In the first part of the talk, I will show that using connectedness structure of matrix elements, while expressing the 2-body sub-channel scattering amplitudes by a tower of isobars one can derive a 3-dimensional integral equation for the 3->3 scattering amplitude. In this context isobars are functions with definite quantum numbers and correct right-hand singularities. The analytic properties of those and other building blocks of the derived integral equation equations are constrained by the 2- and 3-body unitarity, and are derived using dispersion relation.
In the second part of the talk, I will show possible applications of such an amplitude in infinite and in finite volume. The latter is of particular interest for the recent and upcoming Lattice QCD calculations as it allows to derive a simple form of the 3-body quantization condition.