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.