Why are interacting field theories called nonlinear? Explanation for interacting EM field, in particular

Question

The classical equation of motion for the electromagnetic field interacting with a charged fermion field $\psi$ of charge $eq$ is given by $$\Box A^\mu(x)=j^\mu(x)$$ where $j^\mu(x)=eq\bar{\psi}(x)\gamma^\mu\psi(x)$. The equation of motion for $\psi$ reads $$(i\gamma^\mu\partial_\mu-m)\psi(x)=eq\gamma^\mu A_\mu\psi.$$ According to the nomenclature of differential equations, both the equations are linear, inhomogeneous partial differential equations. Then, why are interacting field theories called non-linear?

Answer 1

I think the following is a clean and correct way to see why the equation for $\psi$ is nonlinear. The solution of the first equation is $$A^\mu(x)=\int d^4y ~G(x-y)j^\mu(y)=eq\int d^4y ~G(x-y)\bar{\psi}(y)\gamma_\mu\psi(y)$$ where $$G(x-y)=\int\frac{d^4p}{(2\pi)^4}\frac{e^{ip\cdot(x-y)}}{p^2+i\epsilon}.$$ Substituting it in the equation for $\psi$, we realize that $$(i\gamma^\mu\partial_\mu-m)\psi(x)=(eq)^2\gamma^\mu\left(\int d^4y ~G(x-y)\bar{\psi}(y)\gamma_\mu\psi(y)\right)\psi(x).$$ Now the source term is manifestly non-linear.