The classical equation of motion for the electromagnetic field interacting with a charged fermion field $\psi$ of charge $eq$ is given by $$\Box A^\mu(x)=\mu_0j^\mu(x)$$ 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?

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?

The classical equation of motion for the electromagnetic field interacting with a charged fermion field $\psi$ of charge $eq$ is given by $$\Box A^\mu(x)=\mu_0j^\mu(x)$$ where $j^\mu(x)=eq\bar{\psi}(x)\gamma^\mu\psi(x)$. According to the nomenclature of differential equations, this is a linear, inhomogeneous partial differential equation because the RHS is independent of $A^\mu$. Then, why are interacting field theories called non-linear?

The classical equation of motion for the electromagnetic field interacting with a charged fermion field $\psi$ of charge $eq$ is given by $$\Box A^\mu(x)=\mu_0j^\mu(x)$$ 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?