Spinors in Three Dimensions - Isotropic Vectors

Isotropic Vectors

Spinors can be constructed directly from isotropic vectors in 3-space without using the quaternionic construction. To motivate this introduction of spinors, suppose that X is a matrix representing a vector x in complex 3-space. Suppose further that x is isotropic: i.e.,

Then, from the properties of these matrices, X2 = 0. Any such matrix admits a factorization as an outer product

This factorization yields an overdetermined system of equations in the coordinates of the vector x:

(1)

subject to the constraint

(2)

This system admits the solutions

(3)

Either choice of sign solves the system (1). Thus a spinor may be viewed as an isotropic vector, along with a choice of sign. Note that because of the logarithmic branching, it is impossible to choose a sign consistently so that (3) varies continuously along a full rotation among the coordinates x. In spite of this ambiguity of the representation of a rotation on a spinor, the rotations do act unambiguously by a fractional linear transformation on the ratio ξ₁:ξ₂ since one choice of sign in the solution (3) forces the choice of the second sign. In particular, the space of spinors is a projective representation of the orthogonal group.

As a consequence of this point of view, spinors may be regarded as a kind of "square root" of isotropic vectors. Specifically, introducing the matrix

the system (1) is equivalent to solving X = 2 ξ tξ C for the undetermined spinor ξ.

A fortiori, if the roles of ξ and x are now reversed, the form Q(ξ) = x defines, for each spinor ξ, a vector x quadratically in the components of ξ. If this quadratic form is polarized, it determines a bilinear vector-valued form on spinors Q(μ, ξ). This bilinear form then transform tensorially under a reflection or a rotation.