Scalar and gravity quantum field theories with fractional operators.

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.