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The concept of spinor is now important in theoretical physics but it is a difficult topic to gain acquaintance with. Spinors were defined by Elie Cartan, the French mathematician, in terms of three dimensional vectors whose components are complex. The vectors which are of interest are the ones such that their dot product with themselves is zero.
Let X=(x_{1}, x_{2}, x_{3}) be an element of the vector space C^{3}. The dot product of X with itself, X·X, is (x_{1}x_{1}+x_{2}x_{2}+x_{3}x_{3}. Note that if x=a+ib then x·x=x^{2}=a^{2}+b^{2} + i(2ab), rather that a^{2}+b^{2}, which is x times the conjugate of x.
A vector X is said to be isotropic if X·X=0. Isotropic vectors could be said to be orthogonal to themselves, but that terminology causes mental distress.
It can be shown that the set of isotropic vectors in C^{3} form a two dimensional surface. This two dimensional surface can be parameterized by two coordinates, z_{0} and z_{1} where
The complex two dimensional vector Z=(z_{0}, z_{1}) Cartan calls a spinor. But a spinor is not just a two dimensional complex vector; it is a representation of an isotropic three dimensional complex vector. A vector in C^{2} has associated with it the isotropic vector
For any isotropic vector in C^{3} there will be two vectors in C^{2}, corresponding to X; i.e., (z_{0}, z_{1}) and (z_{0}, z_{1}). Both of these will map into the same isotropic X.
When operations such as rotations are carried out on the isotropic
vectors the results in terms of the spinor representations are quite
interesting. For example, suppose X = (1, i, 0). This is an isotropic
vector and its spinors are Z=(1,0) and Z=(1,0). If X is rotated
about
the x_{3} axis through an angle θ it becomes
(cos(θ)isin(θ), sin(θ)+icos(θ, 0).
This is the same as
The components of the spinor for X become
Thus Z becomes exp^{iθ/2}Z, a rotation of θ/2.
When X is rotated through an angle 2π the spinors for X get rotated through an angle of π and thus Z goes to Z. It takes a rotation of 4π of the isotropic vector to rotate Z back to Z.
It is impossible to visual depict isotropic vectors and spinors because three dimensional complex vectors involve six dimensions and spinors as two dimensional complex vectors involve four dimensions.
For other interesting properties of vectors with complex components see Bezout's Theorem.
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