Gauging of 1d-space translations for nonrelativistic point particles

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 Lagrangian. 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 Lagrangian 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.