Macroscopic statistical field from the spectral function of vacuum fluctuations and its gravitational implications

This paper investigates the possibility that gravity emerges as a macroscopic statistical effect of energy--momentum tensor vacuum fluctuations in flat-spacetime quantum field theory. The spectral function of the energy--momentum tensor commutator serves as the fundamental object, circumventing the vacuum energy catastrophe that plagues induced-gravity approaches. Through momentum-space coarse-graining and the spectral representation, a macroscopic symmetric rank-2 tensor field is constructed at each coarse-graining scale. The coarse-graining scale is promoted from an external control parameter to a dynamical order parameter. Its Langevin dynamics is governed by two renormalization-group fixed points: an ultraviolet repellor at the Planck scale and an infrared attractor at the Hubble scale. The irreversibility of coarse-graining yields an effective temperature inversely proportional to the scale, and the fluctuation--dissipation theorem closes the micro--macro dynamics into a self-consistent loop. The condition for the macroscopic field to acquire a massless spin-2 propagator is the emergence of an isolated pole at zero momentum in the total spin-2 spectral density. This pole is absent in perturbation theory, but we argue that it can arise as a solution compatible with the energy--momentum tensor Ward identity under critical slowing-down of the coarse-graining flow. When the pole is present, Weinberg's low-energy theorem fixes the nonlinear self-coupling uniquely to the Einstein--Hilbert action. Newton's constant and the cosmological constant appear as emergent quantities, and the low-energy limit of the framework is consistent with standard general relativity. Whether the spectral pole actually exists is the central question on which the framework depends; it is in principle testable by lattice gauge theory or nonperturbative functional renormalization-group calculations.