2 - Gaussian Ensembles. The Joint Probability Density Function for the Matrix Elements

Abstract

This chapter discusses the consequences of time-reversal invariance and also presents Gaussian ensembles as a mathematical idealization. They are implied if the hypothesis of maximum statistical independence is allowed under the symmetry constraints. Mathematically a much simpler ensemble is the Gaussian unitary ensemble E2G, which applies to systems without invariance under time reversal. Such systems are easily created in principle by putting an ordinary atom or nucleus, for example, into an externally generated magnetic field. The external field is not affected by the time-reversal operation. However, for the unitary ensemble to be applicable, the splitting of levels by the magnetic field must be at least as large as the average level spacing in the absence of the magnetic field. The magnetic field must, in fact, be so strong that it will completely mix up the level structure that would exist in zero field. This state of affairs could never occur in nuclear physics. In atomic or molecular physics, a practical application of the unitary ensemble may perhaps be possible. A system without time-reversal invariance has a Hamiltonian that may be an arbitrary Hermitian matrix not restricted to be real or self-dual.