A question related to a previous question I've asked.
I am wondering why in QFT the arguments of the Lagrangian only go up to the first derivative? I remember hearing someone mention that it has to do with Lorentz invariance, but I can't be sure and I don't understand why it has to be the case for Lorentz invariance to hold?
Probably the reason is that in what you are reading the theories considered only require a first derivative. However, there certainly are higher derivative theories and the Lagrangian formalism is extremely general. Mathematical physicists work out results on a Jet Space which allows for derivatives of arbitrary order. Essentially, the idea is not far removed from what we do in ODEs to reduce a $n$-th order problem to a first order system. Add dimensions and work in the extended space, then at the end of it all project back down to obtain results.
In any event, I haven't seen anyone write clearly that higher order derivatives are incompatible with Lagrangian formalism. Quantization is another matter, but your post really just concerns classical field theory so I'll leave it at that.