Heisenberg’s S matrix for a system of many particles

When creation and annihilation of particles is forbidden, the general features of a manyparticle collision are illustrated by a scattering system containing two particles and a fixed scattering centre. The term in the wave function, which is called by Moller the ‘outgoing’ part, is shown to contain a term representing a totally outgoing wave, but also terms describing the interference effects between the incident wave of one particle and the outgoing wave of the other. Corresponding to the latter terms, there exist singular eigenstates of the S matrix which are simultaneous eigenstates of the separate kinetic energies of the particles. The remaining eigenstates are called non-singular, and for these only the sum of the kinetic energies can be given a definite value. Analytic continuation of the non-singular eigenstates in the complex plane of total kinetic energy shows that the corresponding eigenvalues of S can be used to determine the energy levels of states with both particles bound to the centre of force. The eigenvalues for the singular eigenstates will lead to the bound energy of a single particle in the scattering field of the other. The formalism is extended to include singular eigenstates which describe the scattering of one particle on a compound centre made up of the other particle bound to the scatterer.