First, it's not true that the Lagrangian density is restricted to have first-order space and time derivatives. The example of the scalar field,
\begin{align}
\mathcal{L}=\tfrac{1}{2}\partial_\mu\phi\partial^\mu\phi - \tfrac{1}{2}m^2\phi^2 + \mathcal{L}_{int}(\phi)
\end{align}
is a clear counterexample.

Now, following Weinberg, *QFT I*, section 10.7 on the Kallen-Lehmann representation, we can show (I won't reproduce the full derivation here) that inclusion of higher-order derivatives in the $\mathcal{L}-\mathcal{L}_{int}$ (the free Lagrangian) is inconsistent with the *positivity postulate of quantum mechanics*. The mathematical statement is that the exact two-point function -- the $\phi$-propagator -- must behave as
\begin{align}
\Delta'(p) \underset{p^2\to\infty}\longrightarrow \frac{1}{p^2}.
\end{align}

From the perspective of effective field theory, however, it is possible to have derivative couplings of any finite order in a local theory (infinite orders are inherently non-local), as long as we introduce a dimensional (usually mass) scale of the appropriate dimension at each order. (See Motl's answer to Why are differential equations for fields in physics of order two? for a full explanation.)

The answers given earlier are all incorrect for one or more reasons. See the comments.

This post imported from StackExchange Physics at 2014-07-13 04:40 (UCT), posted by SE-user MarkWayne