Critique of Feynman Propagator, the $\E \cdot x$ gauge

Abstract

Consider M\o{}ller scattering. Electrons with momentum $p$ and $-p$ scatter by exchange of photon say in $z$ direction to $p+q$ and $-(p+q)$. The scattering amplitude is well known, given as Feynman propagator $ \M = \frac{(e \hbar c)^2}{\epsilon_0 V} \frac{\bar{u}(p+q) \gamma^{\mu} u(p) \ \bar{u}(-(p+q)) \gamma_{\mu} u(-p)}{q^2}$, where $V$ is the volume of the scattering electrons, $e$ elementary charge and $\epsilon_0$ permitivity of vacuum. But this is not completely correct. Since we exchange photon momentum in $z$ direction, we have two photon polarization $x,y$ and hence the true scattering amplitude should be $$ \M_1 = \frac{(e \hbar c)^2}{\epsilon_0 V} \frac{ \bar{u}(p+q) \gamma^{x} u(p) \ \bar{u}(-(p+q)) \gamma_{x} u(-p)\ \ + \bar{u}(p+q) \gamma^{y} u(p) \ \bar{u}(-(p+q)) \gamma_{y} u(-p) \ }{q^2}. $$ But when electrons are non-relativistic, $\M_1 \sim 0$. This is disturbing, how will we ever get the coulomb potential, where $\M \sim \frac{(e \hbar c)^2}{\epsilon_0 V q^2}$. Where is the problem ? The problem is with the gauge in Dirac equation. For a plane wave along $z$ direction, with electric field $E_x \sin (kz - \omega t)$, the Lorentz gauge is $$ (A_0, A_x, A_y, A_z) = \frac{E_x}{\omega} \cos(kz-\omega t)(0, 1, 0, 0)$$. But this gauge is not suited for calculating optical transitions, because we don't recover the Rabi frequency $q E_x d$ ($d$ electric dipole moment). What we find is something orders of magnitude smaller. Nor is it suitable for calculating electron electron scattering because we don't recover Coulomb potential. What we find is something orders of magnitude smaller. Instead, we work with $\E \cdot x$ gauge $$ (A_0, A_x, A_y, A_z) = \frac{-E_x}{2} ( x\ \sin(kz-\omega t), -\frac{\cos(kz-\omega t)}{\omega}, 0, \frac{x}{c} \sin(kz-\omega t) ) $$ ($c$ light velocity) to find everything correct. What we get is new propagator.