When you have a crystal, a regular lattice of atoms, then you often describe it in a continuous approximation. So you interpolate for example the displacement o

When you have a crystal, a regular lattice of atoms, then you often describe it in a continuous approximation. So you interpolate for example the displacement of each atom from its resting position by a displacement field. When you then quantize the atoms, then you will also quantize the displacement, and you end up with a quantum field theory of phonons. The nice thing about looking at such systems is, that we have experimental access at all scales, so we can study the breakdown of a quantum field theory in detail. The unfortunate thing is, that there may or may not be an analogy between a condensed matter system and fundamental physics, that is just as much conjecture as is naturalness.