Universal fine grained asymptotics of free and weakly coupled Quantum Field Theory

TL;DR

This paper rigorously proves a universal asymptotic formula for the density of states in free and weakly coupled quantum field theories with finite group symmetries, extending previous results to higher dimensions.

Contribution

It establishes a universal high-energy density of states formula for free and weakly coupled QFTs with finite group symmetries in higher dimensions, generalizing earlier 1+1D results.

Findings

Universal density of states formula proven for free QFTs.

Extension of results to weakly coupled theories with proper coupling scaling.

Comparison of behaviors between finite and continuous symmetry groups.

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 $\rho_\alpha(E)$ for each irreducible representation $\alpha$ 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.