A third representation, the Majorana representation, is given by (now no
longer writing in 2 by 2 block form, but as 4 by 4 matrices)
γM 0 =
0 0 0 − 1
0 0 1 0
0 −1 0 0
1 0 0 0
, γM
1 =
1 0 0 0
0 −1 0 0
0 0 1 0
0 0 0 − 1
γ 2 M=
0 0 0 1
0 0 −1 0
0 − 1 0 0
1 0 0 0
, γM
3 =
0 − 1 0 0
− 1 0 0 0
0 0 0 − 1
0 0 − 1 0
with
γM 5 =i
0 − 1 0 0
1 0 0 0
0 0 0 1
0 0 −1 0
The importance of the Majorana representation is that it shows the interesting
possibility of having (in signature (3,1)) a spinor representation on a real vector
spaceR^4 , since one sees that the Clifford algebra matrices can be chosen to be
real. One has
γ 0 γ 1 γ 2 γ 3 =
0 − 1 0 0
1 0 0 0
0 0 0 1
0 0 −1 0
and
(γ 0 γ 1 γ 2 γ 3 )^2 =− 1
The Majorana spinor representation is onSM =R^4 , withγ 0 γ 1 γ 2 γ 3 a real
operator on this space with square−1, so it provides a complex structure on
SM. Recall that a complex structure on a real vector space gives a splitting of
the complexification of the real vector space into a sum of two complex vector
spaces, related by complex conjugation. In this case this corresponds to
SM⊗C=SL⊕S∗Rthe fact that complexifying Majorana spinors gives the two kinds of Weyl
spinors.
41.3 For further reading
Most quantum field theory textbook have extensive discussions of spinor rep-
resentations of the Lorentz group and gamma matrices, although most use the
opposite convention for the signature of the Minkowski metric. Typical exam-
ples are Peskin-Schroeder [67] and chapter II.3 and Appendix E of Zee [105].