Formulating and indirectly proving the quantum fluctuations of space-time and a justification for applying fractional quantum gravity

Abstract: It is well known that in quantum gravity, the very geometry of space and time is subject to continual fluctuation. The mathematical formulation for this old theory is still lacking. This article formulates this more than forty-year-old theory of quantum gravity, the result of which is that the equations of quantum gravity become fractional. The existence of such fluctuations is proved indirectly. On the other hand, recent attention has been paid to fractional gravity. Although this type of gravity leads to brilliant results, we have no root and logical reason for applying it other than that it works. We actually simultaneously demonstrate that fractional gravity is generated by stochastic quantum fluctuations of space-time. For clarification, Einstein-Hilbert theory along with a scalar field is investigated in deformed and non-deformed minisuperspaces. The results illustrate a transition from decelerated to accelerated expansion (late-time-accelerated expansion).