  The Lieb-Robinson theorem states that locality is approximately preserved in
the dynamics of quantum lattice systems. Whenever one has finite-dimensional
constituents, observables evolving in time under a local Hamiltonian will
essentially grow linearly in their support, up to exponentially suppressed
corrections. In this work, we formulate Lieb-Robinson bounds for general
harmonic systems on general lattices, for which the constituents are
infinite-dimensional, as systems representing discrete versions of free fields
or the harmonic approximation to the Bose-Hubbard model. We consider both local
interactions as well as infinite-ranged interactions, showing how corrections
to locality are inherited from the locality of the Hamiltonian: Local
interactions result in stronger than exponentially suppressed corrections,
while non-local algebraic interactions result in algebraic suppression. We
derive bounds for canonical operators, Weyl operators and outline
generalization to arbitrary operators. As an example, we discuss the
Klein-Gordon field, and see how the approximate locality in the lattice model
becomes the exact causality in the field limit. We discuss the applicability of
these results to quenched lattice systems far from equilibrium, and the
dynamics of quantum phase transitions.
