Spinor fields in general Newton-Cartan backgrounds

We give a covariant construction of Lagrangians for spinor fields in generic Newton-Cartan backgrounds. A nonrelativistic Dirac or L\'evy-Leblond operator and the associated fields are obtained from relativistic analogues by a limiting procedure. The relativistic symmetries induce the complete set of nonrelativistic symmetries, including Milne boosts and local Galilean transformations. The resulting L\'evy-Leblond operator includes nonminimal 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.