Revisiting non-relativistic limits

TL;DR

This work addresses how non-relativistic Galilean theories inherit their full Newton–Cartan structure from a relativistic parent. By adding a chemical potential equal to the rest energy and taking the covariant limit $c \to \infty$, the authors derive Newton–Cartan geometry, Milne boosts, and the corresponding NR Ward identities and hydrodynamics from relativistic backgrounds. They demonstrate the method with perturbative scalar field theory and relativistic hydrodynamics, including magnetic moment terms and their impact on Milne transformations, and they connect to the hydrostatic partition function. The approach provides a physical interpretation of Milne boosts, constrains NR effective actions via their relativistic origins, and suggests paths for holographic extensions and broader applicability to gapped systems and Hall-type responses.

Abstract

We show that the full spurionic symmetry of Galilean-invariant field theories can be deduced when those theories are the limits of relativistic parents. Under the limit, the non-relativistic daughter couples to Newton-Cartan geometry together with all of the symmetries advocated in previous work, including the recently revived Milne boosts. Our limit is a covariant version of the usual one, where we start with a gapped relativistic theory with a conserved charge, turn on a chemical potential equal to the rest mass of the lightest charged state, and then zoom in to the low energy sector. This procedure gives a simple physical interpretation for the Milne boosts. Our methods even apply when there is a magnetic moment, which is known to modify the non-relativistic symmetry transformations. We focus on two examples, taking the non-relativistic limits of scalar field theory and hydrodynamics.