Note: I am not asking about why time is in 2nd derivative: that makes perfect intuitive sense given the relativistic need to treat space and time in equal footing.
We often hear how the Klein-Gordon is the simplest example of equation of motion in (classical/quantum) field theory. So my question is: why is the simplest case of equation of motion in field theory expressed in second derivatives, and not in first derivatives?
My first guess is that it might still be related to Lorentz invariance. Is it so, and why? Do you have physical/intuitive/geometric examples of why a first derivative Klein-Gordon equation does not work?
