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In physical terms, a spinor should determine a probability amplitude for the quantum state. A manner of regarding the product ψ ϕ as a vector. This is an essential feature of Dirac's theory, which ties the spinor formalism to the geometry of physical space. A manner of regarding a spinor as acting upon a vector, by an expression such as ψv ψ.
In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos.It appears in the plane-wave solution to the Dirac equation, and is a certain combination of two Weyl spinors, specifically, a bispinor that transforms "spinorially" under the action of the Lorentz group.
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.
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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.
In quantum mechanics, the spinor spherical harmonics [1] (also known as spin spherical harmonics, [2] spinor harmonics [3] and Pauli spinors [4]) are special functions defined over the sphere. The spinor spherical harmonics are the natural spinor analog of the vector spherical harmonics .
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for all tangent vectors X, where is the spinor covariant derivative, is Clifford multiplication and is a constant, called the Killing number of . If λ = 0 {\displaystyle \lambda =0} then the spinor is called a parallel spinor.