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

TL;DR

This paper develops a canonical formulation for a non-relativistic particle in a Newton-Cartan background, clarifying its equations of motion and paving the way for quantization.

Contribution

It presents a detailed canonical analysis of a new action for a non-relativistic particle coupled to gravity, including Hamiltonian formulation and quantization steps.

Findings

Equation of motion satisfies the geodesic equation in Newton-Cartan background

Hamiltonian analysis performed in gauge independent and fixed approaches

Explicit derivation of the Schrödinger equation for quantization

Abstract

A detailed canonical treatment of a new action for a nonrelativistic particle coupled to background gravity, recently given by us, 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 quantisation is outlined by explicitly deriving the Schroedinger equation.