Quantum chaos and chiral symmetry at the QCD and QED phase transition

We investigate the eigenvalue spectrum of the staggered Dirac matrix in SU( 3) gauge theory and in full QCD as well as in quenched U(1) theory. As a me asure of the fluctuation properties of the eigenvalues, we consider the nea rest-neighbor spacing distribution. We find that in all regions of their ph ase diagrams, compact lattice gauge theories have bulk spectral correlation s given by random matrix theory, which is an indication for quantum chaos. In the confinement phase, the low-lying Dirac spectrum of these quantum fie ld theories is well described by random matrix theory, exhibiting universal behavior. Related results for gauge theories with minimal coupling are now discussed also in the chirally symmetric phase.