= Time-reversal reciprocity of the scattering matrix
{title2=$\langle f|S|i\rangle=\langle\hat Ti|S|\hat Tf\rangle$}
If a Hermitian interaction-picture Hamiltonian satisfies $\hat TV(t)\hat T^{-1}=V(-t)$ for an <antiunitary operator>, its <Dyson series> conjugates $(-i)^n$ to $i^n$. The substitution $t_j\mapsto-t_j$ converts ordered integration to anti-ordered integration, giving $\hat TS\hat T^{-1}=S^\dagger$. <Antiunitarity> then gives the displayed reciprocity: the time-reversed process has the reversed outgoing state as its incoming state and vice versa. It does not merely reverse the momenta while leaving their incoming/outgoing roles fixed.
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