  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.
