A Spectral Criterion for Emergent Gravity

Abstract

This paper proposes that gravity emerges as a statistical phase transition in the spectral function of the energy-momentum tensor commutator. The Weinberg-Witten theorem forbids a fundamental massless spin-2 particle, yet the energy-momentum tensor of quantum field theory carries an unsuppressed spin-2 channel in its vacuum fluctuation spectrum. Any massless spin-2 excitation in this channel can only be a composite collective mode. Because the spectral function is defined from the commutator, static vacuum contributions are absent by construction. The cosmological constant problem is sidestepped at the structural level. Coarse-graining in momentum space constructs, at each scale, a macroscopic rank-2 tensor field. The irreversibility of this operation promotes the scale itself 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. Entropy production and energy transfer define an effective temperature, and the fluctuation-dissipation theorem closes the dynamics self-consistently. The criterion advanced in this paper is whether the spin-2 spectral density can develop an isolated zero-momentum pole, a nonperturbative effect absent in perturbation theory but required by unitarity and locality. Should such a pole emerge, Weinberg's low-energy theorem then fixes the effective action to Einstein-Hilbert form, with Newton's constant and the cosmological constant determined respectively by the pole residue and the infrared fixed point. The criterion is in principle testable. Lattice gauge theory or functional renormalization group methods can decide the question.