General coordinate invariance in quantum many-body systems

TL;DR

This paper generalizes the concept of coordinate invariance to nonrelativistic many-body systems, providing a systematic framework that clarifies the distinction between coordinate freedom and physical symmetries, with applications to superfluids.

Contribution

It introduces a formalism for implementing general coordinate invariance in nonrelativistic systems, extending previous ideas and clarifying the role of internal symmetries.

Findings

Derived transformation rules for background fields from the formalism

Demonstrated the framework with nonrelativistic superfluid example

Showed that coordinate invariance is independent of Galilei or Poincare symmetry

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.