Consider some interacting QFT on a lattice (just to avoid infinitely large momentums). The size of the lattice is assumed to be much smaller than the size of the emergent particles (like in our world). Lets assume there are scientists living in such system and they try to measure the field (which is an "observable" by the standards of QM) at a lattice site. Would they be able to do this? Wouldn't the fact that all particles are much larger than the size of the lattice restrict probing the field at a single site? In other words would the observables in such system actually be observable (at least in theory)?