Revisiting non-relativistic limits

TL;DR

This paper demonstrates how the full symmetry structure of non-relativistic theories, including Milne boosts, can be derived from relativistic theories through a covariant limiting process involving Newton-Cartan geometry.

Contribution

It introduces a covariant limiting procedure from relativistic theories to non-relativistic ones, clarifying the origin of symmetries like Milne boosts and extending applicability to theories with magnetic moments.

Findings

Full spurionic symmetry of Galilean-invariant theories derived from relativistic limits

Non-relativistic theories couple to Newton-Cartan geometry with all symmetries

Method applies to scalar fields and hydrodynamics, even with magnetic moments

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.