Spinor fields in general Newton-Cartan backgrounds

TL;DR

This work constructs covariant non-relativistic spinor actions on general Newton–Cartan geometries by deriving a Levy–Leblond operator from a relativistic Dirac theory via a $c\to\infty$ limit, ensuring the resulting theory inherits Milne boosts, local Galilean invariance, and $U(1)$ gauge symmetry from the parent spacetime symmetries. A non-minimal coupling to the background gauge field is required, with a specific strength $\alpha=-\tfrac{1}{4}$ that cancels divergent terms in the NR limit and fixes the gyromagnetic ratio to $g=1$ (except in three dimensions, where a different structure appears). The non-relativistic Dirac operator $\slashed{\mathcal{D}}$ and the NR Lagrangian $L_{\text{nr}}$ are shown to transform covariantly under Milne boosts and local Galilean transformations, and the formalism naturally reduces to a NR theory on flat space with the expected magnetic coupling. In causal backgrounds, one can integrate out auxiliary components to obtain a NR theory described entirely by $\Psi_+$. The results illuminate the relationship between relativistic parent symmetries and NR Newton–Cartan symmetries, with potential applications to condensed matter systems, holography, and non-relativistic supergravity.

Abstract

We give a covariant construction of Lagrangians for spinor fields in generic Newton-Cartan backgrounds. A non-relativistic Dirac/Levy-Leblond operator and the associated fields are obtained from relativistic analogues by a limiting procedure. The relativistic symmetries induce the complete set of non-relativistic symmetries, including Milne boosts and local Galilean transformations. The resulting Levy-Leblond operator includes non-minimal couplings to the Newton-Cartan structure as well as to the gauge field, and with these couplings it transforms covariantly. Phenomenologically, this fixes the gyromagnetic ratio to g=1. Three-dimensional spacetimes are an exception: generic g is possible but results in modified Milne transformations, which - upon gauge fixing - reproduces the anomalous diffeomorphisms found in earlier approaches.