Thayer Watkins
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The Nature of Spinors

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. A vector in C2 has associated with it the isotropic vector

x1 = z02 - z12
x2 = i(z02 + z12)
x3 = -z0z1.

For any isotropic vector in C3 there will be two vectors in C2, corresponding to X; i.e., (z0, z1) and (-z0, -z1). 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 x3 axis through an angle θ it becomes
(cos(θ)-isin(θ), sin(θ)+icos(θ, 0). This is the same as

(exp(-iθ), iexp(-iθ), 0) =
exp(-iθ)(1, i, 0) = e-iθX.

The components of the spinor for X become

z0 = [(exp(-iθ) - i(iexp(-iθ)))/2]1/2 = e-iθ/2
and z1=0.

Thus Z becomes exp-iθ/2Z, 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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