Model for the motion of a particle in a quantum background

Abstract

We are studying the dynamics of a one-dimensional field in a noncommutative Euclidean space. The noncommutative space we consider is the one that emerges in the context of three-dimensional Euclidean quantum gravity: it is a deformation of the classical Euclidean space E{sup 3} and the Planck length l{sub P} plays the role of the deformation parameter. The field is interpreted as a particle which evolves in a quantum background. When the dynamics of the particle are linear, the resulting motion is similar to the standard motion in the classical space E{sup 3}. However, nonlinear dynamics on the noncommutative space are different from the corresponding nonlinear dynamics on the classical space. These discrepancies are interpreted as ''quantum gravity'' effects. Finally, we propose a background independent description of the propagation of the particle in the quantum geometry.