Renormalized squares of white noise and other non-gaussian noises as Levy processes on real Lie algebras

It is shown how the relations of the renormalized squared white noise defined by Accardi, Lu, and Volovich [ALV99] can be realized as factorizable current representations or Levy processes on the real Lie algebra sl(2). This allows to obtain its Ito table, which turns out to be infinite-dimensional. The linear white noise without or with number operator is shown to be a Levy process on the Heisenberg-Weyl Lie algebra or the oscillator Lie algebra. Furthermore, a joint realization of the linear and quadratic white noise relations is constructed, but it is proved that no such realizations exist with a vacuum that is an eigenvector of the central element and the annihilator. Classical Levy processes are shown to arise as components of Levy processes on real Lie algebras and their distributions are characterized. In particular the square of white noise analogue of the quantum Poisson process is shown to have a chi2 probability density and the analogue of the field operators to have a density proportional to \Gamma(m0+ix/2)\(2), where Gamma is the usual Gamma-function and m(0) a real parameter.