Principles of Advanced Mathematical Physics

This item is printed on demand - Print on Demand Titel. Neuware -Inhaltsangabe18 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 Finitedimensional representations of a group. 20.2 Vector and tensor transformation laws. 20.3 Other group representations in physics. 20.4 Infinitedimensional 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 finitedimensional case. 22.5 Local representations. 22.6 Origin of the twovalued 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 finitedimensional 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 CampbeSpringer-.…