Quadratic terms in QED lagrangian density

I recently learned that when we speak about a "free lagrangian", this actually means that the lagrangian is quadratic in the fields. When considering the Lagrangian density describing the coupling to electro-magnetic field:

${\mathcal{L}=\bar{\Psi}(i\gamma^{\mu}D_{\mu}-m)\Psi}-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$, (where $D_{\mu}=\partial_{\mu}+ieA_{\mu}$)

one must then encounter some terms that are not quadratic in the field.

My question is what are the terms that are not quadratic in the QED lagrangian and why.

• It's the $\bar \Psi \gamma^\mu A_\mu \Psi$ term, which is not quadratic in the fields, as there occur three fields multiplied together. – Sebastian Riese Oct 20 '15 at 7:53
• $D_\mu$ depends on a field and modifies the kinetic term by a cubic interaction. – Arnold Neumaier Nov 8 '15 at 18:31