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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.

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.

The flavour-diagonal weak neutral current coupled to the real Z boson has real vector and axial current couplings. With ${\eta }_C(Z)=-1$, part (e) shows that this tree-level interaction is invariant under CP symmetry, although the simultaneous presence of the Dirac current and axial current generally violates charge conjugation and parity symmetry in quantum field theory separately. This conclusion concerns the tree-level weak neutral current, rather than every interaction or radiative effect in the Standard Model.

The weak charged current couples distinct quark flavours and involves the Cabibbo-Kobayashi-Maskawa matrix:Here $u_{Li}$ and $d_{Lj}$ are left-handed quark fields. A CP symmetry transformation exchanges the two conjugate-field operators; because it acts through a unitary operator, it does not complex-conjugate their numerical coefficients. Equality with the original interaction therefore requires a Cabibbo-Kobayashi-Maskawa matrix that can be made real by quantum field rephasings. With three generations, a physical CP-violating phase remains in general, measured by the Jarlskog invariant. Therefore the W boson interactions can violate CP symmetry. Merely seeing a complex coefficient is insufficient: the phase must survive quantum field rephasing, unlike the real two-generation Cabibbo angle rotation.

In four spacetime dimensions $[\psi ]=3/2$, $[\varphi ]=1$, and $[{\partial }_{\mu }]=1$. HenceThe coupling is an irrelevant coupling by power counting in quantum field theory, so it is a nonrenormalizable interaction that would ordinarily define only an effective theory with a cutoff. Here it is also a redundant operator: integration by parts gives $\lambda \varphi {\partial }_{\mu }{j}^{\mu }$ up to a boundary term, and the Dirac current $j^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$ is conserved. This derivative coupling to a conserved current is removed exactly by the local phase redefinition $\psi =e^{-i\lambda \varphi }\chi$.

Every scalar-fermion vertex contracts the scalar momentum with the Dirac current. Between on-shell external spinors,and the analogous particle-antiparticle identity also vanishes. Equivalently, the field redefinition in the previous part turns the theory into a free theory. Consequently every putative tree channel for $\psi \bar{\psi }\to \psi \bar{\psi }$ has zero amplitude and

Write ${\psi }^C=\mathcal{C}{\bar{\psi }}^T$ and ${\bar{\psi }}^C={\psi }^T\mathcal{C}$. Anticommuting the spinor fields and using ${\mathcal{C}}^{-1}({\gamma }^{\mu }{)}^T\mathcal{C}=-{\gamma }^{\mu }$ givesThe free Dirac terms are invariant up to the total derivative used to move the derivative between the two anticommuting fields. The QED interaction $-e\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }$ is also invariant because both the Dirac current and $A_{\mu }$ are odd. Together with $F_{\mu \nu }{F}^{\mu \nu }$ invariance, this proves charge-conjugation invariance of quantum electrodynamics.

Choose $C=i/4$. Expandingwith the Clifford algebra, applying the product rule and then using the Dirac equation and its adjoint givesThis is the Belinfante-Rosenfeld stress-energy tensor written as an improvement of the canonical tensor.

The translation charges areFor spatial $\nu$, their difference is the integral of ${\partial }_i{S}^{i0\nu }-{\partial }^{\nu }(\bar{\psi }{\gamma }^0\psi )$. For $\nu =0$, use conservation of the Dirac current to replace ${\partial }^0(\bar{\psi }{\gamma }^0\psi )$ by a spatial divergence; also $S^{00\nu }=0$. Hence ${\hat{P}}^{\nu }-P^{\nu }$ is always a spatial boundary integral. Under the usual decay boundary condition it vanishes, so both currents generate the same four-momentum.

Complete asymptotically flat spin initial data obeying the Einstein constraints and the dominant energy condition have future-causal ADM four-momentum, under the appropriate decay and boundary hypotheses. A spinorial proof solves the Witten spinor equation and integrates the Nester two-form. Its divergence is a sum of a nonnegative spinor-gradient term and a matter-energy contraction with a future-causal Dirac current. Positivity for every asymptotic constant spinor yields the full inequality, rather than merely nonnegative energy in one frame.

The vector Dirac current has an even temporal component and odd spatial components under time-reversal symmetry. Invariance of its real minimal coupling therefore gives the displayed transformation of the gauge field, up to a gauge transformation. Differentiation at $x_T=(-x^0,\mathbf{x})$ yields $F_{\mu \nu }(x)\mapsto -t_{\mu }{t}_{\nu }{F}_{\mu \nu }(x_T)$, with $t=(1,-1,-1,-1)$. Thus the electric field is even and the magnetic field is odd.

A fermion bilinear transforming as a Lorentz four-vector. For a single Dirac field this is the Dirac current; distinct flavours can give a charged or flavour-changing current.

In mostly-plus signature orient the commuting Dirac current to the future by $\overline{ϵ}=-{ϵ}^{†}{\gamma }^{\hat{0}}$. For the Nester two-form, put $b^i=n_{\mu }{B}^{\mu i}$. Its covariant divergence has a spinor-gradient term and the Einstein tensor contraction of spin curvature. Projecting onto the hypersurface gives the displayed identity with positive spinor norm. The Witten spinor equation removes the negative square, and the dominant energy condition makes the matter term nonnegative after the Einstein constraints are imposed.