I'd like to know how to solve the dirac equation with some small gauge potential $\epsilon \gamma^\mu{A}_\mu(x)$ by applying perturbation theory. The equations reads as $$(\gamma^\mu\partial_\mu-m+\epsilon\gamma^\mu A_\mu(x))\psi(x) = 0.$$

Now I use the ansatz $\psi(x) = \psi_0(x)+\epsilon\psi_1(x)$ where $\psi_0$ is the free solution of the dirac equation. Putting the ansatz into the dirac equation we get for different orders:

$$\epsilon^0: (\gamma^\mu\partial_\mu-m)\psi_0(x) = 0,\\ \epsilon^1: (\gamma^\mu\partial_\mu-m)\psi_1(x) = -\gamma^\mu A_\mu(x) \psi_0(x),\\ \epsilon^2: \gamma^\mu A_\mu (x) \psi_1(x) = 0.$$

At this point I'm not sure how to solve the second equation such that $\psi_1 \in \ker(\gamma^\mu A_\mu)$.


The perturbation $\psi_1$ doesn't need to lie in the kernel of $\gamma^\mu A_\nu$. The second of your equations should be solved by using the free-electron Green's function (i.e. solve the Dirac equation in the presence of a delta-source), essentially obtaining $$ \psi_1(x) = -\left(\gamma^\mu \partial_\mu - m \right)^{-1}\circ \gamma^\mu A_\mu \psi_0 $$ where the inverse of the free-electron Dirac equation is understood as representing the appropriate Green's function (i.e. with suitable boundary and temporal conditions).

The third equation (which is why I believe you are asking for the solution to lie in the kernel) is not correct, because you have not included the quadratic correction to $\psi$. That is, you have expanded to order $\epsilon^2$ in terms of the first-order correction $\psi_1$ but in order to maintain consistency in your expansion, you should write $\psi =\psi_0 + \epsilon \psi_1 + \epsilon^2 \psi_2+...$, and $\psi_2$ should enter in your equation at quadratic order in $\epsilon$. So the corrected version of your equation 3 is that $$ \epsilon^2 (\gamma^\mu \partial_\mu - m)\psi_2 + \epsilon^2 \gamma^\mu A_\mu \psi_1 = 0 $$ which can then be solved using your solution for $\psi_1$. Proceeding to higher orders in $\epsilon$ will amount to expanding the Dyson equation for multiple scatterings to the desired order.


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