General coordinate invariance in quantum many-body systems

TL;DR

This work builds a comprehensive framework for general coordinate invariance in nonrelativistic many-body systems by employing the vielbein formalism to realize diffeomorphism covariance while treating physical spacetime symmetries as internal, coset-realized structures. By combining the vielbein approach with the coset construction, the authors derive manifestly invariant actions, relate background fields to conserved currents, and elucidate how Galilei and Poincaré symmetries can be implemented without requiring these symmetries as fundamental spacetime diffeomorphisms. The paper provides explicit realizations for a Galilei-invariant microscopic theory and its superfluid EFT, including the correct gauging of energy, momentum, and particle-number currents, the role of spin connection and torsion, and a systematic procedure to integrate auxiliary Goldstone modes. The framework clarifies the interplay between diffeomorphism invariance and internal symmetries, connects to Newton–Cartan geometry, and opens paths to extend NR EFTs to anisotropic systems and non-Abelian internal symmetries with potential applications in condensed matter and quantum field theory in curved backgrounds.

Abstract

We extend the notion of general coordinate invariance to many-body, not necessarily relativistic, systems. As an application, we investigate nonrelativistic general covariance in Galilei-invariant systems. The peculiar transformation rules for the background metric and gauge fields, first introduced by Son and Wingate in 2005 and refined in subsequent works, follow naturally from our framework. Our approach makes it clear that Galilei or Poincare symmetry is by no means a necessary prerequisite for making the theory invariant under coordinate diffeomorphisms. General covariance merely expresses the freedom to choose spacetime coordinates at will, whereas the true, physical symmetries of the system can be separately implemented as "internal" symmetries within the vielbein formalism. A systematic way to implement such symmetries is provided by the coset construction. We illustrate this point by applying our formalism to nonrelativistic s-wave superfluids.