Complex-number vectors {spinor}| have rotation around an axis. Specifically, complex-number vectors are second-rank Hermitean spinors. Spinors have direction, amplitude, and frequency. Spinors can be hypercomplex-number vectors. Spinors are like flagpoles, plus flags with lengths, plus orientation-entanglement relations. Though flagpole and flag are like two vectors, spinors are not bivectors, which have real numbers only. Complex-number bivectors are bispinors. Complex-number trivectors are trispinors.
chirality
Spinors have right-handed or left-handed orientation (chirality).
axis
Vectors can rotate around own axis, coordinate axis, or any axis.
quaternions
Quaternions have form a + b*i + c*j + d*k, where a, b, c, and d are real numbers, and i, j, and k are orthogonal unit vectors, so quaternions are vectors in three-dimensional space but with added scalar. Rotating quaternions are real-number spinors. Multiplying quaternions gives i*j = k, j*k = i, k*i = j, j*i = -k, k*j = -i, and i*k = -j, so quaternion multiplication is non-commutative. Multiplying quaternions describes quaternion rotations. Rotation transforms quaternion coordinates {spinor transformation}.
spin matrix
Matrices {spin matrix} describe quaternion and spinor rotations. Spin matrices are scalar products of spinor matrix and rotation matrix: new spin matrix = (rotation matrix) * (old spin matrix) * (rotation-matrix conjugate transpose).
rotation
Spinors reverse sign for 360-degree rotation, because loop cannot shrink to point. Spinors reverse sign twice for 720-degree rotation (4 * pi radians). Rotation sums are vector sums. Two rotations make double-twist that is equivalent to no twist {topological torsion}, because loop can shrink to point.
Complex-plane pi/2 radian rotations rotate pi radians in spherical representations, along oriented great-circle arcs.
If space has n dimensions, rotations are always about axes with n - 2 dimensions. Reflections are always through n - 1 dimension plane. Two reflections through perpendicular planes are equivalent to rotation through pi radians.
history
Elie Cartan discovered spinors [1913] and invented orthogonal-group representation theory.
purposes
Spinors relate geometry, topology, and analysis. Spinors describe fermion and boson spin. Spin matrices (Pauli) and relativistic electron-spin theory (Dirac) use spinors. Spinors are in index theorems for elliptic operators, characteristic number integrability, positive scalar curvature metric existence, twistor spaces, Seiberg-Witten theory, Clifford algebras, spin groups, manifold spin structures, Dirac operators, supersymmetry, four-manifold invariants, and superstring theory.
Mathematical Sciences>Calculus>Vector>Kinds
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Date Modified: 2022.0224