Chiral symmetry and spectrum of Euclidean Dirac operator in QCD

Abstract

Some exact relations for the spectral density $ρ(λ)$ of the Euclidean Dirac operator in $QCD$ are derived. They follow directly from the chiral symmetry of the $QCD$ lagrangian with massless quarks. New results are obtained both in thermodynamic limit when the Euclidean volume $V$ is sent to infinity and also in the theory defined in finite volume where the spectrum is discrete and a nontrivial information on $ρ(λ)$ in the region $λ\sim 1/(|<\bar{q}q>_0|V)$ (the characteristic level spacing) can be obtained. These exact results should be confronted with &#34;experimental&#34; numerical simulations on the lattices and in some particular models for $QCD$ vacuum structure and may serve as a nontrivial test of the validity of these simulations.