Why are the poles of the propagator the masses of bound states and not the masses of unbound states as well? For a real scalar field $\phi$, the Kallén-Lehmann representation of a two-particle propagator in a translation invariant theory in QFT is:
$G(x_1,x_2) = \sum_{m, \; other \; quantum \; numbers} \int \frac{d^4p}{\left( 2 \pi \right)^4} e^{ip(x_1-x_2)} \frac{i}{p^2-m^2+i \epsilon} \lvert <\Omega| \phi(0) | m,\; \pmb{p}=\pmb{0}, \; other \; quantum \; numbers> \rvert^2$,
where the sum over $m$ is performed over all the invariant masses allowed by the theory.
I get that if the theory allowed bound states then I should sum over the masses of bound states as well, which would then show up as poles of the propagator, but why should't I sum over the masses of unbound states as well? After all a state of two unbound particles would have a mass of $2m_{\phi}$, which should then appear as a pole in the propagator. Is it because if the two particles are unbound $\lvert <\Omega| \phi(0) | m, \; \pmb{p}=\pmb{0}, \; other \; quantum \; numbers> \rvert^2$ is just zero, therefore it cancels the pole?
 A: The sum over states that you've written is supposed to be literally the sum over all states, with the restriction that the total spatial momentum of each state should be $\vec p=0$. You are in principle summing over bound and unbound states. For example if there was a bound state of two $\phi$ particles which had Lorentz-invariant mass $M^2$, this state would show up in the sum.
I don't understand why you think there is some preference for bound states in that sum, and that we're somehow neglecting unbound states. For example, in that sum, we could have a state with two unbound $\phi$ particles with opposite spatial-momenta so that $\vec p_{\textrm{total}}=0$. The energy of such a state would be $E_{\textrm{total}}=2\sqrt{m^2+|\vec p|^2}$ and the mass-squared would be $m^2$. However the state of a single $\phi$ particle having with $\vec p\neq 0$ would not be present in the sum.
Edit: Take a look at Figure 7.2 on page 214 of Peskin & Schroeder. It shows a typical plot of the spectral density function, which is essentially what you're asking about. It  illustrates the idea that I conveyed in this answer.
