  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.
