Canonical Formulation of a New Action for a Non-relativistic Particle Coupled to Gravity

A detailed canonical treatment of a new action for a nonrelativistic particle coupled to background gravity, recently given by R. Banes jee and P. Mukheijee [Phys. Lett. B 797, 134484 (2019); 797, 134834 (2019)], is performed both in the Lagrangian and Hamiltonian formulations. The equation of motion is shown to satisfy the geodesic equation in the Newton-Cartan background, thereby clearing certain confusions. The Hamiltonian analysis is done in the gauge-independent as well as gauge fixed approaches, following Dirac's analysis of constraints. The physical (canonical) variables are identified, and the path to canonical quantization is outlined by explicitly deriving the Schrodinger equation. Usual flat space results are correctly reproduced.