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arXiv:math-ph/0007035

On q-deformed quantum stochastic calculus

Piotr Śniady

Published Jul 26, 2000• 10 citations
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Abstract

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.

DOI
10.48550/arxiv.math-ph/0007035

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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.
We first study a class of fundamental quantum stochastic processes induced by the generators of a six dimensional non-solvable Lie †-algebra consisting of all linear combinations of the generalized Gross Laplacian and its adjoint, annihilation operator, creation operator, conservation, and time, and then we study the quantum stochastic integrals associated with the class of fundamental quantum stochastic processes, and the quantum Itô formula is revisited. The existence and uniqueness of solution of a quantum stochastic differential equation is proved. The unitarity conditions of solutions of quantum stochastic differential equations associated with the fundamental processes are examined. The quantum stochastic calculus extends the Hudson-Parthasarathy quantum stochastic calculus.
Annihilation and creation operators in quantum mechanics are constructed in the framework of Nelson's stochastic mechanics. Employing Ito's stochastic calculus, annihilation and creation operators were found to be written as (1/\ensuremath{\surd}2\ensuremath{\omega})[d/dt,\ensuremath{\partial}/\ensuremath{\partial}B\ifmmode \dot{}\else \.{}\fi{}(t)] and (1/\ensuremath{\surd}2\ensuremath{\omega})[d/dt,\ensuremath{\partial}/\ensuremath{\partial}B${\ifmmode \dot{}\else \.{}\fi{}}^{\mathrm{*}}$(t)], respectively, where B\ifmmode \dot{}\else \.{}\fi{}(t) is a Gaussian white noise and B${\ifmmode \dot{}\else \.{}\fi{}}^{\mathrm{*}}$(t) is a time-reversed Gaussian white noise. The operators were shown to have proper commutation relations and to provide a Hamiltonian which has an eigenvalue equation desired in the {N} representation as should exist in conventional quantum mechanics. Therefore, the present formulation provides a correspondence between stochastic mechanics and conventional quantum mechanics.
The basic integrator processes of quantum stochastic calculus, namely, creation, conservation, and annihilation, are introduced in the Hilbert space of square integrable Brownian functionals. Stochastic integrals with respect to these processes and a quantum Itô’s formula are described. As an application two examples of quantum stochastic differential equations are discussed. A continuous time version of Stinespring’s theorem on completely positive maps in C *-algebras is exploited to formulate the notion of a quantum Markov process and indicate how classical Markov processes are woven into the fabric of the Schodinger-Heisenberg dynamics of quantum theory.
Quantum stochastic calculus is extended in a new formulation in which its stochastic integrals achieve their natural and maximal domains. Operator adaptedness, conditional expectations and stochastic integrals are all defined simply in terms of the orthogonal projections of the time filtration of Fock space, together with sections of the adapted gradient operator. Free from exponential vector domains, our stochastic integrals may be satisfactorily composed yielding quantum Itô formulas for operator products as sums of stochastic integrals. The calculus has seen two reformulations since its discovery---one closely related to classical Itô calculus; the other to noncausal stochastic analysis and Malliavin calculus. Our theory extends both of these approaches and may be viewed as a synthesis of the two. The main application given here is existence and uniqueness for the Attal--Meyer equations for implicit definition of quantum stochastic integrals.