Spinors in Three Dimensions - Reality

Reality

The above considerations apply equally well whether the original euclidean space under consideration is real or complex. When the space is real, however, spinors possess some additional structure which in turn facilitates a complete description of the representation of the rotation group. Suppose, for simplicity, that the inner product on 3-space has positive-definite signature:

(4)

With this convention, real vectors correspond to Hermitian matrices. Furthermore, real rotations preserving the form (4) correspond (in the double-valued sense) to unitary matrices of determinant one. In modern terms, this presents the special unitary group SU(2) as a double cover of SO(3). As a consequence, the spinor Hermitian product

(5)

is preserved by all rotations, and therefore is canonical.

If, however, the signature of the inner product on 3-space is indefinite (i.e., non-degenerate, but also not positive definite), then the foregoing analysis must be adjusted to reflect this. Suppose then that the length form on 3-space is given by:

(4′)

Then the construction of spinors of the preceding sections proceeds, but with x₂ replacing i x₂ in all the formulas. With this new convention, the matrix associated to a real vector (x₁,x₂,x₃) is itself real:

.

The form (5) is no longer invariant under a real rotation (or reversal), since the group stabilizing (4′) is now a Lorentz group O(2,1). Instead, the anti-Hermitian form

defines the appropriate notion of inner product for spinors in this metric signature. This form is invariant under transformations in the connected component of the identity of O(2,1).

In either case, the quartic form

is fully invariant under O(3) (or O(2,1), respectively), where Q is the vector-valued bilinear form described in the previous section. The fact that this is a quartic invariant, rather than quadratic, has an important consequence. If one confines attention to the group of special orthogonal transformations, then it is possible unambiguously to take the square root of this form and obtain an identification of spinors with their duals. In the language of representation theory, this implies that there is only one irreducible spin representation of SO(3) (or SO(2,1)) up to isomorphism. If, however, reversals (e.g., reflections in a plane) are also allowed, then it is no longer possible to identify spinors with their duals owing to a change of sign on the application of a reflection. Thus there are two irreducible spin representations of O(3) (or O(2,1)), sometimes called the pin representations.