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 andthe angular momentum axes axis bases basis bilinear transformation binary rotations bythe called canbe Clifford algebra column commute complex numbers components configuration space conjugate convention corresponding defined denote double group eigenvalues eigenvectors entails equal equation Euler angles EulerÂRodrigues parameters factor system follows fromthe functions g i g gauge geometrical given Hamilton Hermitian homotopy identity image image image imaginary unit improper rotations infinitesimal rotations inthe invariant inversion irreducible representations mapping multiplication rules negative normalized notation Notice obtained ofthe operation g orthogonal parametric ball parametric point path Pauli Pauli matrices phase factors plane point groups pole Problem projective factors projective representations Prove pure quaternion quaternion parameters quaternion units representation of SO(3 respectively result Rodrigues rotation group rotation matrix scalar spinor representations standard subgroup symbol symmetry operation tensor thatthe theorem tothe unitary vector representations Verify weshall whence