Casimir chaos in a boson Fock space
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 \(\int_{0<t_ 1< t_ 2<\dots <t_ n} d\Lambda^{i_{\pi (1)}}_{i_{\sigma (1)}}\dots d\Lambda^{i_{\pi (m)}}_{i_{\sigma (m)}}\). Indeed, the operator \({\mathcal G}^ n_ m (t)\) is a linear combination of the operators \(Z^ n_ j (t)\) for \(1\leq j\leq m\), with coefficients which do not depend on \(t\). This striking fact reflects the deep relationship between Itô's formula for conservation processes on the one hand and representation theory for \(U(n)\) on the other. An explicit spectral decomposition for the processes \({\mathcal G}^ n_ m(t)\) and \(Z^ n_ j(t)\) is given next, and some hints are given towards the description of the classical stochastic process underlying them in a coherent state.
scientific article; zbMATH DE number 599815