$\mathbf{Q}$-adapted quantum stochastic integrals and differentials in Fock scale
Volume 96 / 2011
Abstract
In this paper we first introduce the Fock–Guichardet formalism for the quantum stochastic (QS) integration, then the four fundamental processes of the dynamics are introduced in the canonical basis as the operator-valued measures, on a space-time $\sigma$-field $\mathfrak{F}_\mathbb{X}$, of the QS integration. Then rigorous analysis of the QS integrals is carried out, and continuity of the QS derivative $\mathbf{D}$ is proved. Finally, $\mathrm{Q}$-adapted dynamics is discussed, including Bosonic ($\mathrm{Q}=\mathrm{I}$), Fermionic ($\mathrm{Q}=-\mathrm{I}$), and monotone ($\mathrm{Q}=\mathrm{O}$) quantum dynamics. These may be of particular interest to quantum field theory, quantum open systems, and quantum theory of stochastic processes.
