Principles of Advanced Mathematical Physics

Inhaltsverzeichnis

• 18 Elementary Group Theory.
• 18.1 The group axioms; examples.
• 18.2 Elementary consequences of the axioms; further definitions.
• 18.3 Isomorphism.
• 18.4 Permutation groups.
• 18.5 Homomorphisms; normal subgroups.
• 18.6 Cosets.
• 18.7 Factor groups.
• 18.8 The Law of Homomorphism.
• 18.9 The structure of cyclic groups.
• 18.10 Translations, inner automorphisms.
• 18.11 The subgroups of ?4.
• 18.12 Generators and relations; free groups.
• 18.13 Multiply periodic functions and crystals.
• 18.14 The space and point groups.
• 18.15 Direct and semidirect products of groups; symmorphic space groups.
• 19 Continuous Groups.
• 19.1 Orthogonal and rotation groups.
• 19.2 The rotation group SO(3); Euler’s theorem.
• 19.3 Unitary groups.
• 19.4 The Lorentz groups.
• 19.5 Group manifolds.
• 19.6 Intrinsic coordinates in the manifold of the rotation group.
• 19.7 The homomorphism of SU(2) onto SO(3).
• 19.8 The homomorphism of SL(2, ?) onto the proper Lorentz group ?
  p.
• 19.9 Simplicity of the rotation and Lorentz groups.
• 20 Group Representations I: Rotations and Spherical Harmonics.
• 20.1 Finite-dimensional representations of a group.
• 20.2 Vector and tensor transformation laws.
• 20.3 Other group representations in physics.
• 20.4 Infinite-dimensional representations.
• 20.5 A simple case: SO(2).
• 20.6 Representations of matrix groups on X?.
• 20.7 Homogeneous spaces.
• 20.8 Regular representations.
• 20.9 Representations of the rotation group SO(3).
• 20.10 Tesseral harmonics; Legendre functions.
• 20.11 Associated Legendre functions.
• 20.12 Matrices of the irreducible representations of SO(3); the Euler angles.
• 20.13 The addition theorem for tesseral harmonics.
• 20.14 Completeness of the tesseral harmonics.
• 21 Group Representations II: General; Rigid Motions; Bessel Functions.
• 21.1 Equivalence; unitary representations.
• 21.2 The reduction of representations.
• 21.3 Schur’s Lemma and its corollaries.
• 21.4 Compact and noncompact groups.
• 21.5 Invariant integration; Haar measure.
• 21.6 Complete system of representations of a compact group.
• 21.7 Homogeneous spaces as configuration spaces in physics.
• 21.8 M2 and related groups.
• 21.9 Representations of M2.
• 21.10 Some irreducible representations.
• 21.11 Bessel functions.
• 21.12 Matrices of the representations.
• 21.13 Characters.
• 22 Group Representations and Quantum Mechanics.
• 22.1 Representations in quantum mechanics.
• 22.2 Rotations of the axes.
• 22.3 Ray representations.
• 22.4 A finite-dimensional case.
• 22.5 Local representations.
• 22.6 Origin of the two-valued representations.
• 22.7 Representations of SU(2) and SL(2, ?).
• 22.8 Irreducible representations of SU(2).
• 22.9 The characters of SU(2).
• 22.10 Functions of z and z?.
• 22.11 The finite-dimensional representations of SL(2, ?).
• 22.12 The irreducible invariant subspaces of X? for SL(2, ?).
• 22.13 Spinors.
• 23 Elementary Theory of Manifolds.
• 23.1 Examples of manifolds; method of identification.
• 23.2 Coordinate systems or charts; compatibility; smoothness.
• 23.3 Induced topology.
• 23.4 Definition of manifold; Hausdorff separation axiom.
• 23.5 Curves and functions in a manifold.
• 23.6 Connectedness; components of a manifold.
• 23.7 Global topology; homotopic curves; fundamental group.
• 23.8 Mechanical linkages: Cartesian products.
• 24 Covering Manifolds.
• 24.1 Definition and examples.
• 24.2 Principles of lifting.
• 24.3 Universal covering manifold.
• 24.4 Comments on the construction of mathematical models.
• 24.5 Construction of the universal covering.
• 24.6 Manifolds covered by a given manifold.
• 25 Lie Groups.
• 25.1 Definitions and statement of objectives.
• 25.2 The expansions of m(·, ·) and l(·, ·).
• 25.3 The Lie algebra of a Lie group.
• 25.4 Abstract Lie algebras.
• 25.5 The Lie algebras of linear groups.
• 25.6 The exponential mapping; logarithmic coordinates.
• 25.7 An auxiliary lemma on inner automorphisms; the mappings Ad?.
• 25.8 Auxiliary lemmas on formal derivatives.
• 25.9 An auxiliary lemma on the differentiation of exponentials.
• 25.10 The Campbell-Baker-Hausdorf (CBH) formula.
• 25.11 Translation of charts; compatibility; G as an analytic manifold.
• 25.12 Lie algebra homomorphisms.
• 25.13 Lie group homomorphisms.
• 25.14 Law of homomorphism for Lie groups.
• 25.15 Direct and semidirect sums of Lie algebras.
• 25.16 Classification of the simple complex Lie algebras.
• 25.17 Models of the simple complex Lie algebras.
• 25.18 Note on Lie groups and Lie algebras in physics.
• Appendix to Chapter 25—Two nonlinear Lie groups.
• 26 Metric and Geodesics on a Manifold.
• 26.1 Scalar and vector fields on a manifold.
• 26.2 Tensor fields.
• 26.3 Metric in Euclidean space.
• 26.4 Riemannian and pseudo-Riemannian manifolds.
• 26.5 Raising and lowering of indices.
• 26.6 Geodesies in a Riemannian manifold.
• 26.7 Geodesies in a pseudo-Riamannian manifold M.
• 26.8 Geodesies; the initial-value problem; the Lipschitz condition.
• 26.9 The integral equation; Picard iterations.
• 26.10 Geodesies; the two-point problem.
• 26.11 Continuation of geodesies.
• 26.12 Affmely connected manifolds.
• 26.13 Riemannian and pseudo-Riemannian covering manifolds.
• 27 Riemannian, Pseudo-Riemannian, and Affinely Connected Manifolds.
• 27.1 Topology and metric.
• 27.2 Geodesic or Riemannian coordinates.
• 27.3 Normal coordinates in Riemannian and pseudo-Riemannian manifolds.
• 27.4 Geometric concepts; principle of equivalence.
• 27.5 Covariant differentiation.
• 27.6 Absolute differentiation along a curve.
• 27.7 Parallel transport.
• 27.8 Orientability.
• 27.9 The Riemann tensor, general; Laplacian and d’Alembertian.
• 27.10 The Riemann tensor in a Riemannian or pseudo-Riemannian manifold.
• 27.11 The Riemann tensor and the intrinsic curvature of a manifold.
• 27.12 Flatness and the vanishing of the Riemann tensor.
• 27.13 Eisenhart’s analysis of the Stäckel systems.
• 28 The Extension of Einstein Manifolds.
• 28.1 Special relativity.
• 28.2 The Einstein gravitational field equations.
• 28.3 The Schwarzschild charts.
• 28.4 The Finkelstein extensions of the Schwarzschild charts.
• 28.5 The Kruskal extension.
• 28.6 Maximal extensions; geodesic completeness.
• 28.7 Other extensions of the Schwarzschild manifolds.
• 28.8 The Kerr manifolds.
• 28.9 The Cauchy problem.
• 28.10 Concluding remarks.
• 29 Bifurcations in Hydrodynamic Stability Problems.
• 29.1 The classical problems of hydrodynamic stability.
• 29.2 Examples of bifurcations in hydrodynamics.
• 29.3 The Navier-Stokes equations.
• 29.4 Hilbert space formulation.
• 29.5 The initial-value problem; the semiflow in ?.
• 29.6 The normal modes.
• 29.7 Reduction to a finite-dimensional dynamical system.
• 29.8 Bifurcation to a new steady state.
• 29.9 Bifurcation to a periodic orbit.
• 29.10 Bifurcation from a periodic orbit to an invariant torus.
• 29.11 Subharmonic bifurcation.
• Appendix to Chapter 29—Computational details for the invariant torus.
• 30 Invariant Manifolds in the Taylor Problem.
• 30.1 Survey of the Taylor problem to 1968.
• 30.2 Calculation of invariant manifolds.
• 30.3 Cylindrical coordinates.
• 30.4 The Hilbert space.
• 30.5 Separation of variables in cylindrical coordinates.
• 30.6 Results to date for the Taylor problem.
• Appendix to Chapter 30—The matrices in Eagles’ formulation.
• 31 The Early Onset of Turbulence.
• 31.1 The Landau-Hopf model.
• 31.2 The Hopf example.
• 31.3 The Ruelle-Takens model.
• 31.4 The co-limit set of a motion.
• 31.5 Attractors.
• 31.6 The power spectrum for motions in ? n.
• 31.7 Almost periodic and aperiodic motions.
• 31.8 Lyapounov stability.
• 31.9 The Lorenz system; the bifurcations.
• 31.10 The Lorenz attractor; general description.
• 31.11 The Lorenz attractor; aperiodic motions.
• 31.12 Statistics of the mapping f and g.
• 31.13 The Lorenz attractor; detailed structure I.
• 31.14 The symbols [i, j] of Williams.
• 31.15 Prehistories.
• 31.16 The Lorenz attractor; detailed structure II.
• 31.17 Existence of 1-cells in F.
• 31.18 Bifurcation to a strange attractor.
• 31.19 The Feigenbaum model.
• Appendix to Chapter 31 (Parts A-H)—Generic properties of systems:.
• 31. A Spaces of systems.
• 31. B Absence of Lebesgue measure in a Hilbert space.
• 31. C Generic properties of systems.
• 31. D Strongly generic; physical interpretation.
• 31. E Peixoto’s theorem.
• 31. F Other examples of generic and nongeneric properties.
• 31. G Lack of correspondence between genericity and Lebesgue measure 308 31. H Probability and physics.
• References.