Higher-order derivatives than second-order differential equations From https://doi.org/10.1063/1.2155755

he limited himself to second-order differential equations.


Our experience in elementary-particle physics has taught us that any term in the field equations of physics that is allowed by fundamental principles is likely to be there in the equations

I guess the author means from the effective field theory point of view. Namely, effective actions include non-renormalizable terms, which can lead to higher derivatives. I try to see an example beyond second-order differential equations.
Let me start from $\phi^4$. The effective Lagrangian is, e.g., Peskin & Schroeder eq. (12.23)
$$
\int d^d x \mathcal{L}_{\mathrm{eff}}  = \int d^d x' \left[ \frac{1}{2} \left( \partial'_{\mu}  \phi'  \right)^2 + \frac{1}{2} m'^2 \phi'^2 + \frac{1}{4} \left( \lambda' \phi'^4 + C \left(  \partial'_{\mu} \phi' \right)^4 + D' \phi'^6 +\cdots \right)  \right] \tag{1}
$$
I suppose
$$
\mathcal{L}_{\mathrm{eff}}  =  \frac{1}{2} \left( \partial'_{\mu}  \phi'  \right)^2 + \frac{1}{2} m'^2 \phi'^2 + \frac{1}{4} \left( \lambda' \phi'^4 + C \left(  \partial'_{\mu} \phi' \right)^4 + D' \phi'^6 +\cdots \right)  \tag{2} 
$$
Try to cook a classical equation of motion. From the Euler-Lagrangian equation,
$$ 
 \frac{ \partial \mathcal{L} }{ \partial \phi}  - \partial_{\mu}  \frac{ \partial \mathcal{L} }{ \partial \left( \partial_{\mu} \phi \right) }  = 0\tag{3}
$$
plug in the effective lagrangian, we should get some extra terms than the Klein-Gordon equation
$$
\square \phi' - m^2 \phi' + C \partial'_{\mu} \left[ \left(  \partial'^{\mu} \phi' \right)  \left( \partial'_{\mu} \phi' \right)^2     \right] +\cdots = 0.\tag{4}
$$
So far the extra term with prefactor $C$ still looks like a second-order differential equation, as one first-order derivative outside the square bracket, $\partial'_{\mu}$, acting on one first-order derivative term $\left(  \partial'^{\mu} \phi' \right)  $ times the other first-order derivative term (a first-order derivative times itself) $\left( \partial'_{\mu} \phi' \right)^2$, i.e., $(fg)' = f'g + fg'$. If I further organize the inside square bracket part of the extra term by $f' g = （fg)' - f g' $,
$$
 C \partial'_{\mu} \left[ \left(  \partial'^{\mu} \phi' \right)  \left( \partial'_{\mu} \phi' \right)^2     \right]  \\
\equiv C \partial'_{\mu} \left\{ \left(  \partial'^{\mu} \phi' \right)  \left( \partial'_{\mu} \phi' \right)^2     \right\}  \\
= C \partial'_{\mu} \left\{  \partial'^{\mu} \left[   \phi' \left( \partial'_{\mu} \phi' \right)^2  \right] -  \phi'    \partial'^{\mu}\left[ \left( \partial'_{\mu} \phi' \right)^2 \right]  \right\}  \\
= C \underline{\partial'_{\mu}} \left\{  \partial'^{\mu} \left[   \phi' \left( \partial'_{\mu} \phi' \right)^2  \right] - 2 \phi'    \left[ \left( \partial'^{\mu} \phi' \right)   \left(  \underline{ \partial'^{\mu} \partial'_{\mu}} \phi' \right)  \right]  \right\}.\tag{5}  
$$
It seems I get a third-order differential equation from the underline part of the above equation. Is my reasoning right?
I think I did not impose any quantization in getting the equation of motion (except effective action from path integrals), since I think the view in the physics today essay is not much about quantization. Or I am not even wrong?
Or a second-order differential equation should be counted as the total number of the derivatives terms than taking a second-order differentiation on a single term?
 A: *

*OP is right that if the Lagrangian density remains of 1st order, then the Euler-Lagrange (EL) equations will only be of 2nd order. See also e.g. this & this related Phys.SE posts.


*However, the Wilsonian effective action
$$\begin{align}
\exp&\left\{ -\frac{1}{\hbar}W_c[J^H,\phi_L] \right\}\cr
~:=~~~&\int \! {\cal D}\frac{\phi_H}{\sqrt{\hbar}}~\exp\left\{ \frac{1}{\hbar} \left(-S[\phi_L+\phi_H]+J^H_k \phi_H^k\right)\right\} \end{align}$$
is defined by integrating out heavy/high modes $\phi^k_H$ and leaving the light/low modes $\phi^k_L$. Here $J^H_k$ denotes sources for the heavy modes. The (possibly non-local!) Wilsonian effective action $W_c[J^H,\phi_L]$ is the generating functional of connected $\phi_H$ Feynman diagrams in a background $J^H,\phi_L$.


*Nevertheless, the heavy propagators are exponentially suppressed, so the non-locality is mild, and can be taking into account by a Taylor expansion, cf. e.g. my Phys.SE answer here.


*The upshot is that, in the Wilsonian renormalization group flow, the Wilsonian Lagrangian density will in principle contain all possible terms that are not excluded by symmetry, e.g.
$$ \ldots 
+ \ldots 
+\frac{E}{2} (\partial_{\mu}\partial_{\nu}\phi)(\partial^{\mu}\partial^{\nu}\phi) 
+ \frac{F}{2} (\partial_{\mu}\phi)(\partial^{\mu}\partial^{\nu}\phi)(\partial_{\nu}\phi) 
+ \ldots ,$$
i.e., the Lagrangian density becomes of higher order.


*For higher-order Lagrangian theories,
the EL equations (3) become
$$ 0~\approx~\frac{\delta S}{\delta \phi}
~=~\frac{\partial {\cal L}}{\partial \phi} 
-\sum_{\mu} \frac{d}{dx^{\mu}} \frac{\partial {\cal L}}{\partial (\partial_{\mu}\phi)} + \sum_{\mu\leq \nu} \frac{d}{dx^{\mu}} \frac{d}{dx^{\nu}} \frac{\partial {\cal L}}{\partial (\partial_{\mu}\partial_{\nu}\phi)} - \ldots.  $$
Here the $\approx$ symbol means equality modulo eoms, and the ellipsis $\ldots$ denotes possible higher-derivative terms.


*In general, if the Lagrangian density is of $n$'th order, then the EL equations will be of $2n$'th order.
