Appendix A: Spinors and Gamma Matrix Algebra

A.1 Introduction to the Spinor Representations of SO(1, D − 1)

The spinor representations of the orthogonal and pseudo-orthogonal groups have different structure in various dimensions. Starting from the representation of the Dirac gamma matrices one begins with a complex representation whose dimension is equal to the dimension of the gammas. A vector in this complex linear space is named a Dirac spinor. Typically Dirac spinors do not form irreducible representations. Depending on the dimensions, one can still impose SO(1, D − 1) invariant conditions on the Dirac spinor that separate it into irreducible parts. These constraints can be of two types:

(a)A reality condition which maintains the number of components of the spinor but relates them to their complex conjugates by means of linear relations. This reality condition is constructed with an invariant matrix C , named the charge conjugation matrix whose properties depend on the dimensions D.

(b)A chirality condition constructed with a chirality matrix ΓD+1 that halves the number of components of the spinor. The chirality matrix exists only in even dimensions.

Depending on which conditions can be imposed, besides Dirac spinors, in various dimensions D, one has Majorana spinors, Weyl spinors and, in certain dimensions, also Majorana-Weyl spinors. In this appendix we discuss the properties of gamma matrices and we present the various types of irreducible spinor representations in all relevant dimensions from D = 4 to D = 11. The upper bound D = 11 is dictated by supersymmetry since supergravity, i.e. the supersymmetric extension of Einstein gravity, can be constructed in all dimensions up to D = 11, which is maximal in this respect. In the present volume, supergravity is not discussed, but some glances of it will occur in the second volume and for this reason, while discussing the necessary topic of gamma matrix algebra, we present a complete description of the available spinors in the various relevant dimensions.

A.2 The Clifford Algebra

In order to describe spinors one needs the Dirac gamma matrices. These form the Clifford algebra:

where ηab is the invariant metric of SO(1, D − 1), that we always choose according to the mostly minus conventions, namely:

To study the general properties of the Clifford algebra (A.2.1) we use a direct construction method.

We begin by fixing the following conventions. Γ 0 = Γ0 corresponding to the time direction is Hermitian:

while the matrices Γi = −Γ i corresponding to space directions are anti-Hermitian:

In the study of Clifford algebras it is necessary to distinguish the case of even and odd dimensions.

A.2.1 Even Dimensions

When D = 2ν is an even number the representation of the Clifford algebra (A.2.1) has dimension:

In other words the gamma matrices are 2ν × 2ν . The proof of such a statement is easily obtained by iteration. Suppose that we have the gamma matrices γa corresponding to the case ν = ν − 1, satisfying the Clifford algebra (A.2.1) in D − 2 dimensions and that they are 2ν -dimensional. We can write down the following

representation for the gamma matrices in D-dimension by means of the following 2ν × 2ν matrices:

(A.2.6)

which satisfy the correct anticommutation relations and have the correct hermiticity properties specified above. This representation admits the following interpretation in terms of matrix tensor products:

To complete the proof of our statement we just have to show that for ν = 2, corresponding to D = 4 we have a 4-dimensional representation of the gamma matrices.