Adding energies vs. adding momenta Suppose we have a (time and space) translationally invariant system with the Fock space for a Hilbert space. Temporal and spatial translation invariance implies that energy and momentum are good quantum numbers. In the free theory, the momenta and energies of one-particle states simply add to get the total momentum and total energy of the system. Why is it that, for an interacting theory, we can (always?) add the momenta of the one-particle states to obtain the total momentum given that we can't do the same for the energy?
 A: In quantum mechanics, the linear momentum is represented by a one-body operator: the momentum of the $i$-particle only depends on the variables of this particle, $\hat{\bf p}_i = -i\hbar \nabla_i.$ So, the total momentum of the system is simply defined by the sum of the momentum of each particle. Of course, if the particles are immersed in some field (like an electromagnetic one), it would be necessary to sum the field momentum to the momenta of the particles. 
On the other hand, the interaction energy terms correspond to two-body operators, for example, the repulsion between two electrons corresponds to $\hat{V}({\bf \hat{r}}_1, {\bf \hat{r}}_2) = \frac{e^2}{|{\bf \hat{r}}_1-{\bf \hat{r}}_2|},$ and it involves, naturally, the variables of both particles. For these reason, the total energy is not additive, as the interaction terms "belong" to the two-particles, and not to each particle separately. It doesn't matter if the particles form bound states.
Other energies are additive (one-body operators), for example, the kinetic energy, the potential energy in an external field. 
