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We are studying the dynamics of a one-dimensional field in a non-commutativeEuclidean space. The non-commutative space we consider is the one that emergesin the context of three dimensional Euclidean quantum gravity: it is adeformation of the classical Euclidean space and the Planck length plays therole of the deformation parameter. The field is interpreted as a particle whichevolves in a quantum background. When the dynamics of the particle is linear,the resulting motion is similar to the standard motion in the classical space.However, non-linear dynamics on the non-commutative space are different fromthe corresponding non-linear dynamics on the classical space. Thesediscrepencies are interpreted as "quantum gravity" effects. Finally, we proposea background independent description of the propagation of the particle in thequantum geometry.

A model for the motion of a particle in a quantum background