 1.21.1 Definition, 44.—1.21.2 Characterization in terms of orthonormal basis,47. 1.22 Orthonormal (orthogonal) transformation 48 1.23 Permutation 48 1.24 Projection or projection operator 49 1.24.1 Definition, 50.—1.24.2 Orthogonal (perpendicular) projections, 51.— 1.24.3 Construction of orthogonal projections from single unit vectors, 53.— 1.24.4 Examples of oblique projections which are not orthogonal projections, 55. 1.25 Proper value or eigenvalue 56 1.25.1 Definition, 56.—1.25.2 Determination,56. 1.26 Normal transformation 60 1.27 Spectrum 60 1.27.1 Spectral theorem, 60.—1.27.2 Composition of the spectral form, 62. 1.28 Functions of normal transformations 64 1.29 Decomposition of operators 65 1.29.1 Standard decomposition, 65.—1.29.2 Polar decomposition, 66.—1.29.3 Decomposition of isometries, 66.—1.29.4 Singular value decomposition, 67.—1.29.5 Schmidt decomposition of the tensor product of two vectors, 67. 1.30 Purification 68 1.31 Commutativity 69 1.32 Measures on closed subspaces 73 1.32.1 Gleason’s theorem,74.—1.32.2 Kochen-Specker theorem, 75. 2 Multilinear Algebra and Tensors 79 2.1 Notation 80 2.2 Change of Basis 81 2.2.1 Transformation of the covariant basis, 81.—2.2.2 Transformation of the contravariant coordinates, 82.—2.2.3 Transformation of the contravariant (dual) basis, 83.—2.2.4 Transformation of the covariant coordinates, 85.— 2.2.5 Orthonormal bases,85. Mathematical Methods of Theoretical Physics v 2.3 Tensor as multilinear form 85 2.4 Covariant tensors 86 2.4.1 Transformation of covariant tensor components, 86. 2.5 Contravariant tensors 87 2.5.1 Definition of contravariant tensors, 87.—2.5.2 Transformation of con- travariant tensor components, 87. 2.6 General tensor 87 2.7 Metric 88 2.7.1 Definition, 88.—2.7.2 Construction from a scalar product, 88.— 2.7.3 What can the metric tensor do for you?, 89.—2.7.4 Transformation of the metric tensor, 90.—2.7.5 Examples,90. 2.8 Decomposition of tensors 94 2.9 Form invariance of tensors 94 2.10 The Kronecker symbol ± 100 2.11 The Levi-Civita symbol " 100 2.12 Nabla, Laplace, and D’Alembert operators 101 2.13 Tensor analysis in orthogonal curvilinear coordinates 102 2.13.1 Curvilinear coordinates, 102.—2.13.2 Curvilinear bases, 104.— 2.13.3 Infinitesimal increment, line element, and volume, 104.—2.13.4 Vector differential operator and gradient, 106.—2.13.5 Divergence in three dimen- sional orthogonal curvilinear coordinates, 107.—2.13.6 Curl in three dimen- sional orthogonal curvilinear coordinates, 108.—2.13.7 Laplacian in three di- mensional orthogonal curvilinear coordinates, 108. 2.14 Index trickery and examples 109 2.15 Some common misconceptions 117 2.15.1 Confusion between component representation and “the real thing”, 117.—2.15.2 Matrix as a representation of a tensor of type (order, degree, rank) two, 118. 3 Groups as permutations 119 3.1 Basic definition and properties 119 3.1.1 Group axioms, 119.—3.1.2 Discrete and continuous groups, 120.— 3.1.3 Generators and relations in finite groups, 120.—3.1.4 Uniqueness of identity and inverses, 121.—3.1.5 Cayley or group composition table, 121.— 3.1.6 Rearrangement theorem, 122. 3.2 Zoology of finite groups up to order 6 123 3.2.1 Group of order 2, 123.—3.2.2 Group of order 3, 4 and 5, 123.— 3.2.3 Group of order 6, 124.—3.2.4 Cayley’s theorem, 124. 3.3 Representations by homomorphisms 125 3.4 Lie theory 126 3.4.1 Generators, 126.—3.4.2 Exponential map, 126.—3.4.3 Lie algebra, 126. 3.5 Zoology of some important continuous groups 126 3.5.1 General linear group GL(n,C), 126.—3.5.2 Orthogonal group over the reals O(n,R) O(n), 126.—3.5.3 Rotation group SO(n), 127.—3.5.4 Unitary Æ group U(n,C) U(n), 127.—3.5.5 Special unitary group SU(n), 128.— Æ 3.5.6 Symmetric group S(n), 128.—3.5.7 Poincaré group, 128. vi Karl Svozil 4 Projective and incidence geometry 129 4.1 Notation 129 4.2 Affine transformations map lines into lines as well as parallel lines to parallel lines 129 4.2.1 One-dimensional case, 132. 4.3 Similarity transformations 132 4.4 Fundamental theorem of affine geometry revised 132 4.5 Alexandrov’s theorem 132 Part II: Functional analysis 135 5 Brief review of complex analysis 137 5.1 Geometric representations of complex numbers and functions thereof 139 5.1.1 The complex plane, 139.—5.1.2 Multi-valued relationships, branch points, and branch cuts, 139. 5.2 Riemann surface 140 5.3 Differentiable, holomorphic (analytic) function 141 5.4 Cauchy-Riemann equations 141 5.5 Definition analytical function 142 5.6 Cauchy’s integral theorem 143 5.7 Cauchy’s integral formula 143 5.8 Series representation of complex differentiable functions 144 5.9 Laurent series 145 5.10 Residue theorem 147 5.11 Some special functional classes 150 5.11.1 Criterion for coincidence, 150.—5.11.2 Entire function, 150.— 5.11.3 Liouville’s theorem for bounded entire function, 150.—5.11.4 Picard’s theorem, 151.—5.11.5 Meromorphic function, 151. 5.12 Fundamental theorem of algebra 152 6 Brief review of Fourier transforms 153 6.0.1 Functional spaces, 153.—6.0.2 Fourier series, 154.—6.0.3 Exponential Fourier series, 156.—6.0.4 Fourier transformation, 157. 7 Distributions as generalized functions 161 7.1 Heuristically coping with discontinuities and singularities 161 7.2 General distribution 162 7.2.1 Duality, 163.—7.2.2 Linearity, 163.—7.2.3 Continuity, 164. 7.3 Test functions 164 7.3.1 Desiderata on test functions, 164.—7.3.2 Test function class I, 165.— Mathematical Methods of Theoretical Physics vii 7.3.3 Test function class II, 166.—7.3.4 Test function class III: Tempered dis- tributions and Fourier transforms, 166.—7.3.5 Test function class C 1, 168. 7.4 Derivative of distributions 168 7.5 Fourier transform of distributions 169 7.6 Dirac delta function 169 7.6.1 Delta sequence, 170.—7.6.2 ±£'¤ distribution, 172.—7.6.3 Useful formulæ involving ±, 172.—7.6.4 Fourier transform of ±, 178.— 7.6.5 Eigenfunction expansion of ±, 178.—7.6.6 Delta function expansion, 179. 7.7 Cauchy principal value 179 1 7.7.1 Definition, 179.—7.7.2 Principle value and pole function x distribution, 180. 7.8 Absolute value distribution 180 7.9 Logarithm distribution 181 7.9.1 Definition, 181.—7.9.2 Connection with pole function, 181. 1 7.10 Pole function xn distribution 182 1 7.11 Pole function x i® distribution 182 § 7.12 Heaviside or unit step function 183 7.12.1 Ambiguities in definition, 183.—7.12.2 Unit step function sequence, 184.—7.12.3 Useful formulæ involving H, 185.—7.12.4 H £'¤ distribution, 186.—7.12.5 Regularized unit step function, 186.—7.12.6 Fourier transform of the unit step function, 186. 7.13 The sign function 187 7.13.1 Definition, 187.—7.13.2 Connection to the Heaviside function, 187.— 7.13.3 Sign sequence, 188.—7.13.4 Fourier transform of sgn, 188.
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