Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 305 / 1 / c

For a normal ordering of a fermion bilinear, the result of part (a) and the fermionic sign from interchanging the fields giveFor the Dirac current, $(C^{-1}{\gamma }^{\mu }C{)}^T=-{\gamma }^{\mu }$. For the axial current, the definition of ${\gamma }^5$ and the gamma matrix identities implyThus the charge conjugation of fermion bilinears givesThe Dirac current is odd under charge conjugation, and the axial current is even under charge conjugation. The normal ordering specifies the usual treatment of coincident-field contact terms.