The lowest dimensional representations play an ubiquitousrole in physics,
(0,0) dim = 1 real scalar
(1
2
,0) dim = 2 complex left Weyl spinor(0,1
2
) dim = 2 complex right Weyl spinor(1
2
,
1
2
) dim = 4 real vector
(1,0) dim = 3 complex self−dual antisymmetric tensor
(0,1) dim = 3 complex anti−self−dual antisymmetric tensor
(1
2
,1) dim = 6 complex left gravitino(1,1
2
) dim = 6 complex right gravitino
(1,1) dim = 9 real graviton (21.87)In this course, we shall only need the scalar, the Weyl spinors, and the vector. Note that the
combination
(1
2
,0)⊕(0,
1
2
) (21.88)
corresponds to a 4-dimensional (complex) Dirac spinor if the two component representations are
unrelated, while it corresponds to a Majorana spinor if the representations are complex conjugates
of one another.