Principles of Advanced Mathematical Physics : Volume II

R. D. Richtmyer

ISBN 10: 3642510787 / ISBN 13: 9783642510786

Published by Springer Berlin Heidelberg, 2012

About this Item

Druck auf Anfrage Neuware - Printed after ordering - 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 Campbe.

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Bibliographic Details

• Title: Principles of Advanced Mathematical Physics : Volume II
• Publisher: Springer Berlin Heidelberg
• Publication Date: 2012
• Language: English
• ISBN 10: 3642510787
• ISBN 13: 9783642510786
• Binding: Taschenbuch
• Condition: Neu
• Dimensions: 235W x 155H x 19D millimeters 235 millimeters width by 155 millimeters height by 19 millimeters depth
• Weight: 511 grams

About this title

Synopsis

A first consequence of this difference in texture concerns the attitude we must take toward some (or perhaps most) investigations in "applied mathe­ matics," at least when the mathematics is applied to physics. Namely, those investigations have to be regarded as pure mathematics and evaluated as such. For example, some of my mathematical colleagues have worked in recent years on the Hartree-Fock approximate method for determining the structures of many-electron atoms and ions. When the method was intro­ duced, nearly fifty years ago, physicists did the best they could to justify it, using variational principles, intuition, and other techniques within the texture of physical reasoning. By now the method has long since become part of the established structure of physics. The mathematical theorems that can be proved now (mostly for two- and three-electron systems, hence of limited interest for physics), have to be regarded as mathematics. If they are good mathematics (and I believe they are), that is justification enough. If they are not, there is no basis for saying that the work is being done to help the physicists. In that sense, applied mathematics plays no role in today's physics. In today's division of labor, the task of the mathematician is to create mathematics, in whatever area, without being much concerned about how the mathematics is used; that should be decided in the future and by physics.