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My question is: what's the construction of these multiple-particle states? We say the energy of two-particle states starts from 2m because we're using the approximation that these two particles are very far away from each other, thus the total energy starts from 2m. However, this is only an approximation. It's hard to imagine that one uses only "two-particle states" where the two particles are far apart, and yet get a complete set. In my opinion, the complete set should be constructed using truly eigenvectors of the Hamiltonian, and in this case we don't even know what a truly eigenvector is, except the vacuum state and the one-particle states. And the fact that we know the properties of the vacuum and one-particle states, is by virtue of physical arguments. In particular, we don't even have a definition for "multiparticle states".

Please help me understand this. Thanks.