The Centrally Extended Heisenberg Algebra andIts Connection with the Schrodinger Galilei and Renormalized Higher Powers ofQuantum White Noise Lie Algebras

摘要

In previous papers we have shown that the one mode Heisenberg algebra Heis(1) admitsa unique non-trivial central extensions CeHeis(1) which can be realized as a sub-Lie-algebra of theSchrOdinger algebra, in fact the Galilei Lie algebra. This gives a natural family of unitary represen-tations ofCeHeis(1) and allows an explicit determination of the associated group by exponentiation.In contrast with Heis(1), the group law for CeHeis(1) is given by nonlinear (quadratic) functionsof the coordinates. The vacuum characteristic and moment generating functions of the classicalrandom variables canonically associated to CeHeis(1) are computed. The second quantization ofCeHeis(1) is also considered.