Principles of advanced mathematical physics, Volume 2

Preface to Volume II 
Elementary Group Theory 
18.l The group axioms; examples 
18.2 Elementary consequences of the axioms; further definitions 3 
18.3 Isomorphism 5 
18.4 Permutation groups 6 
18.5 Homomorphisms; normal subgroups 8 
18.6 eosets 10 
18.7 Factor groups 10 
18.8 The Law of Homomorphism 1J 
18.9 The structure of cyclic groups II 
18.10 Translations, inner automorphisms 12 
18.l1 The subgroups of /1\'4 13 
18.l2 Generators and relations; free groups IS 
18.13 Multiply periodic functions and crystals 16 
18.l4 The space and point groups 17 
18.15 Direct and semidirect products of groups; symmorphic space 
groups 20 
Continuous Groups 
19.1 Orthogonal and rotation groups 25 
19.2 The rotation group SO(3); Euler\'s theorem 27 
19.3 Unitary groups 28 
19.4 The Lorentz groups 29 
19.5 Group manifolds 34 
19.6 Intrinsic coordinates in the manifold of the rotation group 35 
19.7 The homomorphism of SU(2) onto SO(3) 37 
19.8 The homomorphism of SL(2, q onto the proper Lorentz 
group ~p 38 
19.9 Simplicity of the rotation and Lorentz groups 38 
20 Group Representations I: Rotations and Spherical Harmonics 40 
20.1 Finite-dimensional representations of a group 41 
20.2 Vector and tensor transformation laws 41 
20.3 Other group representations in physics 44 
20.4 Infinite-dimensional representations 45 
20.5 A simple case: SO(2) 46 
20.6 Representations of matrix groups on Xoo 47 
20.7 Homogeneous spaces 48 
20.8 Regular representations 49 
20.9 Representations of the rotation group SO(3) 50 
20.10 Tesseral harmonics; Legendre functions 53 
20.11 Associated Legendre functions 55 
20.12 Matrices of the irreducible representations of SO(3); the 
Euler angles 57 
20.13 The addition theorem for tesseral harmonics 59 
20.14 Completeness of the tesseral harmonics 60 
Group Representations II: General; Rigid Motions; 
Bessel Functions 
21.1 Equivalence; unitary representations 62 
21.2 The reduction of representations 63 
21.3 Schur\'s Lemma and its corollaries 65 
21.4 Compact and noncompact groups 66 
21.5 Invariant integration; Haar measure 67 
21.6 Complete system ofrepresentations of a compact group 71 
21.7 Homogeneous spaces as configuration spaces in physics 72 
21.8 M 2 and related groups 73 
21.9 Representations of M 2 73 
21.10 Some irreducible representations 74 
21.11 Bessel functions 75 
21.12 Matrices of the representations 76 
21.13 Characters 77 
Group Representations and Quantum Mechanics 
22.1 Representations in quantum mechanics 80 
22.2 Rotations of the axes 81 
22.3 Ray representations 82 
22.4 A finite-dimensional case 83 
22.5 Local representations 83 
22.6 Origin of the two-valued representations 84 
22.7 Representations of SU(2) and SL(2, IC) 85 
22.8 Irreducible representations of SU(2) 87 
22.9 The characters of SU(2) 89 
22.10 Functions of z and z 89 
22.11 The finite-dimensional representations of SL(2, IC) 90 
22.12 The irreducible invariant subspaces of xro for SL(2, IC) 92 
22.13 Spinors 93 
Elementary Theory of Manifolds 
23.1 Examples of manifolds; method of identification 96 
23.2 Coordinate systems or charts; compatibility; smoothness 98 
23.3 Induced topology 101 
23.4 Definition of manifold; Hausdorff separation axiom 101 
23.5 Curves and functions in a manifold 103 
23.6 Connectedness; components of a manifold 104 
23.7 Global topology; homotopic curves; fundamental group 105 
23.8 Mechanical linkages: Cartesian products 111 
Covering Manifolds 
24.1 Definition and examples 114 
24.2 Principles of lifting 117 
24.3 Universal covering manifold 119 
24.4 Comments on the construction of mathematical models 121 
24.5 Construction of the universal covering 123 
24.6 Manifolds covered by a given manifold 125 
Lie Groups 
25.1 Definitions and statement of objectives 130 
25.2 Theexpansions ofm(\" .) andI(\" .) 132 
25.3 The Lie algebra of a Lie group 133 
25.4 Abstract Lie algebras 135 
25.5 The Lie algebras of linear groups 135 
25.6 The exponential mapping; logarithmic coordinates 136 
96 
114 
129 
25.7 An auxiliary lemma on inner automorphisms; the mappings Ad p 139 
25.8 Auxiliary lemmas on formal derivatives 141 
25.9 An auxiliary lemma on the differentiation of exponentials 143 
25.10 The Campbell-Baker-Hausdorf (CBH) formula 144 
25.11 Translation of charts; compatibility; G as an analytic manifold 146 
25.12 Lie algebra homomorphisms 149 
25.13 Lie group homomorphisms 151 
25.14 Law of homomorphism for Lie groups 155 
25.15 Direct and semidirect sums of Lie algebras 160 
25.16 Classification of the simple complex Lie algebras 162 
25.17 Models of the simple complex Lie algebras 167 
25.18 Note on Lie groups and Lie algebras in physics 170 
Appendix to Chapter 25-Two nonlinear Lie groups 171 
Metric and Geodesics on a Manifold 
26.1 Scalar and vector fields on a manifold 175 
26.2 Tensor fields 180 
26.3 Metric in Euclidean space 182 
26.4 Riemannian and pseudo-Riemannian manifolds 183 
26.5 Raising and lowering of indices 185 
26.6 Geodesics in a Riemannian manifold 186 
26.7 Geodesics in a pseudo-Riamannian manifold 9Ji 190 
26.8 Geodesics; the initial-value problem; the Lipschitz condition 190 
26.9 The integral equation; Picard iterations 192 
26.10 Geodesics; the two-point problem 193 
26.11 Continuation of geodesics 194 
26.12 Affinely connected manifolds 195 
26.13 Riemannian and pseudo-Riemannian covering manifolds 197 
Riemannian, Pseudo-Riemannian, and Affinely 
Connected Manifolds 
27.1 Topology and metric 199 
27.2 Geodesic or Riemannian coordinates 199 
27.3 Normal coordinates in Riemannian and pseudo-Riemannian 
manifolds 202 
27.4 Geometric concepts; principle of equivalence 203 
27.5 Covariant differentiation 206 
27.6 Absolute differentiation along a curve 208 
27.7 Parallel transport 209 
27.8 Orientability 210 
27.9 The Riemann tensor, general; Laplacian and d\'Alembertian 211 
27.10 The Riemann tensor in a Riemannian or pseudo-Riemannian 
manifold 214 
27.11 The Riemann tensor and the intrinsic curvature of a manifold 216 
27.12 Flatness and the vanishing of the Riemann tensor 218 
27.13 Eisenhart\'s analysis of the Stackel systems 221 
The Extension of Einstein Manifolds 
28.1 Special relativity 223 
28.2 The Einstein gravitational field equations 224 
28.3 The Schwarzschild charts 227 
28.4 The Finkelstein extensions of the Schwarzschild charts 231 
28.5 The Kruskal extension 233 
28.6 Maximal extensions; geodesic completeness 235 
28.7 Other extensions of the Schwarzschild manifolds 235 
28.8 The Kerr manifolds 237 
28.9 The Cauchy problem 240 
28.10 Concluding remarks 243 
Bifurcations in Hydrodynamic Stability Problems 
29.1 The classical problems of hydrodynamic stability 244 
29.2 Examples of bifurcations in hydrodynamics 245 
29.3 The Navier-Stokes equations 247 
29.4 Hilbert space formulation 248 
29.5 The initial-value problem; the semiflow in,5 248 
29.6 The normal modes 249 
29.7 Reduction to a finite-dimensional dynamical system 250 
29.8 Bifurcation to a new steady state 254 
29.9 Bifurcation to a periodic orbit 255 
29.10 Bifurcation from a periodic orbit to an invariant torus 257 
29.11 Subharmonic bifurcation 261 
Appendix to Chapter 29-Computational details for the invariant torus 261 
Invariant Manifolds in the Taylor Problem 
30.1 Survey of the Taylor problem to 1968 263 
30.2 Calculation of invariant manifolds 265 
30.3 Cylindrical coordinates 268 
30.4 The Hilbert space 270 
30.5 Separation of variables in cylindrical coordinates 27l 
30.6 Results to date for the Taylor problem 272 
Appendix to Chapter 30-The matrices in Eagles\' formulation 274 
263 
31 
The Early Onset ofTurbulence 
276 
31.1 The Landau~Hopfmodel 276 
31.2 The Hopf example 278 
31.3 The Ruelle~ Takens model 279 
31.4 The w-limit set of a motion 280 
31.5 Attractors 282 
31.6 The power spectrum for motions in [Rn 283 
31.7 Almost periodic and aperiodic motions 284 
31.8 Lyapounov stability 285 
31.9 The Lorenz system; the bifurcations 286 
31.10 The Lorenz attractor; general description 288 
31.11 The Lorenz attractor; aperiodic motions 290 
31.12 Statistics of the mapping! and 9 293 
31.13 The Lorenz attractor; detailed structure I 294 
31.14 The symbols [i,j] of Williams 297 
31.15 Prehistories 299 
31.16 The Lorenz attractor; detailed structure II 300 
31.17 Existence of I-cells in F 301 
31.18 Bifurcation to a strange attractor 302 
31.19 The Feigenbaum model 303 
Appendix to Chapter 3I (Parts A~H)-Generic properties of systems: 304 
31.A Spaces of systems 304 
31.B Absence of Lebesgue measure in a Hilbert space 304 
31.C Generic properties of systems 305 
31.D Strongly generic; physical interpretation 305 
31.E Peixoto\'s theorem 306 
Other examples of generic and nongeneric properties 306 
Lack of correspondence between genericity and Lebesgue measure 308 
Probability and physics 308 
References 313 
Index 317