The solution of the Dirac equation with the interaction term*

With the best knowledge of the authors, it is very difficult, if it is not impossible, to find the exact solution of the Dirac equation with the interaction term [1,2]. In this article, we carefully analyze the mathematical theory of the quantum field and try to find the exact solution of the field equation with the interaction term. For simplicity, we take the Dirac field equation with the interaction term as a template of the gauge field equation. We will show how to obtain the exact solution of the Dirac equation with the interaction term. The results can be applied to other fields as well, such as QCD.