Spinors in Three Dimensions - Reality Structures

Reality Structures

The differences between these two signatures can be codified by the notion of a reality structure on the space of spinors. Informally, this is a prescription for taking a complex conjugate of a spinor, but in such a way that this may not correspond to the usual conjugate per the components of a spinor. Specifically, a reality structure is specified by a Hermitian 2 × 2 matrix K whose product with itself is the identity matrix: K2 = Id. The conjugate of a spinor with respect to a reality structure K is defined by

The particular form of the inner product on vectors (e.g., (4) or (4′)) determines a reality structure (up to a factor of -1) by requiring

, whenever X is a matrix associated to a real vector.

Thus K = i C is the reality structure in Euclidean signature (4), and K = Id is that for signature (4′). With a reality structure in hand, one has the following results:

• X is the matrix associated to a real vector if, and only if, .
• If μ and ξ is a spinor, then the inner product

determines a Hermitian form which is invariant under proper orthogonal transformations.