History-Dependent Gravity as an Emergent Renormalization Group Object: Spectral Flow of the Effective Average Action (Paper LXIV)

Abstract

History-Dependent Gravity (HDG) introduces nonlocal memory kernels to capture the macroscopic imprint of quantum gravitational dynamics. However, the spectral density underlying these kernels is typically introduced phenomenologically. In this work, we demonstrate that this spectral density is not a free input, but emerges dynamically as the infrared limit of the functional renormalization group (FRG) flow. By applying a spectral projection to the Wetterich equation, we derive an exact flow equation for the scale-dependent spectral density ρ_k(μ). Within a controlled scalar toy truncation with a gravity-inspired regulator, we show that the interplay between mass-flow (tadpole) and continuum-generation (sunset) dynamically shapes the spectral measure. In the infrared limit k → 0, the resulting spectral density naturally defines the HDG memory kernel. This establishes HDG as a rigorous, nonlocal effective manifestation of RG dynamics, bridging the gap between macroscopic memory models, nonlocal gravity, and Asymptotic Safety.