New York: Oxford University Press, 1986. — 312 p.Notation-conventions-how to use this book Introduction The Rodrigues programme Rotations by 2π Spinor representations All you need to know about symmetries, matrices, and groups Symmetry operators in configuration space. Description of the point-symmetry operations. Specification of the symmetry operations. Composition of symmetry operations Eigenvectors of configuration space operators Symmetry operators in function space Matrices and operators Groups All about matrix properties. Orthogonal matrices. Unitary matrices. Hermitian and skew-Hermitian matrices. Supermatrices and the direct product. Commutation of matrices. Matrix functions Quantal symmetry. Observables and infinitesimal operators. Symmetries and observables. Infinitesimal operators and observables A primer on rotations and rotation matrices Euler angles. Rotation matrices in terms of the Euler angles Angle and axis of an orthogonal matrix The matrix of a rotation R(φn) Euler angles in terms of the angle and axis of rotation A rotation in terms of rotations about orthogonal axes Comments on the parametrization of rotations Rotations and angular momentum Infinitesimal rotations The infinitesimal generator: angular momentum Rotation matrices Commutation Shift operators The eigenfunctions of Iz The irreducible bases for SO(3). Spherical and solid harmonics The Condon and Shortley convention Applications. Matrices for j = 1 and j =1/2 (Pauli matrices). The Pauli matrices, j =1/2 Tensor bases: introduction to spinors Vectors and spherical vectors Tensor bases and tensor products. Symmetrization of tensors Half-integral bases: spinors The bilinear transformation: introduction to SU(2), SU'(2), and rotations, more about spinors The bilinear transformation. The inverse Special unitary matrices. The SU(2) group Rotations and SU(2): a first contact Binary rotations as the group generators Do we have a representation of SO(3)? SU(2) plus the inversion: SU'(2). Inversion and parity Spinors and their invariants Rotations and SU(2). The stereographic projection The stereographic projection Geometry and coordinates of the projection The homomorphism between SU(2) and SO(3). The spinor components Projective representations The group D2 and its SU(2) matrices. Definition of projective representations Bases of the projective representations. Bases and energy levels The factor system The representations. Characters Direct products of representations The covering group. Remarks The geometry of rotations The unit sphere and the rotation poles. Conjugate poles. Improper rotations The Euler construction Spherical trigonometry revisited The Euler construction in formulae. The Euler-Rodrigues paranleters. Remarks The conical transformation The topology of rotations The parametric ball Paths Programme: continuity Homotopy The projective factors Operations, turns, and connectivity The spinor representations Determination of the projective factors The intertwining theorem The character theorem The irreducible representations The projective factors from the Euler-Rodrigues parameters Inverses and conjugates in the Euler-Rodrigues parametrization. Conjugation and the choice of the positive hemisphere The character theorem proved in the Euler-Rodrigues parametrization The SU(2) representation of SO(3) Ci and the irreducible representations of O(3). The SU'(2) representation of O(3). The representations of Ci . The irreducible representations of O(3). The SU'(2) representation of O(3). The factor system for O(3) Improper point groups The algebra of rotations: quaternions An entertainment on binary rotations The definition of quaternions Inversion of quaternions. Characterization of their scalar and vector parts Conjugate and normalized quaternions. Inverse quaternions The quaternion units SO(3), SU(2), and quaternions Exponential form of quaternions The conical transformation The rectangular transformation Quaternion algebra and the Clifford algebra. In praise of mirrors Applications: angle and axis of rotation and SU(2) matrices in terms of Euler angles Double groups Introduction and example The double group in the quaternion parametrization Notation and operational rules. Intertwining Class structure: Opechowski Theorem The irreducible representations of SO(3) More about spinor bases The irreducible representation The bases of the representations Examples and applications The choice of the positive hemisphere Parametrization of the group elements for D6, D3, C3v . Multiplication tables and factor systems The standard representation The irreducible projective and vector representations. The representations of D3. The representations of C3v. The double group D3 Some applications Solutions to problems References Index
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