How are Bethe-Salpeter type bound state equations derived in light-front QFT?

Question

In light-front QCD, one is often interested in computing the lightcone Hamiltonian \(P^-\) constructed from the stress tensor. Then one goes on to expand the fields in an oscillator basis and then normal order the Hamiltonian. Finally, one arrives at an infinite set of coupled integral equations by considering the action of the Hamiltonian on a superposition of multi particle states, e.g. of the form Eq. (18) in http://arxiv.org/pdf/hep-th/9705045.pdf .

My question is: how is this derived? Naively, I should think that the wavefunctions (which in the above reference are denoted by \(f_n\)) should not change by acting with the light cone Hamiltonian, since it is basically Wick contractions. What gives rise to the fact that the wavefunctions in Eq. (18) involve different particle numbers?

Answer 1

(18) is obtained by substituting (12) and (14) into (8). Since (14) contains different numbers of creation and annihilation operators, the mass square operator doesn't preserve particle number.