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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.

Starting from a matter-antimatter symmetric ensemble, exact charge conjugation would pair every baryon-producing process with an equally probable antibaryon-producing process. Their contributions to the net baryon number cancel. Thus C violation is necessary. Violating charge conjugation alone is insufficient: if CP symmetry is exact, the parity-reflected conjugate channels still have equal probabilities and cancel in a CP symmetry invariant ensemble. One needs CP violation as well.

The remaining Sakharov conditions are baryon number violation and departure from thermal equilibrium. Conserving baryon number cannot turn an initially zero value into a nonzero one. In thermal equilibrium, under the usual CPT theorem hypotheses and with no imposed chemical potentials for charges odd under CPT symmetry, states and their conjugates have equal thermal weights, so the equilibrium expectation of net baryon number vanishes and thermal stationarity prevents its sustained production. A nonequilibrium history allows different forward and reverse populations to generate an asymmetry. These are the standard necessary conditions for dynamical baryogenesis from symmetric initial conditions, not a guarantee of a sufficiently large asymmetry.

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.

Let $|1⟩=|D^0⟩$, $|2⟩=|{\bar{D}}^0⟩$. In the rest frame, the CPT theorem interchanges these states up to irrelevant phases. For the weak Hamiltonian operator $H^{\prime }$, with $H^{\prime }=H^{\prime †}$, and its antiunitary CPT symmetry operator $\Theta$,Its diagonal matrix elements are real by Hermitian conjugation, soIn the specified phase convention, the CP symmetry operator on this two-state subspace is $-{\sigma }_x$. If the weak interaction preserves CP symmetry, ${\sigma }_{x}R{\sigma }_x=R$, givingIndependently, Hermitian conjugation gives $R_{21}=R_{12}^{\ast }$ for this $H^{\prime }$. Hence CP symmetry makes the off-diagonal element real in this convention; that conjugation relation alone does not imply CP symmetry.

For neutral meson mixing described by the decay-effective Hamiltonian operator $R=M-i\Gamma /2$, the effective matrix is generally not a Hermitian matrix. The CPT constraints on neutral-meson mixing still give equality of the complex diagonal entries, expressing equal masses and total decay widths, and CP symmetry gives equality of the off-diagonal entries in the specified convention. One must not additionally impose $R_{21}=R_{12}^{\ast }$ on this decay-effective matrix.