# Perturbation of Diracs equation (first order)

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 $$\psi_1 \in \ker(\gamma^\mu A_\mu)$$.

## 1 Answer

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.