Motivation for abstract index approach
In Chapter 1 we introduced the concept of a spin-vector. We saw that a spin-vector is to be pictured essentially as a null flag in Minkowski vector space, but with the additional property that under rotation through 2π about any axis it is returned, not to its original state, but to another spin-vector associated with the same null flag, called the negative of the original spin-vector. Spin-vectors form a two-dimensional complex vector space, called spin-space, on which a skew-symmetric inner product is also defined. All operations have explicit geometrical, Lorentz invariant interpretations, in space–time terms.
Later in this chapter (in §2.5) we shall develop the algebra of spinors. The essential idea is that spinors may be constructed, starting from the basic concept of spin-space, analogously to the way that tensors are built up starting from the concept of a vector space. It will emerge, moreover (in §3.1), that the world-tensor algebra of space–time is contained in the spinor algebra. Thus spin-space is, in a sense, even more basic than worldvector space. It is, therefore, conceptually very valuable that spin-space has a clear-cut geometrical space–time interpretation. For this removes much of the abstractness which has tended to cloud the spinor concept. It shows, furthermore, that while we shall describe spinors and spinor operations in this (and subsequent) chapters in a largely algebraic way, nevertheless each such object and operation will have an essential geometrical content in space–time terms.