  The "fakeon" is a fake degree of freedom, i.e. a degree of freedom that does
not belong to the physical spectrum, but propagates inside the Feynman
diagrams. Fakeons can be used to make higher-derivative theories unitary.
Moreover, they help us clarify how the Lee-Wick models work. In this paper we
study the fakeon models, that is to say the theories that contain fake and
physical degrees of freedom. We formulate them by (nonanalytically) Wick
rotating their Euclidean versions. We investigate the properties of arbitrary
Feynman diagrams and, among other things, prove that the fakeon models are
perturbatively unitary to all orders. If standard power counting constraints
are fulfilled, the models are also renormalizable. The S matrix is regionwise
analytic. The amplitudes can be continued from the Euclidean region to the
other regions by means of an unambiguous, but nonanalytic, operation, called
average continuation. We compute the average continuation of typical amplitudes
in four, three and two dimensions and show that its predictions agree with
those of the nonanalytic Wick rotation. By reconciling renormalizability and
unitarity in higher-derivative theories, the fakeon models are good candidates
to explain quantum gravity.
