A new formulation of non-relativistic diffeomorphism invariance

TL;DR

The paper addresses the lack of a systematic foundation for nonrelativistic diffeomorphism invariance by localising the global Galilean group in nonrelativistic field theories. It develops a gauge-theoretic, Cartan-inspired procedure that introduces a local basis and compensating fields (including $A_t$, $A_k$, $\theta$, $\Psi^k$, ${\\Sigma_a}^k$) and a measure factor $\\Lambda = M/\\theta$ to construct covariant derivatives $\nabla_t$ and $\nabla_a$, yielding a 3-d diffeomorphism invariant action. The resulting Schrödinger action on curved space, $S = \int dt \int d^3x \sqrt{g}\left[ \frac{i}{2}(\phi^*\nabla_t\phi - \phi\nabla_t\phi^*) - \frac{1}{2m} g^{kl}D_k\phi^* D_l\phi \right]$, reduces in the static limit to the diff-invariant models used in FQHE, while maintaining a route to Newton–Cartan geometry in the fully general case. This provides a systematic, geometrically grounded framework for constructing nonrelativistic diffeomorphism invariant theories with potential applications in condensed matter physics and beyond.

Abstract

We provide a new formulation of nonrelativistic diffeomorphism invariance. It is generated by localising the usual global Galilean Symmetry. The correspondence with the type of diffeomorphism invariant models currently in vogue in the theory of fractional quantum Hall effect has been discussed. Our construction is shown to open up a general approach of model building in theoretical condensed matter physics. Also, this formulation has the capacity of obtaining Newton - Cartan geometry from the gauge procedure.