  A spin system on a lattice can usually be modelled at large scales by an
effective quantum field theory. A key mathematical result relating the two
descriptions is the quantum central limit theorem, which shows that certain
spin observables satisfy an algebra of bosonic fields under certain conditions.
Here, we show that these particular observables and conditions are the relevant
ones for an observer with certain limited abilities to resolve spatial
locations as well as spin values. This is shown by computing the asymptotic
behaviour of a quantum Fisher information metric as function of the resolution
parameters. The relevant observables characterise the state perturbations whose
distinguishability does not decay too fast as a function of spatial or spin
resolution.
