Summary of Research Interests

The main theme in my research activities since my Ph.D. has been the study of nonperturbative effects in Quantum Mechanics and Field Theory. One important question that has been addressed is the effect of complexity and chaotic behavior in quantum systems. We showed in [2] that answer to this question is in the correlations of the eigenvalues. If the system is chaotic the spectral correlations do not depend on its dynamics and are given by Random Matrix Theory. This has been observed in complex systems as varied as atomic nuclei, resonance cavities and the zeros of the Riemann $\zeta$-function. Analytical results can be obtained, for example, with the supersymmetric formalism [1]. More recently, we have applied these ideas to strongly interacting Quantum Field Theories, in particular to QCD, the theory of the Nuclear Forces. Typically, nonperturbative effects in QCD are studied in 4 Euclidean dimensions in which there is no difference between space and time. The motion of a quark can then be interpreted as motion in 4 spatial dimensions and 1 artificial time dimension. One expects that its classical trajectories in the quantum disordered Yang-Mills background field configurations are chaotic. For example, this has been confirmed by microcanonical simulations of lattice QCD. From our experience with simple chaotic systems we thus expect that the correlations of the QCD Dirac eigenvalues are given by Random Matrix Theory. The study of this conjecture has been my main interest during the past five years. The appropriate Random Matrix Theories have been formulated [4] and classified according to the global symmetries of the QCD partition function [6]. Analytical results have been derived [5] and the universality of these results has been understood. These ideas have been verified by explicit Monte-Carlo simulations of the QCD partition function. Of course, the QCD partition function is much richer than chiral Random Matrix Theory. This implies that there exists a scale above which Random Matrix Theory is not applicable. We have identified this scale as the equivalent of the Thouless energy in mesoscopic physics. Other non-perturbative effects in Field Theory I have been interested in are the study of instanton field configurations in QCD (for a summary we refer to the page of Edward Shuryak), the study of nucleon as a topological Skyrmion in the Skyrme model, and the study of particles with fractional statistics (anyons). Specifically, I wish to mention the discovery of the axially symmetric solution with baryon number two in the Skyrme model [3].

Three most cited papers over all:

1. J.J.M. Verbaarschot, H.A. Weidenm\"uller and M.R. Zirnbauer, Grassmann integration in stochastic quantum physics: the case of compound nucleus scattering, Phys. Rep. {\bf 129} (1985) 367 (323 citations)

2. T.H. Seligman, J.J.M. Verbaarschot and M.R. Zirnbauer, Quantum spectra and the transition from regular to chaotic classical motion, Phys. Rev. Lett. {\bf 53} (1984) 215 (180 citations)

3. J.J.M. Verbaarschot, Axial symmetry of bound baryon number two solution of the skyrme model, Phys. Lett. {\bf B195} (1987) 235 (86 citations)

Three most cited papers written during the past ten years:

4. E.V. Shuryak and J.J.M. Verbaarschot, Random matrix theory and spectral sum rules for the Dirac operator in QCD, Nucl. Phys. {\bf A560} (1993) 306 (153 citations).

5. J.J.M. Verbaarschot and I. Zahed, On the spectrum of the Dirac operator in QCD near zero virtuality, Phys. Rev. Lett. {\bf 70} (1993) 3852 (133 citations).

6. J.J.M. Verbaarschot, The spectrum of the QCD Dirac operator and chiral random matrix theory: the threefold way, Phys. Rev. Lett. {\bf 72} (1994) 2531 (140 citations).