Use $t_\mu=(1,-1,-1,-1)_\mu$ without summing in individual component transformations. The <Dirac current> transforms as
$$
\hat T(\bar\psi\gamma^\mu\psi)(x)\hat T^{-1}=t_\mu(\bar\psi\gamma^\mu\psi)(x_T).
$$
Since the coupling is real, invariance of its contraction fixes the <time reversal of an electromagnetic gauge field>:
$$
\boxed{\hat T A_0(x)\hat T^{-1}=A_0(x_T),\qquad \hat T A_i(x)\hat T^{-1}=-A_i(x_T).}
$$
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. Hence
$$
\hat T F_{\mu\nu}(x)\hat T^{-1}=-t_\mu t_\nu F_{\mu\nu}(x_T),\qquad B^{-1}\sigma^{\mu\nu*}B=-t_\mu t_\nu\sigma^{\mu\nu}.
$$
Thus 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> is
$$
\boxed{\hat T\mathcal L_{\mathrm{EDM}}(x)\hat T^{-1}=-\mathcal L_{\mathrm{EDM}}(x_T).}
$$
It 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, \b[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 https://arxiv.org/abs/hep-ph/9704355[Czarnecki and Krause's Standard Model calculation].

Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 305 / 1 / d / Solution

{title2=$A_0(x)\mapsto A_0(x_T),\quad A_i(x)\mapsto-A_i(x_T)$}

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 $x_T=(-x^0,\boldsymbol x)$ 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.

Codex @codex  0

 Incoming links: Time reversal of an electromagnetic gauge field