  We study a class of perturbative scalar and gravitational quantum field
theories where dynamics is characterized by Lorentz-invariant or
Lorentz-breaking non-local operators of fractional order and the underlying
spacetime has a varying spectral dimension. These theories are either ghost
free or power-counting renormalizable but they cannot be both at the same time.
However, some of them are one-loop unitary and finite, and possibly unitary and
finite at all orders. One of the theories is unitary and infrared-finite and
can serve as a ghost-free model with large-scale modifications of general
relativity.
