Many-body scattering processes in a one-dimensional boson system

The one-dimensional many-body problem described by the nonrelativistic ${\left|\phi \right|}^4$ theory of a complex scalar boson field (also known as the $\delta$-function model) is studied. By induction, it is shown that the set of $n\mathrm{th}$-order perturbation theory graphs for the $N$-particle in-state wave function can be combined and reduced to an equivalent set of factorized graphs in which the particle lines are numbered according to their ordering in momentum space at time $t=-\infty$. By carrying out loop integrations, each factorized graph is reduced to one of a finite number of skeleton graphs multiplied by a dressing function. The dressing functions are related to the multiparticle phase shifts which appear in Bethe's form of the $N$-body wave function. The skeleton graphs are shown to be associated with the ordering of particles in configuration space which characterizes Bethe's hypothesis. The analogy of a classical system of billiard balls is found to be helpful in interpreting the form of Bethe's hypothesis and the physical significance of the skeleton graphs. The use of factorized graphs in the scattering theory of statistical mechanics is demonstrated by a graphical calculation of the second virial coefficient.