I am studying the derivation of LSZ reduction formula on the book "A modern introduction to quantum field theory" by Maggiore. Here it is derived considering a single species of neutral scalar particles. The matrix element of the scattering matrix S is given in the Heisenberg representation by $$\langle p_1\dots p_n;T_f|k_1\dots k_m;T_i\rangle$$ at page 113 it is assumed that $$\langle p_1\dots p_n;T_f|a_{k_1}^{\dagger,(out)}|k_1\dots k_m;T_i\rangle=0$$ as $a_{k_1}^{\dagger,(out)}$ acts on its left destroying a particle with momentum $k_1$ and we assume that there isn't a particle in the final state with momentum $p_i$ that coincides with $k_1$, as we consider a situation in which all particles interact and there are not "spectators". In this way I think we are excluding the situation in which for example two particles interact but exchanges their momentum (A, B particles, $p_A^{fin} = p_B^{in}$ and $p_B^{fin} = p_A^{in}$). I think it could make sense to exclude this situation as we are considering identical particles that we can't distinguish, so in this kind of situation, we couldn't even tell that there was an interaction between A and B and hence even this case must be excluded. Is this right or there is another explanation?
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I think you got it right: as the particles are indistinguishable, the situation where particles A and B interact and exchange their momenta would appear the same as the situation where they don't interact. Hence you must set that $p^{in}_i\neq p^{fin}_j$ for all particles if you want no spectators particles.