Renormalized squares of white noise and other non-Gaussian noises as Lévy processes on real Lie algebras
type: Item
instance of: scholarly article
title: Renormalized squares of white noise and other non-Gaussian noises as Lévy processes on real Lie algebras
zbMATH Open document ID: 1006.60064
author: L. Accardi
author: Uwe Franz
author: Michael Skeide
published in: Communications in Mathematical Physics
publication date: 2002-12-01T00:00:00Z
review text: It is shown how the relations of the renormalized squared white noise defined by Accardi, Lu and Volovich can be realized as factorizable current representations of Lévy processes on the real Lie algebra \(\text{sl}_2\). 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 relation 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 \(\chi^2\) probability density and the analogue of the field operators to have a density proportional to \(|\Gamma((m_0+ ix)/2)|^2\), where \(\Gamma\) is the usual \(\Gamma\)-function and \(m_0\) a real parameter.
reviewed by: Zagorka Lozanov-Crvenković
Mathematics Subject Classification ID: 60H40
Mathematics Subject Classification ID: 81S25
Mathematics Subject Classification ID: 60G51
Mathematics Subject Classification ID: 17B99
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 1836933
zbMATH Keywords: white noise
zbMATH Keywords: Lie algebra
zbMATH Keywords: Lévy process
MaRDI profile type: Publication
full work available at URL: s002200200647
OpenAlex ID: W2080211519
DOI: 10.1007/S002200200647
description: scientific article; zbMATH DE number 1836933