Gauging of 1d-space translations for nonrelativistic point particles

TL;DR

This paper explores how gauging space translations in one-dimensional nonrelativistic particles introduces gauge fields, resulting in a Maxwellian Lagrangian and revealing connections to Newtonian gravity and collective rotational motion.

Contribution

It presents a gauge-invariant extension of particle velocity in 1D, introducing gauge fields with a Maxwellian term, and analyzes resulting dynamics including gravitational and rotational behaviors.

Findings

Reduced Lagrangian describes Newtonian gravitational potential.

On a circle, particles exhibit collective rotation with constant angular velocity.

No dilaton field is necessary in the gauge formulation.

Abstract

Gauging of space translations for nonrelativistic point particles in one dimension leads to general coordinate transformations with fixed Newtonian time. The minimal gauge invariant extension of the particle velocity requires the introduction of two gauge fields whose minimal self interaction leads to a Maxwellian term in the Lagrangean. No dilaton field is introduced. We fix the gauge such that the residual symmetry group is the Galilei group. In case of a line the two-particle reduced Lagrangean describes the motion in a Newtonian gravitational potential with strength proportional to the energy. For particles on a circle with certain initial conditions we only have a collective rotation with constant angular velocity.