Gravity and Strings

616 Appendix B
When d is even, both matrices exist because +Ɣ a T and −Ɣ a T generate the same
finite group as Ɣ a . When d is odd, only one of them exists because of the above
definition of Ɣ d−1 in terms of Ɣ 0 ,...,Ɣ d−2 . Furthermore, both matrices, when they
exist, are either symmetric or antisymmetric.
When d is even, the charge-conjugation matrices act on Ɣd+1 as follows:
C±Ɣd+1C −1
± = ϕ2Ɣ T d+1 . (B.31)
Using the C± charge-conjugation matrices and taking the Majorana conjugate of the
Dirac equation, we find that the Majorana conjugate satisfies the following equation:
ψ c (i ←− ∇∓m + e A) = 0, (B.32)
which implies that ψ c has charge opposite to that of ¯ψ (hence the name “chargeconjugation
matrix”). It is obviously desirable that both ψ c and ¯ψ have the same
mass and, thus, in the massive case the only acceptable charge-conjugation matrix is
C−.
By construction ψ cψ is Lorentz-invariant and (ψ c ) c = ψ.
We can now study various types of spinors that are in general associated with special
representations of gamma matrices.
Weyl spinors (Also called chiral spinors.) For even d it is possible to define as before the
chirality matrix Ɣd+1 which anticommutes with all the gamma matrices and therefore
commutes with the generators of the Lorentz group Ɣs(Mab) and with their exponentials,
which span the Spin(1, d − 1) group. Thus (Schur’s lemma) this representation
of the Spin(1, d − 1) group, and Dirac spinors, are reducible even if the gamma matrices
provide an irreducible representation of the d-dimensional Clifford algebra of
ηab. The chirality matrix is traceless and squares to unity and therefore half of its
eigenvalues are +1s and the other half are −1s. It is natural to split the space of
Dirac spinors into the direct sum of the subspaces of spinors with eigenvalues +1
and −1. The elements of each of these subspaces are called Weyl spinors and, by
definition, satisfy the Weyl or chirality condition
1
2 (1 ± Ɣd+1)ψ = ψ. (B.33)
For the positive sign, the spinors are called left-handed (negative chirality); and for
the negative sign they are called right-handed (positive chirality). A Weyl spinor
describes half the degrees of freedom of a Dirac spinor.
Observe that, while Weyl spinors are irreducible representations of the Spin(1, d −
1) group, they are not irreducible representations of the Lorentz group SO(1, d − 1)
because this group contains discrete transformations that interchange the two subspaces
of opposite chiralities. In particular, the parity transformation is implemented
by P = iƔ 0 ,which does not commute but anticommutes with the chirality matrix,
switching the chirality of the spinors.