# Commutator formula in infinite dimensions

The commutator formula states that for A,B elements of a Lie algebra,

$\lim_{n\to \infty}\left\{ \exp\left(-A\tfrac{t}{n}\right)\exp\left(-B\tfrac{t}{n}\right)\exp\left(A\tfrac{t}{n}\right)\exp\left(B\tfrac{t}{n}\right)\right\}^{n^2}=\exp\left(t^2[A,B]\right)$

I am interested in the case where $A=iH_1$ and $B=iH_2$ with $H_i$ self-adjoint. For finite dimensions the above certainly holds, but what happens in infinite dimensions? Under which conditions? Bounded/unbouded operators? I know that Trotter's formula has some complications in infinite dimensions, I'd be very thankful for any hints here.

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Let us start with the Campbell-Hausdorff formula for selfadjoint operators: let $H_j$ be bounded selfadjoint operators on a Hilbert space. Then $$e^{i\tau H_1}e^{i\tau H_2}=e^{i\tau (H_1+H_2)-\frac{\tau ^2}2[H_1,H_2]+\tau^3 R_\tau},$$ where $\sup_{\vert\tau\vert\le \alpha_0}\Vert R_\tau\Vert_{\mathcal B(H)}<+\infty$ for some positive value of $\alpha_0$. Then $$e^{-i\tau H_1}e^{-i\tau H_2}e^{i\tau H_1}e^{i\tau H_2}=e^{-i\tau (H_1+H_2)-\frac{\tau ^2}2[H_1,H_2]-\tau^3 R_{-\tau}} e^{i\tau (H_1+H_2)-\frac{\tau ^2}2[H_1,H_2]+\tau^3 R_{\tau}}$$ so that applying the formula again, you get with $S_\tau$ bounded in operator-norm near the origin $$e^{-i\tau H_1}e^{-i\tau H_2}e^{i\tau H_1}e^{i\tau H_2}=e^{-\tau^2[H_1,H_2]+\tau^3S_\tau}.$$ Replacing $\tau$ by $t/n$ and waiting for $t/n$ to get smaller than $\alpha_0$, you find the sought formula $$\lim_{n\rightarrow+\infty}\bigl( e^{-i\frac tn H_1}e^{-i\frac tn H_2}e^{i\frac tn H_1}e^{i\frac tn H_2}\bigr)^{n^2}=\lim_{n\rightarrow+\infty}e^{- {t^2}[H_1,H_2]+O(n^{-1})}.$$ When the operators are unbounded, there are complications with the domains and the size of the remainders. Note also that the operator $[H_1,H_2]$ is skew-adjoint as the commutator of two selfadjoint operators so the limit is indeed unitary.
The domain condition is that the square roots of $A$ and $B$ should have a dense intersection, then the convergence holds, to the exponential of the form sum of these operators.