Renormalized Squares of White Noise¶and Other Non-Gaussian Noises as Lévy Processes¶on Real Lie Algebras

Abstract:

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 Lévy processes on the real Lie algebra ??₂. This allows to obtain its Itô table, which turns out to be infinite-dimensional. The linear white noise without or with number operator is shown to be a Lévy 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 Lévy processes are shown to arise as components of Lévy 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 χ² probability density and the analogue of the field operators to have a density proportional to , where Γ is the usual Γ-function and m ₀ a real parameter.