Spectral Quantization of History-Dependent Higher-Derivative Gravity: Unitarity, Causality, and FRG Flow of the Spectral Density

This paper presents a self-consistent spectral formulation of history-dependent higher-derivative gravitational (HDG) theories. By reformulating the non-local HDG action as a continuous superposition of local massive field theories via a generalized Källén–Lehmann representation, we bypass the traditional Ostrogradsky instability at the kinematic level. We construct the spectral Hilbert space as a direct integral of Fock spaces, demonstrating that microcausality and positive-definite norms strictly follow from the spectral positivity condition ρ(μ)≥0ρ(μ)≥0. Furthermore, we derive a projected Functional Renormalization Group (FRG) flow equation for the spectral density within the Einstein–Hilbert truncation. In the infrared limit, the flow settles into a universal power-law fixed spectrum ρ∗(μ)∼μρ∗(μ)∼μ, which naturally generates a modified gravitational propagator and a long-tailed temporal memory kernel K(t)∼t−2K(t)∼t−2. We discuss the observable cosmological consequences of this spectral phase, including the suppression of late-time structure growth (fσ8fσ8) and modifications to the ISW effect, providing a theoretically motivated framework to address the S8S8 tension. This work is Paper XLII in the HDG series, establishing the rigorous quantum foundations and RG dynamics of temporal nonlocality.