Phase space invariance groups and relativistic three-particle states

A new approach is proposed to the problem of the classification of the states of three relativistic particles. The method is based on the idea of the existence of a finite group $H$ of transformations that leave invariant not only the equation of the energy surface but also the element of the relativistic three-particle phase volume. Equations are found that determine a one-parametric subgroup of $H$ and, in the case of three identical particles, the group itself is found. An important feature of this group is the fact that the exchange of particles is a particular clement of the group. The Lie algebra of the generators of $H$ are used to construct a complete set of commuting Hermitian operators, including the exchange operator. A complete orthonormalized system of states is obtained; it possesses the necessary symmetry propertics under exchange. The kinematic variables used in the problem map the physical region of the Dalitz plot onto a ring.