We generalize to multi-commutators the usual Lieb-Robinson bounds for
commutators. In the spirit of constructive QFT, this is done so as to allow the
use of combinatorics of minimally connected graphs (tree expansions) in order
to estimate time-dependent multi-commutators for interacting fermions.
Lieb-Robinson bounds for multi-commutators are effective mathematical tools to
handle analytic aspects of the dynamics of quantum particles with interactions
which are non-vanishing in the whole space and possibly time-dependent. To
illustrate this, we prove that the bounds for multi-commutators of order three
yield existence of fundamental solutions for the corresponding non-autonomous
initial value problems for observables of interacting fermions on lattices. We
further show how bounds for multi-commutators of an order higher than two can
be used to study linear and non-linear responses of interacting fermions to
external perturbations. All results also apply to quantum spin systems, with
obvious modifications. However, we only explain the fermionic case in detail,
in view of applications to microscopic quantum theory of electrical conduction
discussed here and because this case is technically more involved.