Random Matrix Theories in Quantum Physics: Common Concepts

TL;DR

This paper reviews recent developments in random-matrix theory (RMT), highlighting its theoretical foundations and diverse applications across quantum physics, suggesting RMT's role in a new universal statistical mechanics framework.

Contribution

It provides a comprehensive overview of RMT's evolution, applications, and emphasizes the universality and common concepts across various quantum physics fields.

Findings

RMT has broad applications in quantum physics.

Universal laws emerge from stochasticity and symmetry in RMT.

RMT signals a new form of statistical mechanics.

Abstract

We review the development of random-matrix theory (RMT) during the last decade. We emphasize both the theoretical aspects, and the application of the theory to a number of fields. These comprise chaotic and disordered systems, the localization problem, many-body quantum systems, the Calogero-Sutherland model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions. The review is preceded by a brief historical survey of the developments of RMT and of localization theory since their inception. We emphasize the concepts common to the above-mentioned fields as well as the great diversity of RMT. In view of the universality of RMT, we suggest that the current development signals the emergence of a new "statistical mechanics": Stochasticity and general symmetry requirements lead to universal laws not based on dynamical principles.