FERMIONIC MEIXNER CLASSES, LIE ALGEBRAS AND QUADRATIC HAMILTONIANS

摘要

We introduce the quadratic Fermi algebra, which is a Lie algebra, and calculate the vacuum distributions of the associated Hamiltonians. 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 multidimensional extensions of the above results.