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Let $H_1,H_2$ be local Hamiltonians (i.e. interactions are finite range). Let us form the product of the exponentials of both. By Baker-Campbell-Hausdorff, this defines a third Hamiltonian,

$$e^{H_1}e^{H_2}:=e^{H_3}$$

Can we say that $H_3$ is at least quasi-local (i.e. interactions exponentially decay with distance)?

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  • $\begingroup$ I think so. If you write the Baker-Campbell-Hausdorff formula explicitly, the contribution of higher order commutators should decay exponentially. $\endgroup$
    – Exhaustive
    Mar 9, 2017 at 12:34
  • $\begingroup$ Hmm interesting - how would I show this explicitly? $\endgroup$
    – David Roberts
    Mar 11, 2017 at 9:30

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