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In QED, one can relate the two-particle scattering amplitude to a static potential in the non-relativistic limit using the Born approximation. E.g. in Peskin and Schroeder pg. 125, the tree-level scattering amplitude for electron-electron scattering is computed, and in the non-relativistic limit one finds the Coulomb potential. If one allows for 1/c^2 effects in the non-relativistic expansion, one also finds spin-dependent interactions (e.g. spin-orbit, see Berestetskii, Lifshitz, Pitaevskii pg. 337).

Are there any alternative methods for calculating a two-particle non-relativistic potential?

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    $\begingroup$ Try Zee, Quantum Field Theory in a Nutshell, chapters 1.4-1.6. Basically compute the path integral with a classical current and read off the energy from $Z \sim e^{i H t}$. $\endgroup$ – Michael Brown Mar 23 '13 at 9:23
  • $\begingroup$ Related: physics.stackexchange.com/q/2244/2451 , physics.stackexchange.com/q/3580/2451 and links therein. $\endgroup$ – Qmechanic Dec 17 '13 at 18:55
  • $\begingroup$ See the power-counting of NRQED, the potential is the leading effect carried by $D_0$ and the rest are suppressed by $1/c$, including the spin-orbital term you mentioned, and higher kinetic corrections and Darwin term. $\endgroup$ – Turgon Dec 3 '18 at 7:08
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If you set $c\to \infty$ in the QED Hamiltonian you obtain a non-relativistic Hamiltonian whose potential only includes a pseudo-Coulomb term $(1/r)$, because this is the only term of order $c^0$ in the QED interaction. No further calculation is needed to obtain the potential.

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