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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.

Thanks in advance.

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    $\begingroup$ 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. $\endgroup$ – Sebastian Riese Oct 20 '15 at 7:53
  • $\begingroup$ $D_\mu$ depends on a field and modifies the kinetic term by a cubic interaction. $\endgroup$ – Arnold Neumaier Nov 8 '15 at 18:31

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