  Understanding the equilibrium properties and out of equilibrium dynamics of
quantum field theories are key aspects of fundamental problems in theoretical
particle physics and cosmology. However, their classical simulation is highly
challenging. In this work, we introduce a tensor network method for the
classical simulation of continuous quantum field theories that is suitable for
the study of low-energy eigenstates and out-of-equilibrium time evolution. The
method is built on Hamiltonian truncation and tensor network techniques,
bridging the gap between two successful approaches. One of the key developments
is the exact construction of matrix product state representations of global
projectors, crucial for the implementation of interacting theories. Despite
featuring a relatively high computational effort, our method dramatically
improves predictive precision compared to exact diagonalisation-based
Hamiltonian truncation, allowing the study of so far unexplored parameter
regimes and dynamical effects. We corroborate trust in the accuracy of the
method by comparing it with exact theoretical results for ground state
properties of the sine-Gordon model. We then proceed with discussing
$(1+1)$-dimensional quantum electrodynamics, the massive Schwinger model, for
which we accurately locate its critical point and study the growth and
saturation of momentum-space entanglement in sudden quenches.
