This chapter of the Handbook of Quantum Gravity aims to illustrate how
nonlocality can be implemented in field theories, as well as the manner it
solves fundamental difficulties of gravitational theories. We review Stelle’s
quadratic gravity, which achieves multiplicative renormalizability successfully
to remove quantum divergences by modifying the Einstein’s action but at the
price of breaking the unitarity of the theory and introducing Ostrogradski’s
ghosts. Utilizing nonlocal operators, one is able not only to make the theory
renormalizable, but also to get rid of these ghost modes that arise from higher
derivatives. We start this analysis by reviewing the classical scalar field
theory and highlighting how to deal with this new kind of nonlocal operators.
Subsequently, we generalize these results to classical nonlocal gravity and,
via the equations of motion, we derive significant results about the stable
vacuum solutions of the theory. Furthermore, we discuss the way nonlocality
could potentially solve the singularity problem of Einstein’s gravity. In the
final part, we examine how nonlocality induced by exponential and
asymptotically polynomial form factors preserves unitarity and improves the
renormalizability of the theory.

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