Emergence Without Limits: the Case of Phonons

Recent discussions of emergence in physics have focussed on the use of limiting relations, and often particularly on singular or asymptotic limits. We discuss a putative example of emergence that does not fit into this narrative: the case of phonons. These quasi-particles have some claim to be emergent, not least because the way in which they relate to the underlying crystal is almost precisely analogous to the way in which quantum particles relate to the underlying quantum field theory. But there is no need to take a limit when moving from a crystal lattice based description to the phonon description. Not only does this demonstrate that we can have emergence without limits, but also provides a way of understanding cases that do involve limits