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## Main Question or Discussion Point

Hi, I'm not very good at QFT, but starting from Dirac's QED Lagrangian (look for example at: http://en.wikipedia.org/wiki/Quantum_electrodynamics" [Broken] )

[tex]

L=\bar{\psi}\left(\gamma^{\mu}\left(i\hbar\frac{\partial}{\partial\mu}-q(A^{ext}_{\mu}+A_{\mu})\right)+

mc\gamma^0 \right)\psi-\frac{1}{4\mu_0}F_{\mu\nu}F^{\mu\nu}

[/tex]

From here we derive Dirac's equation and Maxwell's equations. Now omitting all derivation steps, every thing could be summarized into the non-relativistic limit as:

[tex]

\left(\frac{\hat{p}^2}{2m}+q\Phi_{ext}-

\frac{q^2}{4\pi\epsilon}\int\frac{\mid\psi(\vec{r})\mid^2}{\mid \vec{r}-\vec{r}^{\prime} \mid}d^3\vec{r}^{\prime}\right)\psi=E\psi

[/tex]

This is in fact what the QED Lagrangian result in (ignoring the very small contribution from the magnetic vector potential A for simplicity), but the effective Schrödinger equ looks more like Kohn-Sham equation for a single particle. But is this correct?

The Coulomb integral suggest that the electron is self-interacting with itself! (compare coulomb blocking) I thought that the correct limit has to be Schrödinger equation, but with some funny coupling term (but not so strong as coulomb self-interaction). Is it the field operators that are needed in some way to vanish this term, when it is not "appropriate"?

[tex]

L=\bar{\psi}\left(\gamma^{\mu}\left(i\hbar\frac{\partial}{\partial\mu}-q(A^{ext}_{\mu}+A_{\mu})\right)+

mc\gamma^0 \right)\psi-\frac{1}{4\mu_0}F_{\mu\nu}F^{\mu\nu}

[/tex]

From here we derive Dirac's equation and Maxwell's equations. Now omitting all derivation steps, every thing could be summarized into the non-relativistic limit as:

[tex]

\left(\frac{\hat{p}^2}{2m}+q\Phi_{ext}-

\frac{q^2}{4\pi\epsilon}\int\frac{\mid\psi(\vec{r})\mid^2}{\mid \vec{r}-\vec{r}^{\prime} \mid}d^3\vec{r}^{\prime}\right)\psi=E\psi

[/tex]

This is in fact what the QED Lagrangian result in (ignoring the very small contribution from the magnetic vector potential A for simplicity), but the effective Schrödinger equ looks more like Kohn-Sham equation for a single particle. But is this correct?

The Coulomb integral suggest that the electron is self-interacting with itself! (compare coulomb blocking) I thought that the correct limit has to be Schrödinger equation, but with some funny coupling term (but not so strong as coulomb self-interaction). Is it the field operators that are needed in some way to vanish this term, when it is not "appropriate"?

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