Path Integral Quantization in General Relativity In Ref. 1 I have seen that the action must contain only the first derivative of the metric as required by the path integral approach. I don't understand why. I mean why the path integral approach of quantization requires the action to contain a first derivative of the metric only?
References:


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*Gibbons & Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D 15 (1977) 2752; Section II.

 A: Requiring only first derivatives in an action is required because higher time derivatives generically lead to ghost instabilities in the field theory.  This goes under the name "Ostragradsky instability," see the excellent reference by Richard Woodard discussing the kinds of issues that arise when this sort of instability is present.  
The main problem with the Ostragradsky instability is that the ghost degrees of freedom that are introduced due to the higher time derivatives carry negative kinetic energy, and hence allow the system to decay into highly excited states consisting large amounts of positive and negative energy that cancel against each other.  These infinitely many zero energy states tend to cause the quantum theory to be ill-defined.  In the framework of canonical quantization, once you ensure that the Hilbert space has a positive definite inner product, you find these zero energy states lead to the immediate decay of the vacuum, so that it is probably not possible to define unitary dynamics with your Hamiltonian (sometimes people say that ghost degrees of freedom lead to negative norm states.  This is just a different side of the same problem: if you demand that the states have positive energy, you find that they have negative norm, whereas if you demand they have positive norm (which is essential to quantum mechanics), you find they have negative energy).  
This sickness of the theory cannot be remedied by using the path integral quantization, if you want to define a unitary theory.  However, it has been suggested by Hawking and Hertog that you can allow Ostragradsky ghosts if you define the path integral via Wick rotation from Euclidean space, see here.  The resulting theory, however, is nonunitary.  If you try to demand unitarity in the presence of the ghosts, the path-integral will likely not even be well-defined, due to the tendency of the system to instantly decay (and hence the idea of slicing the evolution into small step to derive the path integral will not converge in any sense).  
A: In quantum field theory, or in field theory, it is assumed that the Lagrangian should only contain the first derivative term, because we require that the field is local. 
