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In QED, on general grounds one can expect that there are mixed electron-photon Fock components contributing to the Lamb shift at 2 loops and beyond—i.e. states of the form $a^{\dagger} c^{\dagger} |0\rangle$ where $a^{\dagger}$ is the creation operator of the photon. These contributions presumably do not arise at 1 loop. Now if one looks at standard books like this one, Eides, Grotch, and Shelyuto, they are discussing high-order calculations, but the Fock states of the above type are never part of the discussion. Ostensibly, they start from Bethe-Salpeter equation, but this equation has to be complemented by higher components beyond 1 loop. (They actually never solve this equation, merely develop perturbation schemes for it.) I would like to understand better this situation.

Buzz
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0x11111
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  • Particle physics is more about dynamics of fundamental interactions and not so much about states. When states are introduced to compute input-output scenarios, the state are rather simple pure product states. In any case, the Bethe-Salpeter equation is a rather difficult equation to solve. – flippiefanus Feb 12 '24 at 02:11
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    Could you be a bit more specific? Do you have e.g. a concrete missing Feynman diagram in mind? In nrQED (the formalism used in the Eides review), an effective Hamiltonian is built which acts on the space of non-relativistic hydrogen-like states only, and not on the photonic Fock space; the effect of the latter is already processed and built into the effective Hamiltonian. In this sense, the handling of photonic Fock states is not really different from the original old-fashioned PT calculation of the Lamb shift (which gives the low-energy part of the one-loop shift). – dennismoore94 Feb 12 '24 at 13:17
  • Specific question: why is it possible to build an NRQED formalism that does not have any photonic degrees of freedom in the Fock space? Certainly photons are created in the vacuum due to polarization near the electron. The concrete diagram is just real photon emission. Although it seems incorrect to think in terms of diagrams in this context - wave functions resum infinitely many terms coming from ever higher order diagrams. I would expect explicit use of functions of the form $\psi_{\mu,\alpha}(x,y)$ where $\mu$ is a vector index and $\alpha$ is a spinor index. – 0x11111 Feb 12 '24 at 19:09
  • But real photon emission does not contribute to the shifting of energy levels. In bound state QED, the shift of hydrogen formally arises from an infinite number of scattering events with only an electron and a proton in both the "in" and "out" states (this infinite number of scatterings being "resummed" in the B-S formalism). Only virtual photons are present as internal lines (self-energy, vacuum polarization, etc.), and their effect is packaged into an effective Hamiltonian (which acts only on the fermions), in such a way that the effective Hamiltonian reproduces the full QED result up... – dennismoore94 Feb 12 '24 at 20:34
  • ...to a certain $\alpha$ order. Have you seen the derivation of the leading QED correction to hydrogen (in external field approximation)? If no, then you should first check it out (in e.g. Greiner-Reinhardt or in Itzykson-Zuber) before trying to read the Eides review; that review is basically a compilation of results, not a proper learning material. – dennismoore94 Feb 12 '24 at 20:35
  • I am not referring to real photons. They are as "bound" as electrons - one would expect the wave function to decrease exponentially with distance. I am familiar with GR and IZ books, but their calculation is so structured that it is valid only at 1 loop, and more work is needed to generalize it beyond 1loop. To make it clear what I am asking about: at even higher orders, there will be e+e- pairs created in vacuum that would contribute to the energy shift. These pairs e.g. will have their own BS-like ladders with the proton.... – 0x11111 Feb 12 '24 at 21:17
  • To account for these, one would need "partonic wave functions" of the form $\psi_{\alpha,\beta,\gamma}(x,y,z)$. The photon-electron WF I was referring before is only one of the infinite tower of wave functions. NRQED framework does not seem to account for these, and this is the essence of my question. Actually, the traditional BS eqn does not account for it either, but in the case of BS, there is at least some discussion among physicists of how to generalize it in gauge invariant way to incorporate these higher components ( it turns out quite hard - full blown BV formalism is needed). – 0x11111 Feb 12 '24 at 21:18
  • The term "partonic wave function" seems to imply some kind of hadronic structure, which is clearly beyond the scope of nrQED. I don't really understand your $e^--e^+$ example: sure, virtual particle-antiparticle pairs do contribute to the energy, but these and the associated ladder effects are taken into account by some (possibly very complicated) irreducible interaction diagram that can be included in the B-S equation. Basically everything that perturbative pure QED can describe is formally covered by the B-S equation, no? (B-S is also perturbative on the level of irreducible interactions.) – dennismoore94 Feb 13 '24 at 19:12

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