In standard textbooks in QFT while discussing e.g. the Kallen-Lehmann formula (see e.g. Section 7.1 in the Peskin-Schroeder book) it is always assumed that bound states of two or more particles have higher mass than the one particle states. Why this should be true?
Let us compare this for example with the classical two particles interacting according to the Coulomb law.They may rotate around each other with fixed distance $R$ such that the center of mass is at rest, thus forming a bound state. The total energy goes to $-\infty$ when $R$ goes to 0.
Analogous problem arises in the case of two non-relativistic quantum particles interacting again according to Coulomb law. While the total energy of the bound state cannot be arbitrarily small as in the classical case, the discrete energy levels (corresponding to bound states) are negative. At the same time if the particles are at rest and are far apart from each other and thus do not interact, their energy vanishes.
Am I wrong?