Supersymmetric Generalization of Dyson’s Brownian Motion (Diffusion)

Abstract

The spectral fluctuation properties of an overwhelmingly rich variety of quantum systems can be modeled with a simple, phenomenological approach, the Theory of Random Matrices ^l,2. The key assumption is that the matrix elements of the Hamilton operator H in Schrödinger’s equation are just random numbers. This idea is due to Wigner. It indeed leads to a very satisfactory description of the fluctuation properties in a wide class of many-body systems, ranging from nuclei to molecules, in disordered systems and also in systems with few degrees of freedom which are classically chaotic. More recently, it has been shown that this statistical concept can also be extended to classical wave phenomena, such as elastomechanics, and to quantum systems described by the Dirac equation, such as Quantum Chromodynamics. The list is still incomplete. A detailed review was recently given in Ref. 3.