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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 ...
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.
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.
The mathematical description of spinors in supergravity and string theory is a particularly subtle open problem, which was recently addressed in references. [ 13 ] [ 14 ] It turns out that the standard notion of spin structure is too restrictive for applications to supergravity and string theory, and that the correct notion of spinorial ...
In mathematics, particularly in the theory of spinors, the Weyl–Brauer matrices are an explicit realization of a Clifford algebra as a matrix algebra of 2 ⌊n/2⌋ × 2 ⌊n/2⌋ matrices. They generalize the Pauli matrices to n dimensions, and are a specific construction of higher-dimensional gamma matrices .
The fact that the Pauli matrices, along with the identity matrix I, form an orthogonal basis for the Hilbert space of all 2 × 2 complex matrices , over , means that we can express any 2 × 2 complex matrix M as = + where c is a complex number, and a is a 3-component, complex vector.
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.
A bispinor is more or less "the same thing" as a Dirac spinor. The convention used here is that the article on the Dirac spinor presents plane-wave solutions to the Dirac equation using the Dirac convention for the gamma matrices. That is, the Dirac spinor is a bispinor in the Dirac convention.