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We generalize to multi-commutators the usual Lieb-Robinson bounds forcommutators. In the spirit of constructive QFT, this is done so as to allow theuse of combinatorics of minimally connected graphs (tree expansions) in orderto estimate time-dependent multi-commutators for interacting fermions.Lieb-Robinson bounds for multi-commutators are effective mathematical tools tohandle analytic aspects of the dynamics of quantum particles with interactionswhich are non-vanishing in the whole space and possibly time-dependent. Toillustrate this, we prove that the bounds for multi-commutators of order threeyield existence of fundamental solutions for the corresponding non-autonomousinitial value problems for observables of interacting fermions on lattices. Wefurther show how bounds for multi-commutators of an order higher than two canbe used to study linear and non-linear responses of interacting fermions toexternal perturbations. All results also apply to quantum spin systems, withobvious modifications. However, we only explain the fermionic case in detail,in view of applications to microscopic quantum theory of electrical conductiondiscussed here and because this case is technically more involved.

Lieb-Robinson Bounds for Multi-Commutators and Applications to Response Theory