The W₃-particle

Summary

Abstract

We show that W₃ is the algebra of symmetries of the “rigid-particle”, whose action is given by the integrated extrinsic curvature of its world line. This is easily achived by showing that its equation of motion can be written in terms of the Boussinesq operator. We also show how to obtain the equations of motion of the standard relativistic particle provided it is consistent to impose the “zero-curvature gauge”, and comment about its connection with the KdV operator.

Introduction

The geometrical interpretation of W-type symmetries has attracted the attention of many mathematical physicists in recent years. Although a plethora of interesting results are now at our disposal it is commonly agreed that we have not yet a complete understanding of the underlying geometry. It is clear that simple mechanical systems enjoying W symmetry could be an unvaluable tool in this difficult task. On one hand they could provide us with some geometrical and/or physical interpretation for W-transformations (W-morphisms), while on the other hand they could give us some hints about which are the relevant structures associated with W-gravity - the paradigmatic example being provided by the standard relativistic particle and difTeomorphism invariance (W2).

It is well known by now the connection between W-morphisms and the extrinsic geometry of curves and surfaces [1]. Therefore, it seems natural to look for a W-particle candidate among the geometrical actions depending on the extrinsic curvature.