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The phase in a field transformation $\hat{C}{V}^{\mu }{\hat{C}}^{-1}={\eta }_C{V}^{\mu †}$. For a real vector relativistic quantum field, Hermitian conjugation restricts this phase to $\pm 1$.

Use the Minkowski metric $g^{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$ throughout. Taking the Hermitian conjugation of the massive Dirac equation and using the Dirac adjoint givesTaking the matrix transpose and multiplying by the charge conjugation matrix $C$ yieldsThus the charge conjugation of a Dirac field preserves the free massive Dirac equation:The unitarity of $C$ and ${\gamma }^0{C}^{-1}{\gamma }^0=-C^{-1}{\gamma }^{0T}$ also giveConsequentlyHere $\hat{C}$ is the unitary operator acting on the relativistic quantum field, whereas $C$ acts on Dirac spinor indices.

For a real vector relativistic quantum field, $V^{\mu }=V^{\mu †}$. Its transformed field must also be a Hermitian operator, so the intrinsic charge-conjugation phase satisfies ${\eta }_C={\eta }_C^{\ast }$; combined with $|{\eta }_C|=1$, this givesThe quantum electrodynamics interaction is $-e\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }$. Since the Dirac electromagnetic current is odd under charge conjugation, charge conjugation invariance of this interaction requires the photon field to be odd too:

Write $J_V^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$ for the Dirac current and $J_A^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$ for the axial current. Under parity symmetry in quantum field theory, the Dirac spinor phases cancel andThe extra sign for the axial current follows from ${\gamma }^0{\gamma }^5{\gamma }^0=-{\gamma }^5$. Combining these transformations with part (c), both currents transform under CP symmetry as $J_{V,A}^{\mu }(x)\mapsto -{\mathbb{P}}^{\mu }{}_{\nu }{J}_{V,A}^{\nu }(x_P)$. The vector relativistic quantum field transforms as $V^{\mu }(x)\mapsto {\eta }_C{\mathbb{P}}^{\mu }{}_{\nu }{V}^{\nu †}(x_P)$. The two parity matrices cancel in the tensor contraction, givingFor a real vector relativistic quantum field with ${\eta }_C=-1$, the interaction is invariant under CP symmetry. A complex vector relativistic quantum field instead maps to the conjugate-field interaction, which must be compared with its Hermitian conjugation partner.