Random Matrix Theories in Quantum Physics: Common Concepts

Random Matrix Theories in Quantum Physics: Common Concepts

(July 10, 1997) | Thomas Guhr, Axel Müller-Groeling, and Hans A. Weidenmüller
This paper reviews the development of Random Matrix Theory (RMT) over the past decade, emphasizing both its theoretical aspects and applications to various fields. The authors highlight the common concepts underlying RMT and its diverse applications, including chaotic and disordered systems, localization problems, many-body quantum systems, the Calogero-Sutherland model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions. The review begins with a historical survey of RMT and localization theory, noting their independent development until the mid-1980s. Key developments include the introduction of supersymmetry techniques and the "Bohigas conjecture," which linked RMT to the spectral fluctuation properties of classically chaotic quantum systems. The paper then delves into the theoretical aspects of RMT, discussing classical results, spectral observables, scattering theory, and wave function statistics. It also explores the application of RMT to many-body systems, quantum chaos, disordered systems, and field theories. The authors argue that the rapid development of RMT signals the emergence of a new "statistical mechanics" based on stochasticity and general symmetry principles, leading to universal laws that are not derived from dynamical principles. The review concludes with a discussion of the universality of RMT and its implications for future research.This paper reviews the development of Random Matrix Theory (RMT) over the past decade, emphasizing both its theoretical aspects and applications to various fields. The authors highlight the common concepts underlying RMT and its diverse applications, including chaotic and disordered systems, localization problems, many-body quantum systems, the Calogero-Sutherland model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions. The review begins with a historical survey of RMT and localization theory, noting their independent development until the mid-1980s. Key developments include the introduction of supersymmetry techniques and the "Bohigas conjecture," which linked RMT to the spectral fluctuation properties of classically chaotic quantum systems. The paper then delves into the theoretical aspects of RMT, discussing classical results, spectral observables, scattering theory, and wave function statistics. It also explores the application of RMT to many-body systems, quantum chaos, disordered systems, and field theories. The authors argue that the rapid development of RMT signals the emergence of a new "statistical mechanics" based on stochasticity and general symmetry principles, leading to universal laws that are not derived from dynamical principles. The review concludes with a discussion of the universality of RMT and its implications for future research.
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