1 Two-component formalism for Spin-1/2 FermionsThat is, the fieldsχ1andχ2obey the standard time reversal transforma-tion laws of a single two-component fermion field [eqs. (1.155)–(1.158)],but withoppositesign phase factorsηTin each case.Had one chosenηT=η∗Tin eqs. (1.175) and (1.176), then the time reversal transformationlaws ofχ1andχ2would have been more complicated [eqs. (1.179) and(1.180)].But as before, one is free to make a further SO(2) rotationto transformχ1andχ2into new fields that do exhibit the simpler timereversal transformation laws [eq. (1.151)].One can now work out the time reversal properties of bilinear covariantsconstructed out of the fields of a charged fermion pair. The results arelisted in Table B.3.1.10 Charge conjugation of two-component spinorsCharge conjugation was introduced in section 1.7. The charge conjugationoperator is a discrete operator that interchanges particles and theirC-conjugates. Here, conjugation refers to some conserved charge operatorQ, whereCQC−1=−Q.The conjugate of the conjugate field is theoriginal field, so thatC2= 1. For a single two-component fermion fieldCχα(x)C−1≡χCα(x) =ηCχα(x),(1.183)Cχ˙α(x)C−1≡χC˙α(x) =η∗Cχ˙α(x),(1.184)where|ηC|= 1. In this case, no conserved charge exists, so that chargeconjugation is trivial. In particular, as noted in section 1.7, the invarianceof the mass termχχ+χχimplies thatη∗C=ηC, and henceηC=±1.27The behavior of the bilinear covariants under C is simple:χC1OχC2=ηC1ηC2χ1Oχ2,(1.185)for anyO=I2, σµ,σµ, σµνandσµν(where bars should appear over theappropriateχidepending on the choice ofO).In theories with multiple two-component fermion fields with no globalsymmetries, the charge conjugation properties of each fermion is givenby eqs. (1.183) and (1.184), withηC=±1.When interactions areincluded, if there is some choice of theηCsuch that the action is invariantunder charge conjugation, then the theory is charge conjugation invariant.If there is an internal global symmetry (χi→Uijχj) under whichthe Lagrangian is invariant, then there are relations among the chargeconjugation transformations of the fermion fields.Here, we consider27In a theory with just one fermion field, no physical quantity can depend on the sign ofηCsince the fermion field must appear quadratically in the Lagrangian. So, withoutloss of generality, we can takeηC= 1.
1.11 CP and CPT conjugation of two-component spinors41the simplest case of two mass-degenerate fermion fields,χ1andχ2, andidentify the conserved charge asQ=J0d3x, where the conservedcurrentJµis given in eq. (1.108).In terms of the linear combinationsξandη[eqs. (1.103) and (1.104)] corresponding to the fields of definitecharge, we define the charge conjugation transformations as follows.