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Use $t_{\mu }=(1,-1,-1,-1{)}_{\mu }$ without summing in individual component transformations. The Dirac current transforms asSince the coupling is real, invariance of its contraction fixes the time reversal of an electromagnetic gauge field:These equations use a compatible gauge; an additional pure gauge transformation would not change the gauge field strength.

Coordinate differentiation at $x_T$ introduces $r_{\mu }=-t_{\mu }$, because the time coordinate is reversed and the spatial coordinates are unchanged. HenceThus the electric field is even and the magnetic field is odd. The chirality matrix is even by part (a). The two tensor signs cancel in the dipole contraction, but antiunitarity conjugates its explicit $i$. Therefore the time-reversal parity of a fermion electric dipole operator isIt is a time-reversal symmetry violating interaction, and, under the usual local relativistic quantum field theory hypotheses of the CPT theorem, it violates CP symmetry. A nonzero coefficient is absent from the renormalizable tree-level Standard Model Lagrangian: the broken-phase fermion electric dipole moment operator has mass dimension five, and its electroweak-invariant completion requires a Higgs field and has dimension six. However, it can arise as a radiatively induced effective interaction in the Standard Model, whose CKM matrix contains a CP-violating phase. It is not forbidden to all orders. An explicit primary calculation of quark dipoles is Czarnecki and Krause's Standard Model calculation.

The operator $ia\bar{\psi }{\sigma }^{\mu \nu }{\gamma }^5\psi F_{\mu \nu }$ is the relativistic fermion electric dipole moment operator. Its nonrelativistic limit contains $\mathbf{S}\cdot \mathbf{E}$: spin angular momentum is an axial vector, whereas the electric field is a polar vector. It is consequently odd under parity symmetry in quantum field theory. Both the pseudotensor fermion bilinear and the electromagnetic field tensor are odd under charge conjugation, so their product is even. ThereforeThe CPT theorem then makes it odd under time-reversal symmetry. Such an interaction can arise only from CP violation. The Cabibbo-Kobayashi-Maskawa matrix therefore induces a nonzero Standard Model electric dipole moment at sufficiently high loop order, but its flavor structure and loop suppressions make the result extraordinarily small.