Casimir chaos in a boson Fock space
type: Item
instance of: scholarly article
title: Casimir chaos in a boson Fock space
zbMATH Open document ID: 0804.60052
author: K. R. Parthasarathy
author: R. L. Hudson
published in: Journal of Functional Analysis
publication date: 1995-01-19T00:00:00Z
review text: Let \(H= L^ 2 (\mathbb{R}_ +) \otimes\mathbb{C}^ n\) and \(\Gamma= \Gamma(H)\) be the associated Fock boson space. For every \(t>0\) one can define a representation \(j_ t\) of the group \(U(n)\) by taking for \(j_ t(g)\), \(g\in U(n)\), the second quantization of the operator \(\chi_{[0,t[} \otimes g\) where \(\chi_{[0,t[}\) is the multiplication operator by the characteristic function of \([0,t[\) on \(L^ 2 (\mathbb{R}_ +)\) and \(g\) acts on \(\mathbb{C}^ n\) by the basic representation of \(U(n)\). The family of representations \((j_ t)_{t\geq 0}\) exponentiate the basic representation of \(U(n)\). In the corresponding infinitesimal representation, the images of the basis elements \(E^ i_ j\) of \(M_ n(\mathbb{C})\) are the conservation processes \(\Lambda^ i_ j (t)\) of quantum stochastic calculus [see e.g., \textit{K. R. Parthasarathy}, An introduction to quantum stochastic calculus (1992; Zbl 0751.60046)]. The authors explore how the center of the enveloping algebra of the Lie algebra \(u(n)\) is mapped by the representations \(j_ t\). Indeed they show that the images of the center by the representations \(j_ t\) lie in a commutative algebra. Furthermore, they introduce ``Casimir processes'' which are families of operators \({\mathcal G}^ n_ m(t)\) on \(\Gamma\), defined by the determinant-like formula \[ {\mathcal G}^ n_ m (t)= \sum_{1\leq i_ 1< \dots <i_ n} \sum_{\pi,\sigma} \varepsilon (\pi\sigma) \Lambda^{i_{\pi(1)}}_{i_{\sigma (1)}}\dots \Lambda^{i_{\pi (m)}}_{i_{\sigma (m)}} \] (where \(\pi\) and \(\sigma\) are permutations of \(1,\dots, m\)), which are images of a complete system of Casimir operators for \(U(n)\), and they show, using Itô's quantum formula, that they can be expressed in terms of ``Casimir chaos processes'' \(Z^ n_ m(t)\) defined by an analogous formula, where the product \(\Lambda^{i_{\pi (1)}}_{i_{\sigma (1)}}\dots \Lambda^{i_{\pi (m)}}_{i_{\sigma (m)}}\) is replaced by the stochastic integral \(\int_{0<t_ 1< t_ 2<\dots <t_ n} d\Lambd…
Mathematics Subject Classification ID: 60H99
Mathematics Subject Classification ID: 81S25
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 20C35
zbMATH DE Number: 599815
zbMATH Keywords: Fock boson space
zbMATH Keywords: second quantization
zbMATH Keywords: characteristic function
zbMATH Keywords: infinitesimal representation
zbMATH Keywords: quantum stochastic calculus
zbMATH Keywords: Casimir operators
zbMATH Keywords: quantum Itô's formula
zbMATH Keywords: spectral decomposition
reviewed by: Philippe Biane
MaRDI profile type: Publication
full work available at URL: jfan.1994.1013
OpenAlex ID: W2028335404
DOI: 10.1006/JFAN.1994.1013
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description: scientific article; zbMATH DE number 599815