Chiral symmetry and the spectrum of the QCD Dirac operator

The order parameter of the chiral phase transition is directly related to the infrared part of the spectrum of the QCD Dirac operator. This part of the spectrum follows from the low energy limit of QCD which is given by the a partition function of weakly interacting Goldstone modes. We find that the slope of the Dirac spectrum is determined by the pion decay constant whereas for λ ≪ 1/L²Λ_QCD the correlations of the Dirac eigenvalues are given by a random matrix theory with the global symmetries of the QCD partition function. The observation of these results in lattice QCD is discussed.