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. We offer an account of emergence which encompasses phonons, and argue both that emergence may thus be found in cases where the use of limits is not required, and that it provides a way of understanding cases that do involve limits