Relating the finite-volume spectrum and the two-and-three-particle S matrix for relativistic systems of identical scalar particles

Working in relativistic quantum field theory, we derive the quantization condition satisfied by coupled two- and three-particle systems of identical scalar particles confined to a cubic spatial volume with periodicity L. This gives the relation between the finite-volume spectrum and the infinite-volume 2→2, 2→3, and 3→3 scattering amplitudes for such theories. The result holds for relativistic systems composed of scalar particles with nonzero mass m, whose center of mass energy lies below the four-particle threshold, and for which the two-particle K matrix has no singularities below the three-particle threshold. The quantization condition is exact up to corrections of the order O(e−mL) and holds for any choice of total momenta satisfying the boundary conditions.