Running of Newton's constant and noninteger powers of the d'Alembertian

The running of Newton's constant can be taken into account by considering covariant, nonlocal generalizations of the field equations of general relativity. These generalizations involve nonanalytic functions of the d'Alembertian, as (-square lg){sup -{alpha}}, with {alpha} a noninteger number, and ln[-square lg]. In this paper we define these nonlocal operators in terms of the usual two point function of a massive field. We analyze some of their properties, and present specific calculations in flat and Robertson Walker spacetimes.