Universal fine grained asymptotics of free and weakly coupled Quantum Field Theory
Weiguang Cao, Tom Melia, Sridip Pal
TL;DR
This work addresses the problem of symmetry-resolved high-energy density of states in quantum field theories with a finite global symmetry $\mathrm{G}$ on compact spaces, proving the Harlow–Ooguri conjecture for free theories via the canonical relation $Z_{\alpha}(\beta) = \frac{\dim(\alpha)^2}{|\mathrm{G}|} Z(\beta)$ in the high-temperature limit. The authors extend the result to weakly coupled QFTs by introducing a synchronized $\beta\to0$, $\lambda\to0$ regime and employing a Plethystic exponential description of the twisted partition function, with a microcanonical version obtained through density-of-states comparisons and an anomalous-dimension analysis. They provide concrete consistency checks through Wilson–Fisher-type fixed points, showing how operator scaling must be tuned (e.g., $\epsilon$-scaling with $\Delta$) to maintain an $O(1)$ bound on anomalous dimensions, thereby preserving the HO universality at high temperature. Overall, the paper generalizes prior $(1+1)$-D results to higher dimensions, clarifies the role of averaging (smearing), contrasts finite versus continuous symmetry sectors, and lays groundwork for symmetry-resolved observables and indices in high-temperature QFT.
Abstract
We give a rigorous proof that in any free quantum field theory with a finite group global symmetry $\mathrm{G}$, on a compact spatial manifold, at sufficiently high energy, the density of states $ρ_α(E)$ for each irreducible representation $α$ of $\mathrm{G}$ obeys a universal formula as conjectured by Harlow and Ooguri. We further prove that this continues to hold in a weakly coupled quantum field theory, given an appropriate scaling of the coupling with temperature. This generalizes similar results that were previously obtained in $(1+1)$-D to higher spacetime dimension. We discuss the role of averaging in the density of states, and we compare and contrast with the case of continuous group $\mathrm{G}$, where we prove a universal, albeit different, behavior.