$\mathbf{Q}$-adapted quantum stochastic integrals and differentials in Fock scale

Volume 96 / 2011

Viacheslav Belavkin, Matthew Brown Banach Center Publications 96 (2011), 51-66 MSC: Primary 60H99; Secondary 60G99. DOI: 10.4064/bc96-0-3

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.

Authors

  • Viacheslav BelavkinSchool of Mathematical Sciences, University Park
    Nottingham, NG7 2RD, UK
    e-mail
  • Matthew BrownSchool of Mathematical Sciences, University Park
    Nottingham, NG7 2RD, UK
    e-mail

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