Do New Quantum Statistics Exist in Nature?

Abstract

We recall the definitions of a Wigner quantum system (WQS) and of A-, (resp B-, C- and D-) quantum (super)statistics. We outline shortly the relation of these new statistics to the classes A, (resp B, C and D) of Lie (super)algebras. We describe in some more details some of the properties of A-oscillator and A-superoscillator. Both of them fall into the category of finite quantum systems and both of them have quite unusual properties. For example let R and P be the position and the momentum operators of an 3D A-superoscillator along the x (or y or z) axes. Then contrary to the canonical uncertainty relations the product of the standard deviations of R and P of the oscillating particle reads: $$ \Delta R .\Delta P \leq {\hbar/2}. $$ i.e., the inequality above $\le$ is just the opposite to the one that appears in the canonical quantum mechanics.