Scattering amplitudes in Quadratic Graivty in a general formalism

In \cite{salvio}, inspired by the works \cite{pauli}-\cite{donogue}, a prescription for calculating the correlation functions in Quadratic Gravity \cite{stelle1}-\cite{stelle2} was presented and further exploited in \cite{salvio2}-\cite{salve}. By construction, it is likely that this procedure does not enter in conflict with unitarity. The corresponding Hamiltonian quantization is based on a covariant and contra-variant distinction in the non positive definite metric in the space of states \cite{gross}. The Gauss-Ostrogradsky method for higher order theories defines two momentum densities $P_1$ and $P_2$ and two coordinate densities $Q_1$ and $Q_2$, one pair is standard, the other ghost like if the creation annihilation algebra is standard. Otherwise it is the opposite. The approach in \cite{salvio} involves the continuation $P_2\to i P_2$ and $Q_2\to i Q_2$ of the ghost variables acting on kets $|>$ after taking mean values. In the present work, following \cite{yomismo}, the LSZ rules are derived, with a formalism adapted to full quartic or higher order theories. Te main technical point is to determine the creation annihilation mode of the modes of the graviton, adapted to the present prescription. This is considered here, in a more general setting than \cite{yomismo}. Two possible quantizations are discussed, which depends on whether the prescription is applied at the begining or at the ned of the LSZ calculation.