Time evolution operator for Hamiltonien with scalar commutator at different times Let $H(t)$ be a time-dependent Hamilton-operator and assume that $[H(t),H(t')] = f(t,t')\, \mathrm{id}_\mathcal{H}$. Is there a closed formula for its time-evolution operator?
I tried deducing an explicit expression from the Dyson series but failed. I'd be grateful for any help or suggestion on good sources.
 A: Here's my attempt:
The time evolution operator can be approximated by
$$U(t=n\Delta) \approx e^{-iH_n~\Delta}e^{-iH_{n-1}~~\Delta}\cdots e^{-iH_1\Delta}~,$$
where $H_{n}=H((n-1)\Delta)$ and $\Delta$ is some small period of time. This approximation is valid if the commutator of Hamiltonians at closely separated instants of time tend to vanish.
Using the Baker-Campbell-Hausdorff formula, we find
$$e^{-iH_n~\Delta}e^{-iH_{n-1}~~\Delta}=e^{-i(H_n~+H_{n-1}~~)~\Delta ~- \frac{\Delta^2}{2}f_{n,n-1}}~~~~,$$
where $f_{n,n-1} = f(n\Delta,(n-1)\Delta)$. This process can be carried out again to include the next factor in the evolution operator:
$$e^{-iH_n~\Delta}e^{-iH_{n-1}~~\Delta}e^{-iH_{n-2}~~\Delta}=e^{-i(H_n~+H_{n-1}~~)~\Delta ~- \frac{\Delta^2}{2}f_{n,n-1}}~~~e^{-iH_{n-2}~~\Delta}~= e^{-i(H_{n}~+H_{n-1}~~+H_{n-2}~~)\Delta-\frac{\Delta^2}{2}(f_{n,n-1}~~~+f_{n,n-2}~~~+~f_{n-1,n-2}~~~~~)}.$$
Repeat this process $n-1$ times, we finally arrive at
$$U(t=n\Delta)=e^{-i\Delta(H_n~+H_{n-1}~~+\cdots+H_1)-\frac{\Delta^2}{2}\sum_{j>i}~~f_{j,i}}~~.$$
Taking the limit of $\Delta\rightarrow 0$,
$$U(t) = \exp\left[{-i\int_{0}^{t}H(t')dt'-\frac{1}{2}\int_{0}^{t}dt'\int_{t'}^{t}dt''f(t',t'')}\right].$$
