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I have a question about a statement I've seen in many a Quantum Field Theory book (e.g. Zee). They say that the general form of the Lagrangian density for a scalar field, once two conditions are imposed:

(1) Lorentz invariance, and

(2) At most two time derivatives,

is:

L = 1/2(d\phi)^2 - V(\phi)

where V(\phi) is a polynomial in \phi.

Why is this? I can understand how the conditions restrict the kinetic energy term to being what it is, but I don't understand why V has to be _polynomial_ in \phi.