  We describe a general procedure to give effective continuous descriptions of
quantum lattice systems in terms of quantum fields. There are two key novelties
of our method: firstly, it is framed in the hamiltonian setting and applies
equally to distinguishable quantum spins, bosons, and fermions and, secondly,
it works for arbitrary variational tensor network states and can easily produce
computable non-gaussian quantum field states. Our construction extends the
mean-field fluctuation formalism of Hepp and Lieb (developed later by Verbeure
and coworkers) to identify emergent continuous large-scale degrees of freedom -
the continuous degrees of freedom are not identified beforehand. We apply the
construction to to tensor network states, including, matrix product states and
projected entangled-pair states, where we recover their recently introduced
continuous counterparts, and also for tree tensor networks and the multi-scale
entanglement renormalisation ansatz. Finally, extending the continuum limit to
include dynamics we obtain a strict light cone for the propagation of
information.
