Random Matrix Theory and Quantum Chaos

Research project P 14435 Random Matrix and Quantum Chaos Harald MARKUM 08.05.2000 Quantum chromodynamics (QCD) is considered to be the correct theory which describes quarks and gluons and, thus, all strong interaction phenomena from the fundamental forces of Nature. However, important properties of QCD as the physical mechanism of color confinement are still not completely understood and under extensive discussion. The role of chaotic field dynamics for the confinement of quarks is a longstanding question. Similarly, the spontaneous breaking of chiral symmetry is a non-perturbative phenomenon with decisive consequences for the hadron spectrum. Analytical calculations are limited, because in the low energy regime where quarks are confined, application of perturbation theory is restricted due to the large gluon coupling. A powerful tool to investigate numerically and analytically the non-perturbative region is provided by the lattice formulation of QCD. Monte Carlo simulations of lattice QCD yield information about quarks and hadrons and their interactions starting from first principles. Mainly from lattice simulations we know that chiral symmetry is restored above a critical temperature of about 150 MeV. As the chiral condensate is connected to the spectral density of the Dirac operator via the Banks-Casher relation, the QCD Dirac spectrum is an interesting object for detailed studies. In search for an analytical expression of the infrared limit of the Dirac spectrum it has been realized that chiral random matrix theory is a suitable tool to compare with the distribution and the correlations of the small Dirac eigenvalues. Further, it has been shown that the correlations of eigenvalues on the scale of mean level spacings are universal for complex physical systems and are given by random matrix theory. This has been formulated as the Bohigas- Giannoni-Schmit conjecture which states that spectral correlations of a classically chaotic system are given by random matrix theory on the quantum level. A very fascinating question is the connection between chaotic behavior and confinement of quarks in QCD. The aim of the project is to clarify the relationship between temperature and chiral phase transitions and chaos to order transitions in quantum field theories. We intend to numerically analyze the eigenvalues of the Dirac operator for compact QED, QCD with chemical potential and for supersymmetric Yang-Mills theory on the lattice. These theories show chiral symmetry breaking and confinement in the low energy region. Although being related, they differ considerably from the point of view of the topological structure, the action density and global symmetry. It is a challenge to compare their eigenvalue spectra to analytical results from random matrix theory and verify possible universalities concerning the chiral structure and the appearance of quantum chaos.