Wigner first predicted that a system of electrons in a uniform potential would crystallize at low densities. Localizing electrons around lattice sites increases their kinetic energy, but at sufficiently low densities the reduction in interaction energy is always greater. The Wigner crystal remains a theoretical prediction in three dimensions ~3D!, but in 2D Wigner crystals have been created on ...

An intriguing aspect of quantum mechanics is the intricate description of the state of a physical system. Therefore, a great deal of interest has been attracted by the recent experimental demonstration that the quantum state of a simple system, namely a single light mode, can be completely characterized in a feasible scheme [1]. The measurement was based on an observation that marginals of the ...

Recently there has been renewed interest in applications of the so-called phase-space formulation of quantum mechanics’) for both equilibrium2) and non-equilibrium phenomena3). Its basic feature is to provide a framework for the treatment of quantum mechanical problems in terms of classical phasespace language. Following the appearance of the well-known Wigner distribution function’), many othe...

A gauge-invariant Wigner quasi-distribution function for charged particles in classical electromagnetic fields is derived in a rigorous way. Its relation to the axial gauge is discussed, as well as the relation between the kinetic and canonical momenta in the Wigner representation. Gauge-invariant quantum analogs of Hamilton-Jacobi and Boltzmann kinetic equations are formulated for arbitrary cl...

Statistical Mechanics and Random Matrices 3 1. Introduction 6 2. Motivations 7 3. The different scales; typical results 12 Lecture 1. Wigner matrices and moments estimates 15 1. Wigner's theorem 16 2. Words in several independent Wigner matrices 23 3. Estimates on the largest eigenvalue of Wigner matrices 25 Lecture 2. Gaussian Wigner matrices and Fredholm determinants 27 1. Joint law of the ei...

Wigner functions, and other quasi-probability distributions, have been an important tool in quantum optics for a long time. A quasi-probability distribution is a representation of a quantum state with certain properties. In particular it is a real-valued representation of the state. Of particular interest are states for which a Wigner function is non-negative. Spekkens showed such sets of state...

1. I n t r o d u c t i o n . In this paper we shall discuss operator splitting methods to discretize the linear Wigner equation and the coupled Wigner-Poisson system. The Wigner formalism, which represents a phase-space description of quantum mechanics, has in recent years a t t racted considerable at tent ion of solid s tate physicists for including quantum effects in the simulation of ul tra-...

Department of Neutron Spectroscopy, Wigner Research Centre for Physics, POB 51, Budapest, H-1121, Hungary, Department of Materials Science by Nuclear Methods, Wigner Research Centre for Physics, POB 51, Budapest, H-1121, Hungary, and Institute of Materials and Environmental Chemistry, Research Centre for Natural Sciences, Magyar tudósok körútja 2, Budapest, H-1117, Hungary. *Correspondence e-ma...

It is shown that Bell’s proof of violation of local realism in phase space is incorrect. Using Bell’s approach, a violation can be derived also for nonnegative Wigner distributions. The error is found to lie in the use of an unnormalizable Wigner function.

We consider N ×N Hermitian random matrices with independent identically distributed entries (Wigner matrices). We assume that the distribution of the entries have a Gaussian component with variance N for some positive β > 0. We prove that the local eigenvalue statistics follows the universal Dyson sine kernel. AMS Subject Classification: 15A52, 82B44 Running title: Universality for Wigner matrices