# Solutions to Dirac equation with Gauge field coupling

Looking at the Dirac equation of the form

$$\Big(i\gamma^{\mu}(\partial_{\mu}-iA_{\mu})-m\Big)\psi=0$$

There is a simple solution to this equation, which is

$$\psi=\exp\Big(i\int^xA_{\mu}dx^{\mu}\Big)\psi_0$$

Where $$\psi_0$$ solves the Dirac equation in the absence of the gauge field.

My question is, are there any other solutions to this equation? Presumably there are, or else it appears the gauge field has almost negligible effect on the dynamics of the solution.

Is there a nice way of categorizing these different types of solutions?