# How are bound states handled in QFT?

QFT seems very well suited to handle scattering amplitudes between particles represented by the fields in the Lagrangian. But what if you want to know something about a bound state without including it as an extra field? For example, suppose we have electron+proton QED (ignoring the proton's structure):

$$\mathcal{L} = -\frac14 (F_{\mu\nu})^2 + \bar{\psi_e} (i\not \partial -m_e)\psi_e + \bar{\psi_p} (i\not \partial -m_p)\psi_p - e \bar{\psi_e} \not A \psi_e + e \bar{\psi_p}\not A \psi_p$$

I can use this with no problem to calculate Rutherford scattering or similar processes. But this Lagrangian should also have the hydrogen atom hidden in it somewhere. For example, I may want to use QFT to calculate the binding energy of hydrogen. Or I might want to calculate the probability of firing an electron at a proton and getting hydrogen plus photons as a result. How can this be done? Obviously this is a broad subject, so I'm just looking for an outline of how it goes.

• Related: Bound states in QED(Phys.SE question), "On bound states in quantum field theory"(arXiv link) – ACuriousMind Nov 9 '15 at 20:32
• @ACuriousMind: I've taken a look at that question, it's related but not quite about the same thing. The paper looks interesting though, I'll be sure to look at it. – Javier Nov 10 '15 at 1:15
• In scattering an electron off of a proton, you should see the hydrogen bound states as poles in the S-matrix. To get hydrogen itself as a final state, you might try an effective field theory approach where you add the hydrogen atom to your Lagrangian. – user2309840 Nov 12 '15 at 5:03