 Time reversal of an electromagnetic gauge field

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.