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Here I cite part from Sidney Coleman's lectures on Quantum Field Theory:

It is a phenomenal fluke that relativistic kinematic corrections for the Hydrogen atom work. If the Dirac equation is used, without considering multi-particle intermediate states, corrections of $O \big(\frac{v}{c}\big)$ can be obtained. This is a fluke caused by some unusually low electrodynamic matrix elements.

What is the fluke about? Also, how can one justify the usage of Pauli-Schrodinger type equations that comes from first quantization of Dirac's equation? Schrodinger's equation is universal postulate valid for any quantum theory, and is equation for wave functionals in field theory. Could one go from non-relativistic QED field theory and then justify the usage of Pauli equation in which $\psi$ is interpreted as "wave function" in certain kinematical conditions (approximation)?

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  • $\begingroup$ Re the second part "Could one go from non-relativistic QED field theory..." the answer is yes in the approximation that particle creation/destruction can be neglected so particle number is conserved. Write the states as $|\Psi\rangle\sim\sum_k \psi(k) a^\dagger_k |0\rangle$ and interpret the c-number function $\psi(k)$ as the single particle (or the obvious generalisation for n-particle) wavefunction. You then derive the equation for this "wavefunction" by acting with the field equation. $\endgroup$
    – Michael
    Commented May 21, 2013 at 13:10
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    $\begingroup$ Note that this "wavefunction" is really just the expansion coefficient of the state in the single (or n) particle sector, but if the number of particles isn't changing this is often good enough. $\endgroup$
    – Michael
    Commented May 21, 2013 at 13:14
  • $\begingroup$ Related: " Why does the Dirac equation work for the hydrogen atom?" physics.stackexchange.com/q/483505/226902 $\endgroup$
    – Quillo
    Commented Dec 7, 2022 at 13:45

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There is a well established formalism for studying the non-relativistic limit of QED (and other relativistic field theories, such as QCD with heavy quarks) known as NRQED. The basic idea is to integrate out anti-particles and construct an effective field theory that contains electrons only. The leading term is the Schroedinger lagrangian coupled to the Coulomb potential $$ {\cal L} =\psi^\dagger \left(i\partial_0-\frac{\nabla^2}{2m}-eA_0\right)\psi + \ldots \,. $$ There are infinitely many higher order terms that have to be computed order by order in $\alpha$. They contain purely kinematic corrections, spin-orbit terms, renormalization constants etc.

This lagrangian conserves particle number and can be used in bound state calculations. Counting powers of $\alpha$ in binding energies is not entirely trivial, because in addition to explicit factors of $\alpha$ in the interaction there are also factors of $\alpha$ hidden in the wave function. For the hydrogen atom the leading term in $E_n$ is $O(\alpha^2)$. Kinematic corrections contain $(v/c)^2=O(\alpha^2)$, so $\Delta E=O(\alpha^4)$. What Coleman is referring to is that it is not obvious that there are no radiative corrections at this order. In fact, the Lamb shift is $O(\alpha^5)$. (One $\alpha$ from radiative corrections, one from the Coulomb Hamiltonian, three from the wave function at the origin).

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