Fermionic Meixner probability distributions, Lie algebras and quadratic Hamiltonians

TL;DR

This paper introduces fermionic Meixner distributions via quadratic Fermi algebra and Lie algebra structures, providing a quantum probabilistic perspective and classifying 1D Meixner laws with potential multi-dimensional extensions.

Contribution

It presents a novel quantum probabilistic derivation of fermionic Meixner distributions using quadratic Fermi algebra and classifies 1D Meixner laws in terms of quadratic Bose operators.

Findings

Fermionic Meixner distributions derived from quadratic Fermi algebra.

Classification of 1D Meixner laws using quadratic Bose operators.

Discussion of potential multi-dimensional extensions.

Abstract

We introduce the quadratic Fermi algebra, which is a Lie algebra, and show that the vacuum distributions of the associated Hamiltonians define the fermionic Meixner probability distributions. In order to emphasize the difference with the Bose case, we apply a modification of the method used in the above calculation to obtain a simple and straightforward classification of the 1--dimensional Meixner laws in terms of homogeneous quadratic expressions in the Bose creation and annihilation operators. There is a huge literature of the Meixner laws but this, purely quantum probabilistic, derivation seems to be new. Finally we briefly discuss the possible multi-dimensional extensions of the above results.