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THE STRUCTURE OF THE QUANTUM SEMIMARTINGALE ALGEBRAS

Stéphane Attal

Published Jan 1, 2007• 7 citations
OpenAlex

Abstract

Abstract. In the theory of quantum stochastic calculus one disposes of two quantum semimartingale algebras S and S ′. The first one is an algebra for the composition of operators and has a quantum functional calculus for analytical functions. The second one is larger and is an algebra for the operations of quantum square and angle brackets. In this article we study the algebraic and analytic properties of these algebras. This study is mainly performed through a remarkable transform of quantum processes which, surprisingly, establishes a bijection in between these two algebras. This bijection allows to define norms on these algebras that equip them with Banach algebra structures.

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Abstract. In the theory of quantum stochastic calculus one disposes of two quantum semimartingale algebras S and S ′. The first one is an algebra for the composition of operators and has a quantum functional calculus for analytical functions. The second one is larger and is an algebra for the operations of quantum square and angle brackets. In this article we study the algebraic and analytic properties of these algebras. This study is mainly performed through a remarkable transform of quantum processes which, surprisingly, establishes a bijection in between these two algebras. This bijection allows to define norms on these algebras that equip them with Banach algebra structures.
We give details of a *‐linear bijection between adapted (in the sense of Hudson and Parthasarathy) and vacuum‐adapted quantum stochastic integrals. This provides new insight into Attal's remarkable transformation of quantum semimartingales, by showing that it factorizes in a natural manner. The Banach *‐algebras of regular quantum and Ω‐semimartingales are consequently isomorphic, and an intrinsic characterisation of Ω‐semimartingales is obtained as an application of this fact. Various formulae occurring in quantum stochastic calculus are shown to have a more natural appearance in the vacuum‐adapted framework. We finish by providing the full generalization of this theory to higher dimensions.
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.
The abstract commutation relations of the algebra of the square of white noise of Accardi, Lu, and Volovich are shown to be realized by operator processes acting on the Fock space of Accardi and Skeide which is very closely related to the Finite Difference Fock space of Boukas and Feinsilver. The processes are shown to satisfy the necessary conditions for inclusion in the framework of the representation free quantum stochastic calculus of Accardi, Fagnola, and Quaegebeur. The connection between the Finite-Difference operators and the creation, annihilation, and conservation operators on usual symmetric Boson Fock space is further studied.
The logical structures studied in this paper are generalizations of the propositional calculus. The classical propositional calculus is essentially Boolean algebra or, alternatively, the theory of functions on an arbitrary set S with values in a two-element set. The generalization consists in allowing partial functions on the set S , i.e., functions defined on certain subsets of S , and defining an equivalence relation among these functions such that any two constant functions with the same constant value belong to the same equivalence class. The generalization is equally natural for functions with values in the field of real numbers and we shall consider this case first. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.