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Set $t_0=1$ and $t_i=-1$ for spatial indices. The complex conjugation of the scalar $i$ supplies a minus sign, while conjugating each of the three spatial gamma matrices supplies three more. In the original order,Consequently the chirality matrix obeysNo rearrangement of the gamma matrices is needed, so no additional anticommutator sign occurs. The star is entrywise complex conjugation, not matrix transpose or Hermitian conjugation.

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.

For real $a$, the Hermitian effective interaction $ia\bar{\psi }{\sigma }^{\mu \nu }{\gamma }^5\psi F_{\mu \nu }$ is odd under time-reversal symmetry. The spinor tensor and gauge field strength both acquire $-t_{\mu }{t}_{\nu }$, while the chirality matrix is even. Their signs cancel, leaving the minus sign from the explicit $i$ under antiunitarity. By the CPT theorem this is also CP violation. It is not a renormalizable tree-level Standard Model term, but a radiatively generated low-energy dipole is allowed because of the CP-violating phase in the CKM matrix.