= Differentiation of the Dyson time-ordered exponential
{title2=$i\partial_tU=H_I(t)U$}
For bounded norm-continuous <Hamiltonian operators> on $[t_0,t]$, the <Dyson series> has an $n$-th term bounded by $M^n|t-t_0|^n/n!$, so it converges in operator norm. In its nested-integral form, differentiating the outer endpoint puts $H_I(t)$ on the left of the $(n-1)$-fold term. Summing yields $\partial_tU=-iH_I(t)U$ and $U(t_0,t_0)=I$. If $H_I$ is Hermitian, differentiating $U^\dagger U$ proves unitarity. For unbounded field operators the same algebra is a formal perturbative identity unless domain and convergence assumptions or a regulator justify the steps. Evolution backward uses inverse, anti-time-ordered evolution.
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