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The spinor/quaternion representation of rotations in 3D is becoming increasingly prevalent in computer geometry and other applications, because of the notable brevity of the corresponding spin matrix, and the simplicity with which they can be multiplied together to calculate the combined effect of successive rotations about different axes.
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 ...
For example, effectively all Riemannian manifolds can have spinors and spin connections built upon them, via the Clifford algebra. [1] The Dirac spinor is specific to that of Minkowski spacetime and Lorentz transformations; the general case is quite similar. This article is devoted to the Dirac spinor in the Dirac representation.
The spinor bundle is defined [1] to be the complex vector bundle = associated to the spin structure via the spin representation: (), where () denotes the group of unitary operators acting on a Hilbert space.
Here the coordinates of physical points are transformed according to ′ =, while , a matrix, is an element of the spinor representation (for spin 1/2) of the Lorentz group. In the Weyl basis, explicit transformation matrices for a boost Λ b o o s t {\displaystyle \Lambda _{\rm {boost}}} and for a rotation Λ r o t {\displaystyle \Lambda _{\rm ...
In theoretical physics, a Fierz identity is an identity that allows one to rewrite bilinears of the product of two spinors as a linear combination of products of the bilinears of the individual spinors.
By counting dimensions, A is a complete 2 k ×2 k matrix algebra over the complex numbers. As a matrix algebra, therefore, it acts on 2 k-dimensional column vectors (with complex entries). These column vectors are the spinors. We now turn to the action of the orthogonal group on the spinors.
A pure spinor is defined to be any element () that is annihilated by a maximal isotropic subspace with respect to the scalar product . Conversely, given a maximal isotropic subspace it is possible to determine the pure spinor that annihilates it, up to multiplication by a complex number, as follows.