Covariant Description of Several Spinless Particles

To describe a state of $n$ particles it is necessary to construct a matrix element or wave function from the momentum vectors of the $n$ particles. It is usually possible to write down a simple function having the correct angular momentum and parity. It is not clear in what sense the choice made is general. This is in contrast to the two-particle states, where the spherical harmonics form a complete orthonormal set over the phase space. The spherical harmonics are homogeneous polynomials in the components of the relative momentum of the two particles. It will be shown that homogeneous polynomials in the $n-1\mathrm{}$ relative momenta of $n$ particles entering an $n$-particle state, with a correction term for relativistic kinematics, form a complete orthonormal set of functions over the $n$-body phase space and provide a basis for a systematic classification of $n$-body states. There are some new quantum numbers (degeneracy indices) that enter and may or may not have physical significance. The application of these notions to $\omega$ decay is briefly considered. The basis of this classification is the determination of a larger invariance group than the rotations for a system of free particles. The Lie algebra of generators of this group furnishes a complete commuting set of operators, and it is exhibited. The eigenfunctions of this set are given.