Gravity and Strings

Gamma matrices and spinors 613
where σ 1,2,3 are the (Hermitian, unitary, 2 × 2) Pauli matrices
that satisfy
σ 1 =
01
10
, σ 2 =
0 −i
i 0
, σ 3 =
1 0
, (B.9)
0 −1
σ i σ j = δ ij + iɛ ijk σ k , (B.10)
is a 2 [d/2] -dimensional representation of the d-dimensional Clifford algebra. A d = 2 representation
of the two-dimensional Clifford algebra is provided by (for instance) I2×2, iσ 2
and this completes the proof for d even.
Now, if d is even and Ɣ 0 ,...,Ɣ d−1 are 2 d/2 × 2 d/2 gamma matrices satisfying the
d-dimensional Clifford algebra, then the gamma matrices Ɣ 0 ,...,Ɣ d−1 ,Ɣ d with
Ɣ d ≡−iϕ(d)Ɣ 0 ···Ɣ d−1 , ϕ(d) = (−1) 1 4 (d−2)+1 , (B.11)
satisfy the (d + 1)-dimensional Clifford algebra. 4 Thus, the even d irreducible representations
determine the d + 1 irreducible representations and this completes the proof. Observe
that this matrix is different from the chirality matrix Q = Ɣd+1 (γ5 in d = 4):
Ɣd+1 = iƔ d = ϕ(d)iƔ 0 ···Ɣd−1 ,
Ɣ2 d+1 =+1, Ɣ† d+1 =+Ɣd+1. Ɣ0Ɣd+1Ɣ0 =−Ɣ †
d+1 .
(B.13)
Observe also that, in odd dimensions, by construction, the product of all gamma matrices
is proportional to a constant whose sign can be chosen at will (by changing the sign of Ɣd ).
The two possible signs give inequivalent representations of the Clifford algebra (which are,
nevertheless, physically equivalent).
Let us now consider equivalent representations of the Clifford algebra, related by a similarity
transformation
Ɣ a ′ = SƔ a S −1 . (B.14)
If d is even, then, if we change the sign of all the gamma matrices, we obtain an equivalent
representation with S = Q, the chirality matrix. If d is odd, changing the signs of all the
gamma matrices does not provide an equivalent representation because it changes the sign
of the product of all the gamma matrices.
For both even and odd d the Hermitian conjugates of the gamma matrices constitute another
representation related to the original one by S = D, where D is the Dirac conjugation
matrix and can be taken to be D = iƔ 0 .
In even d the transposed gamma matrices also provide another equivalent representation
of the Clifford algebra. In that case, by definition, S = C, the charge-conjugation matrix,
which we will use later. For d odd, sometimes it is the transposed gamma matrices that
4 By putting together
Ɣ0 · Ɣd−1 = (−1) [d/2] Ɣd−1 · Ɣ0 ,
Ɣ0 · Ɣd−1Ɣd−1 · Ɣ0 = (−1) d−1 ,
it is easy to check that Ɣ d anticommutes with all the other gammas, squares to −1, and is anti-Hermitian.
(B.12)