Introduction

The problem of N mutually interacting particles in direct interaction has received much attention in recent years [1].

The problem can be stated in the following form: to find N mass constraints which satisfy the following requirements

1. i) they must be first class constraints in the sense of Dirac [2],[3]

2. ii) they must have the cluster decomposition property (separability), defined as the possibility of splitting the system in subsystems (clusters) by switching off some of the interactions [5],[6],[7] and [8].

The two requirements (i) and (ii) are difficult to satisfy for more than two bodies. In particular the cluster decomposition property or separability requires the presence of genuine N body forces [9],[10] and [11].

In the present paper we start an analysis of the spinless case, with harmonic interaction. We have chosen this simplified case, since the classical solution is well known, in order to have some insight into the difficulties of the problem. A separable model for three fermions has been proposed in ref.[12].

We will show that it is possible to give the form of the first class constraints in the general case, but defined in implicit form. In our intentions this analysis would be the starting point of a relativistic theory of small vibrations about a stable configuration of some general potential.

In a particular configuration, when two over three of the coupling constants are equal, there is a great simplification, particularly if one uses a suitable gauge fixing. The analysis of this case is in progress.