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

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.

15 pages, 1 Figure; v2 adds proof on a general manifold, and modifies the proof in Sec 3 to separate out microcanonical and canonical ensemble, 18 pages; v3 is restructured for clarity, 19 pages

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