Covariant Formulation of the Newton-Hooke Particle and its Canonical Analysis

TL;DR

This paper develops a covariant formulation of the Newton-Hooke particle, coupling it with Newton-Cartan geometry, and performs a canonical analysis leading to a Schrödinger equation consistent with the model's symmetries.

Contribution

It introduces a covariant approach to the Newton-Hooke particle, coupling it with Newton-Cartan geometry, and provides a canonical analysis for quantization.

Findings

Covariant formulation coupled with Newton-Cartan geometry.

Canonical analysis in gauge independent and fixed approaches.

Derivation of a consistent Schrödinger equation.

Abstract

A covariant formulation for the Newton-Hooke particle is presented by following an algorithm developed by us \cite{BMM1, BMM2, BMM3}. It naturally leads to a coupling with the Newton-Cartan geometry. From this result we provide an understanding of gravitation in a Newtonian geometric background. Using Dirac's constrained analysis a canonical formulation for the Newton-Hooke covariant action is done in both gauge independent and gauge fixed approaches. While the former helps in identifying the various symmetries of the model, the latter is able to define the physical variables. From this analysis a path to canonical quantisation is traced and the Schroedinger equation is derived which is shown to satisfy various consistency checks. Some consequences of this equation are briefly mentioned.