Why no EOM in QFT with higher than second order derivatives in time and space?

[Phiphy]

When we write down a Lagragian for a quantum field theory, it is said that it should not depend on the second and higher order time and space derivatives of $$\phi$$, because we want the equation of motion(EOM) to be at most second order. Why is it so important. What trouble will a higher order EOM cuase in physics? Could anyone give me some examples? Thanks.

 

[Halcyon-on]

The tale says that the problem is non-locality, that is you'd generate a Taylor series expansion.

 

[Phiphy]

Non-locality happens when there are negative powers of momentum in the lagrangian. A finite series of positive powers of momentum is still local. There must be some other physical reasons to rule out higher powers of momentum.

 

[Mosis]

why do negative powers of momentum yield non-locality and in what sense?

 

[Parlyne]

Effective field theories sometimes have higher derivative powers. But, a field theory with any higher derivative terms will not be renormalizable and, so, would be expected not to be a fundamental theory.