Interacting Galilean and Finite-Energy Carroll Fermions

Abstract

We present a unified derivation of the Galilean and Carrollian limits of the massive Dirac action based on a similarity transformation depending on the mass and the speed of light. The two choices of Dirac conjugation, combined with different mass scalings, generate distinct families of limiting actions. This construction recovers known Galilei and Carroll fermion models and yields new systems, one in each regime. We classify a sufficient set of boost-compatible local (self-)interactions, which includes Nambu--Jona-Lasinio (NJL). The new Galilean free action possesses a local fermionic gauge symmetry that eliminates its local field content. A selected quartic interaction, preserving the Bargmann boost, explicitly breaks the original fermionic gauge symmetry so that the argument that eliminates the field content no longer applies. The NJL limit interaction merely deforms the realization of that gauge symmetry. We show that the energy parameter in several Carroll actions can be eliminated by a time-dependent phase redefinition; conversely, in the new Carrollian extension the energy cannot be removed.