Suppose we have Weyl spinor ψa, which transforms under irreducible representation (12,0) of the Lorentz group,
ψa→(T(g)) baψb,
and complex conjugated spinor
κ˙b, which transforms under
(0,12) representation,
κ˙b→(T(g))˙b ˙aκ˙a
Dirac spinor corresponds to the direct sum
(12,0)⊕(0,12):
Ψ=(ψaκ˙b)
Charge conjugation of Dirac spinor is determined as
Ψ→ˆCΨ=(κaψ˙b)
Sometimes charge conjugation is called "the Dirac version of of complex conjugation".
The question. Why do we need to introduce complex conjugation in a form (1), not in a form of ordinary complex conjugation, Ψ→Ψ∗?