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Some systems, like atoms, are described well by quantum mechanics, where one just gives the Hamiltonian in the form $H=T+V$ and computes the eigenvalues and eigenvectors of this operator to figure out the energies and wave functions.

However particles at a more fundamental level should behave according to a quantum field theory. We should then be able to derive the non relativistic potential from this more fundamental theory. Let us focus on deriving $V$ for the Hydrogen atom.

The way I would do it is by considering the scattering process $$e^- (p_1) + e^- (p_2) \to e^- (p_1') + e^- (p_2')$$ in QED and in NRQM (non-relativistic quantum mechanics).

The former (through the usual Feynman rules) gives: $\mathcal{M}_{QED}$ while the latter gives a similar result through the Lippmann-Schwinger equation where in this latter case we keep $V$ unknown. We should then compare the results to obtain $V$ in the non-relativistic limit.

This is how I would do it at least. I find a little weird that the Lippmann-Schwinger has retarded propagators in the expansion while the QED amplitude has time-ordered ones but other than that it makes sense to me.

On the other hand when talking about the Lamb shift, in his book "Quantum Field Theory and the Standard Model", chapter 16.3, Schwartz defines the effective potential simply by correcting the bare propagator with the vacuum bubble, obtaining the Uehling potential:

$$U_{Ue} (r) = -\frac{e^2}{4 \pi r} - \frac{e^4}{60 \pi^2 m^2} \delta (r).\tag{16.59}$$

He then claims that we could plug this corrected potential into the Schroedinger equation to calculate the Lamb-shift. This doesn't seem right to me, aren't we neglecting all effects coming from vertex corrections? These are also of order $e^4$, so they should in principle be equally important.

Naively I would expect all two-particle-irreducible diagrams to give a contribution to the interaction (two-particle-reducible ones should be obtained automatically by the Schroedinger equation via the Lippmann-Schwinger expansion). What am I missing?

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    $\begingroup$ Indeed, the Uehling term is only part of the Lamb shift, and if you read Schwartz's book carefully, you will see that this is what he says. The modern method for deriving this carefully is based on an effective field theory, NRQED. $\endgroup$
    – Thomas
    May 29, 2022 at 2:01
  • $\begingroup$ @Thomas Could you provide a reference where they show exactly this kind of derivation? Also, is "my" method of derivaing NR potentials correct or should it be done by some other way? Thank you. $\endgroup$ May 29, 2022 at 14:17
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    $\begingroup$ See, fr example, arxiv.org/abs/hep-ph/9611313 . What you say is roughly right, but you typically match amplitudes in the perturbative regime (no bound states), and then compute bound states by solving a Lippman-Schwinger equation. Also, not everything can be absorbed in a potential, there are electron self energy effects, retarded photons, multi-photon emission, etc. $\endgroup$
    – Thomas
    May 29, 2022 at 22:40

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