Stochastic Quantization of General Relativity \`a la Ricci-Flow

Abstract

We follow a new pathway to the definition of the Stochastic Quantization (SQ), first proposed by Parisi and Wu, of the action functional yielding the Einstein equations. Hinging on the functional similarities between the Ricci-Flow equation and the SQ Langevin equations proposed by Rumpf, we push forward a novel approach characterized by a multiplicative noise and a stochastic time that converges to the proper time of a space-like foliation in the equilibrium limit, where quantities have constant averages. We express the starting system of equations using the Arnowitt-Deser-Misner (ADM) variables and their conjugated Hamiltonian momenta. Such a choice is instrumental in understanding the newly derived equations in terms of the breakdown of the diffeomorphism invariance of the classical theory, which instead will hold on average at the steady state. We comment on the physical interpretation of the Ricci flow equations, and argue how they can naturally provide, in a geometrical way, the renormalization group equation for gravity theories. In the general setting, the equation associated to the shift vector yields the Navier-Stokes equation with a stochastic source. Moreover, we show that the fluctuations of the metric tensor components around the equilibrium configurations, far away from the horizon of a Schwarzschild black hole, are forced by the Ricci flow to follow the Kardar-Parisi-Zhang equation, whose probabilistic distribution can yield an intermittent statistics. We finally comment on the possible applications of this novel scenario to the cosmological constant, arguing that the Ricci flow may provide a solution to the Hubble tension, as a macroscopic effect of scale dependence of the quantum fluctuations of the metric tensor.