Rotations, Quaternions, and Double GroupsThis self-contained text presents a consistent description of the geometric and quaternionic treatment of rotation operators, employing methods that lead to a rigorous formulation and offering complete solutions to many illustrative problems. Geared toward upper-level undergraduates and graduate students, the book begins with chapters covering the fundamentals of symmetries, matrices, and groups, and it presents a primer on rotations and rotation matrices. Subsequent chapters explore rotations and angular momentum, tensor bases, the bilinear transformation, projective representations, and the geometry, topology, and algebra of rotations. Some familiarity with the basics of group theory is assumed, but the text assists students in developing the requisite mathematical tools as necessary. |
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algebra angular momentum axes bases basis bilinear transformation binary rotation called Chapter column commute complex numbers components configuration space conjugate corresponding Ď₂ defined denote double group eigenvalues eigenvectors entails equal equation Euler angles Euler-Rodrigues parameters factor system follows functions g₁ geometrical given Hamilton Hermitian identity imaginary unit improper rotations infinitesimal rotations introduced invariant inversion irreducible representations mapping mathematical multiplication rules multiplication table negative normal notation Notice obtained operation g orthogonal parametric ball parametric point path phase factors plane point groups pole Problem projective factors projective representations proper rotations Prove pure quaternion quaternion parameters quaternion units R(on representation of SO(3 respectively result Rodrigues rotation angle rotation group rotation matrix sin² spherical spinor representations standard subgroup symbol symmetry operation tensor theorem theory tion transformation unit sphere unitary unitary matrix vector representations Verify