Comment on Dirac spectral sum rules for QCD_3

Recently Magnea hep-th/9907096 , hep-th/9912207 [Phys.Rev.D61, 056005 (2000); Phys.Rev.D62, 016005 (2000)] claimed to have computed the first sum rules for Dirac operators in 3D gauge theories from 0D non-linear sigma models. I point out that these computations are incorrect, and that they contradict with the exact results for the spectral densities unambiguously derived from random matrix theory by Nagao and myself.Comment: REVTeX 3.1, 2 pages, no figure. (v2) redundant part removed, conclusion unchange

Ratios of characteristic polynomials in complex matrix models

We compute correlation functions of inverse powers and ratios of characteristic polynomials for random matrix models with complex eigenvalues. Compact expressions are given in terms of orthogonal polynomials in the complex plane as well as their Cauchy transforms, generalizing previous expressions for real eigenvalues. We restrict ourselves to ratios of characteristic polynomials over their complex conjugate

Finding the Pion in the Chiral Random Matrix Vacuum

The existence of a Goldstone boson is demonstrated in chiral random matrix theory. After determining the effective coupling and calculating the scalar and pseudoscalar propagators, a random phase approximation summation reveals the massless pion and massive sigma modes expected whenever chiral symmetry is spontaneously broken.Comment: 3 pages, 1 figure, revte

Non-Hermitian Random Matrix Theory and Lattice QCD with Chemical Potential

In quantum chromodynamics (QCD) at nonzero chemical potential, the eigenvalues of the Dirac operator are scattered in the complex plane. Can the fluctuation properties of the Dirac spectrum be described by universal predictions of non-Hermitian random matrix theory? We introduce an unfolding procedure for complex eigenvalues and apply it to data from lattice QCD at finite chemical potential $\mu$ to construct the nearest-neighbor spacing distribution of adjacent eigenvalues in the complex plane. For intermediate values of $\mu$, we find agreement with predictions of the Ginibre ensemble of random matrix theory, both in the confinement and in the deconfinement phase.Comment: 4 pages, 3 figures, to appear in Phys. Rev. Let

Quantum Chaos in Compact Lattice QED

Complete eigenvalue spectra of the staggered Dirac operator in quenched $4d$ compact QED are studied on $8^3 \times 4$ and $8^3 \times 6$ lattices. We investigate the behavior of the nearest-neighbor spacing distribution $P(s)$ as a measure of the fluctuation properties of the eigenvalues in the strong coupling and the Coulomb phase. In both phases we find agreement with the Wigner surmise of the unitary ensemble of random-matrix theory indicating quantum chaos. Combining this with previous results on QCD, we conjecture that quite generally the non-linear couplings of quantum field theories lead to a chaotic behavior of the eigenvalues of the Dirac operator.Comment: 11 pages, 4 figure

On the Phase Diagram of QCD

We analyze the phase diagram of QCD with two massless quark flavors in the space of temperature, T, and chemical potential of the baryon charge, mu, using available experimental knowledge of QCD, insights gained from various models, as well as general and model independent arguments including continuity, universality, and thermodynamic relations. A random matrix model is used to describe the chiral symmetry restoration phase transition at finite T and mu. In agreement with general arguments, this model predicts a tricritical point in the T mu plane. Certain critical properties at such a point are universal and can be relevant to heavy ion collision experiments.Comment: 21 pages, version to appear in Phys. Rev. D (2 references added