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 \(\mathbb{B}({\mathcal H}_ 0)\).
Reviewer: A.Łuczak (Łódź)
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) |