Clifford algebras and spinors

So I asked her the nearest way to Learning's home…. Ask for the direct road, she said, from here to Suffer-both-weal-and-woe-if you are willing to learn that lesson. Then ride past Riches, and dont stop there, for if you become attached to them you will never reach Learning. And avoid the lecherous meadow that is called Lust; leave it a good mile or more to the left, and continue till you come to a mansion called Keep-your-toungue-fromlying- and-slander-and-your-mount-from-spicy-drinks. There you will meet Sobriety and Simplicity of-Speech, and while they are with you every man will be glad to show you his wisdom. So you will come to Learning, who knows most of the answers. Taken from the 14th century text Piers the Ploughman by William Langland In this chapter we define a Clifford algebra in a flat space-time of arbitrary dimension and find its irreducible representations and their properties. This enables us to find which types of spinors are allowed in a flat space-time of a given dimension and signature. We will find that in a space-time of even dimension D, a generic, that is, Dirac, spinor has 2 D /2 complex components; however, for certain dimensions and signatures this representation is not irreducible. In these cases we can place restrictions on the Dirac spinor and find spinors which are called Majorana, Weyl or Majorana-Weyl spinors. The starting point for the construction of a supersymmetric theory is the supersymmetry algebra which underlies it. Supersymmetry algebras contain supercharges which transform as spinors under the appropriate Lorentz group. Hence, even to construct the supersymmetry algebras, we must first find out what types of spinors are possible in a given dimension and what are their properties. We will find in section 5.5 and chapter 10 that supersymmetric algebras and the supersymmetric theories on which they are based rely for their existence in an essential way on the detailed properties of Clifford algebras, which we will derive in this chapter.