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# Altmann L.S. Rotations, quaternions, and double groups

• Файл формата pdf
• размером 16,01 МБ
• Добавлен пользователем
• Отредактирован 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)