×

Unitary quantum stochastic evolutions. (English) Zbl 0751.46040

Let \({\mathcal H}_ 0\) be a Hilbert space, \({\mathcal H}\) the Bose-Fock space built over \(L^ 2(-\infty,\infty)\), and let \(\Lambda(t)\), \(A(t)\), \(A^ +(t)\), \(t\geq 0\) be the gauge, annihilation and creation processes, respectively, on \({\mathcal H}\). Consider the quantum stochastic evolution equation \[ U(t)=I+\int^ t_ 0 U(s)(L_ 1 d\Lambda(s)+L_ 2 dA(s)+L_ 3 dA^ +(s)+L_ 4 ds),\tag{*} \] where \(L_ 1,\dots,L_ 4\) are linear operators on \({\mathcal H}_ 0\). The main aim of the paper is to give a necessary and sufficient condition for a solution of \((*)\) to be a unitary process in the case when \(L_ 1,\dots,L_ 4\), \(L_ 1^ +,\dots,L_ 4^ +\) are unbounded linear operators defined on a certain commonly invariant dense domain in \({\mathcal H}_ 0\). Moreover, let \(\psi(0)\) be the vacuum vector in \({\mathcal H}\) and define \(E_ 0: \mathbb{B}({\mathcal H}_ 0)\to\mathbb{B}({\mathcal H}_ 0)\) by \[ \langle u,E_ 0(A)v\rangle=\langle u\otimes \psi(0),Av\otimes \psi(0)\rangle, \qquad A\in\mathbb{B}({\mathcal H}_ 0), \quad u,v\in{\mathcal H}_ 0. \] Put \[ T_ t=E_ 0(U(t)), \quad {\mathcal T}_ t(A)=E_ 0(U(t)AU(t)^ +), \qquad A\in\mathbb{B}({\mathcal H}_ 0), \] where \(U(t)\) is the unitary solution of \((*)\). Then \((T_ t:\;t\geq 0)\) is a strongly continuous contraction semigroup in \({\mathcal H}_ 0\) with generator \(\overline {L}_ 4\), and \(({\mathcal T}_ t:\;t\geq 0)\) is a \(C_ 0^*\)-semigroup of completely positive normal contractions on \(\mathbb{B}({\mathcal H}_ 0)\).

MSC:

46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
46L60 Applications of selfadjoint operator algebras to physics
47D06 One-parameter semigroups and linear evolution equations
46L55 Noncommutative dynamical systems
47A20 Dilations, extensions, compressions of linear operators
81P20 Stochastic mechanics (including stochastic electrodynamics)
PDFBibTeX XMLCite
Full Text: DOI