Itô’s stochastic calculus and Heisenberg commutation relations

Johan Quaegebeur

Luigi Accardi, Andreas Boukas

Piotr Śniady

In this paper we investigate a quantum stochastic calculus build of creation, annihilation and number of particles operators which fulfill some deformed commutation relations. Namely, we introduce a deformation of a number of particles operator which has simple commutation relations with well known q-deformed creation and annihilation operators. Since all operators considered in this theory are bounded we do not deal with some difficulties of a non deformed theory of Hudson and Parthasarathy. We define stochastic integrals and estimate them in the operator norm. We prove Ito's formula as well.

Kalyanapuram Rangachari Parthasarathy

R. L. Hudson, J. Martin Lindsay, Philippe Biane +1 more

Extensions of Quantum Stochastic Calculus (S Attal) Quantum Ito Algebras: Axioms, Representations, Decompositions (V Belavkin) Free Probability for Probabilists (P Biane) Conditional Expectations on von Neumann Algebras (C Cecchini) Classical Probability Theory: An Outline of Stochastic Integrals and Diffusions (M Emery) Quantum Stochastic Differential Equations (F Fagnola) Canonical Commutation and Anticommutation Relations (M Fannes) Quantum and Classical Stochastic Calculus (A Holevo) An Introduction to Quantum Stochastic Calculus and Some of Its Applications (R Hudson) Stationary Processes in Quantum Probability (B Kummerer)

Omar Alzeley, Aymen Ettaieb, Hafedh M. S. Rguigui

ABSTRACT We establish the commutation relations of the quantum Lévy operators, consisting of the generalized Lévy Gross Laplacian and its adjoint, annihilation operator, creation operator, and the Lévy number operator. In particular, such quantum Lévy operators span a nonsolvable ‐Lie algebra. We then give a generalization of the Hudson–Parthasarathy stochastic integrals in terms of the quantum Lévy operators and derive a noncommutative Itô formula.