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Thursday, August 5, 2010


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=(x1, x2, x3) be an element of the vector space C3. The dot product of X with itself, X·X, is (x1x1+x2x2+x3x3. Note that if x=a+ib then x·x=x2=a2+b2 + i(2ab), rather that a2+b2, 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 C3 form a two dimensional surface. This two dimensional surface can be parameterized by two coordinates, z0 and z1 where

z0 = [(x1-ix2)/2]1/2
z1 = i[(x1+ix2)/2]1/2.

The complex two dimensional vector Z=(z0, z1) 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.

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