  We review the formulation of quantum field theories with purely virtual
particles, a new type of degrees of freedom that can mediate interactions
without ever appear as external on-shell states. This property allows to solve
the problem of ghosts in higher-derivative quantum gravity, leading to a
renormalizable and unitary theory. The main steps for the BRST quantization of
gravity are recalled and renormalizability is discussed. Then, we introduce
purely virtual particles in a general quantum field theory and show the
derivation of the so-called spectral identities, which are a key ingredient to
prove unitarity. Finally, phenomenological consequences and predictions in
inflationary cosmology are presented.
