An overall sign in the Lagrangian does not affect the classical equations of motion. However, at the quantum level, if we want to preserve causality by keeping the optical theorem to be valid, then the ghost can be interpreted as a particle which propagates with negative energy, as already stated above. In other words, in special relativity, the ghost would have a four-momentum with . However it would still be a timelike particle as , whether is negative or not. The problem arises when this particle is coupled to some other normal particle, because in this case the process with can be allowed. This means in general that for such a theory one would expect the pair creation of ghost and normal particles out of the vacuum. Notice that the energy is still conserved, but the energy is pumped out of the ghost particle.
Since all the particles are coupled at least to gravity, one would think that out of the vacuum particles could be created via the decay of a couple of gravitons emitted in the vacuum into ghosts and non-ghosts particles. This process does lead to an infinite contribution unless one introduces a cutoff for the theory [145, 161], for which one can set observational constraints.
We have already seen that, for metric f (R) gravity, the kinetic operator in the FLRW background reduces to given in Eq. (7.60) with the perturbed action (7.80). Since the sign of is determined by , one needs to impose in order to avoid the propagation of a ghost mode.
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