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The universal cover of SO(3) is a Lie group called Spin(3). The group Spin(3) is isomorphic to the special unitary group SU(2); it is also diffeomorphic to the unit 3-sphere S 3 and can be understood as the group of versors (quaternions with absolute value 1).
The rotation group SO(3) is a subgroup of O(3), the full point rotation group of the 3D Euclidean space. ... The map Spin(3) → SO(3) is the double cover of the ...
Construction of the Spin group often starts with the construction of a Clifford algebra over a real vector space V with a definite quadratic form q. [3] The Clifford algebra is the quotient of the tensor algebra TV of V by a two-sided ideal.
Given a unit vector in 3 dimensions, for example (a, b, c), one takes a dot product with the Pauli spin matrices to obtain a spin matrix for spin in the direction of the unit vector. The eigenvectors of that spin matrix are the spinors for spin-1/2 oriented in the direction given by the vector. Example: u = (0.8, -0.6, 0) is a unit vector ...
Unlike SO(3), the group of Lorentz transformations SO(3,1) is non-compact and therefore does not have any faithful, unitary, finite-dimensional representations. In case of spin- 1 / 2 particles, it is possible to find a construction that includes both a finite-dimensional representation and a scalar product that is preserved by this ...
The spin group is the group of rotations keeping track of the homotopy class. Spinors are needed to encode basic information about the topology of the group of rotations because that group is not simply connected, but the simply connected spin group is its double cover. So for every rotation there are two elements of the spin group that ...
In geometry the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R 3 under the operation of composition. [1] By definition, a rotation about the origin is a linear transformation that preserves length of vectors (it is an isometry) and preserves orientation (i.e. handedness) of space.
One often says that the restricted Lorentz group and the rotation group are doubly connected. This means that the fundamental group of the each group is isomorphic to the two-element cyclic group Z 2. Twofold coverings are characteristic of spin groups. Indeed, in addition to the double coverings Spin + (1, 3) = SL(2, C) → SO + (1, 3)