All the Lagrangian densities I have seen have always been polynomials of the fields. Is this a coincidence or is there a reason which forbids, say, Lagrangians with logarithms or exponentials of the fields?
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There is a reason that forbids functions other than polynomials of low degree, but sometimes we do not care about that reason, and then we do have complicated functions of the fields in the Lagrangian. Let me first give the reason, then explain when we disregard it, and then an example.
Let us use units where action is dimensionless, i.e., $\hbar = 1$, and also $c = 1$. Then the action is $$S = \int d^4 x \, \mathcal L.$$ and $d^4x $ has the units of $\text{mass}^{-4}$; it's mass dimension is $-4$. Hence the Lagrangian $\mathcal L$ must have units of $\text{mass}^4$, i.e., mass dimension $4$. In these units, the electromagnetic field tensor $F_{\mu\nu}$ has mass dimension $2$ and a Dirac field $\psi$ mass dimension $\frac 3 2$.
Now it turns out that in a renormalizable theory, any coupling constants must have mass dimension $d \ge 0$ -- the proof is in Weinberg, Ch. 12. Hence a term like $\frac{1}{M} \bar \psi \gamma^\mu\gamma^\nu F_{\mu\nu} \psi$ or $\frac{1}{M^4}(F_{\mu\nu}F^{\mu\nu})^2$ cannot appear in a renormalizable Lagrangian. A forteriori, this rules out functions like $\exp(F_{\mu\nu}F^{\mu\nu})$. [This is also why general relativity and QFT don't mix: the curvature has dimension of $\text{length}^{-2} = \text{mass}^2$, so $G$ has mass dimension $-2$.]
However, if we drop the requirement of renormalizabilty, then in an effective field theory there is no limit to what we can put in our Lagrangian. In fact a term like $\frac{1}{M} \bar \psi \gamma^\mu\gamma^\nu F_{\mu\nu} \psi$ appears in the effective theory of electrons in external fields -- it's the famous anomalous magnetic moment. While this is still a polynomial, Heisenberg and Euler calculated the effective Lagrangian for QED in a strong background field already in the 30s. It is $${\displaystyle {\mathcal {L}}=-{\mathcal {F}}-{\frac {1}{8\pi ^{2}}}\int _{0}^{\infty }\exp \left(-m^{2}s\right)\left[(es)^{2}{\frac {\operatorname {Re} \cosh \left(es{\sqrt {2\left({\mathcal {F}}+i{\mathcal {G}}\right)}}\right)}{\operatorname {Im} \cosh \left(es{\sqrt {2\left({\mathcal {F}}+i{\mathcal {G}}\right)}}\right)}}{\mathcal {G}}-{\frac {2}{3}}(es)^{2}{\mathcal {F}}-1\right]{\frac {ds}{s^{3}}}}$$ where $\mathcal F = \frac{1}{4}F_{\mu\nu} F^{\mu\nu}$ and $\mathcal G = \frac{1}{4}\epsilon_{\mu\nu\rho\sigma} F^{\mu\nu}F^{\rho\sigma}$. Quite the complicated function of the fields!
The Heisenberg-Euler Lagrangian has been the subject of much study since, and I'm sure others can give other examples of effective field theories with non-polynomial actions.
Robin Ekman
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