# Linear/ non linear Scalar field theory

How do I understand that the action for the free relativistic scalar field theory is non linear? What will be the associated interaction potential of that equation?

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The actions (or Hamiltonians) have a higher order (by one, in simplest cases) than the equations of motion because the equations of motion are obtained by differentiating the action (or Hamiltonians) with respect to fields (or other degrees of freedom) and the derivative of $\phi^n$ is $n\phi^{n-1}$, where the exponent grew by one. The bilinear/quadratic actions are "natural" because they are what is minimized or maximized in linear regression and similar situations, too.
Usually there are also higher-than-second-order terms, the interaction terms, that are responsible for the interactions between the fields (or a field with itself). There are many choices what these interactions may be. In quantum field theory, we usually consider "renormalizable" interactions which means in $d=4$ "at most fourth-order", roughly speaking.