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arXiv:math-ph/0512071

On Quantum Ito Algebras and their decompositions

V. P. Belavkin

Published Dec 21, 2005• 2 citations
arXivHTMLPDF

Abstract

A simple axiomatic characterization of the noncommutative Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. It is proved that every quotient Ito algebra has a faithful representation in a Minkowski space and is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise of a finite number of degrees of freedom is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Levy-Khinchin theorem in the classical case. Two basic examples of non-commutative Ito finite group algebras are considered.

DOI
10.48550/arxiv.math-ph/0512071

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A simple axiomatic characterization of the noncommutative Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. It is proved that every quotient Ito algebra has a faithful representation in a Minkowski space and is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise of a finite number of degrees of freedom is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Levy-Khinchin theorem in the classical case. Two basic examples of non-commutative Ito finite group algebras are considered.
A simple axiomatic characterization of the general (infinite dimensional, noncommutative) Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. The notion of Ito B*-algebra, generalizing the C*-algebra is defined to include the Banach infinite dimensional Ito algebras of quantum Brownian and quantum Levy motion, and the B*-algebras of vacuum and thermal quantum noise are characterized. It is proved that every Ito algebra is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Levy-Khinchin Theorem in the classical case .
A simple axiomatic characterization of the general (infinite dimensional, noncommutative) Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. The notion of Ito B*-algebra, generalizing the C*-algebra is defined to include the Banach infinite dimensional Ito algebras of quantum Brownian and quantum Levy motion, and the B*-algebras of vacuum and thermal quantum noise are characterized. It is proved that every Ito algebra is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Levy-Khinchin theorem in the classical case.
Quantum chaotic states over a noncommutative monoid, a unitalization of a noncommutative Ito algebra parametrizing a quantum stochastic Levy process, are described in terms of their infinitely divisible generating functionals over the monoid-valued processes on an atomless `space-time' set. A canonical decomposition of the logarithmic conditionally posive-definite generating functional is constructed in a pseudo-Euclidean space, given by a quadruple defining the monoid triangular operator representation and a cyclic zero pseudo-norm state in this space. It is shown that the exponential representation in the corresponding pseudo-Fock space yields the infinitely-divisible generating functional with respect to the exponential state vector, and its compression to the Fock space defines the cyclic infinitly-divisible representation associated with the Fock vacuum state. The structure of states on an arbitrary Ito algebra is studied with two canonical examples of quantum Wiener and Poisson states. A generalized quantum stochastic nonadapted multiple integral is explicitly defined in Fock scale, its continuity and quantum stochastic differentiability is proved. A unified non-adapted and functional quantum Ito formula is discovered and established both in weak and strong sense, and the multiplication formula on the exponential Ito algebra is found for the relatively bounded kernel-operators in Fock scale. The unitarity and projectivity properties of nonadapted quantum stochastic linear differential equations are studied, and their solution is constructed for the locally bounded nonadapted generators in terms of the chronological products in the underlying kernel algebra canonically represented by triangular operators in the pseudo-Fock space.
We review the basic concepts of quantum stochastics using the universal Itô *-algebra approach.The main notions and results of classical and quantum stochastics are reformulated in this unifying approach.The thermal quantum Lévy process with commuting increments is defined in terms of the modular *-Itô algebra.The quasi-Markov quantum stochastic processes over increasing W*-algebras generated by a quantum Lévy process is characterized in terms of their quantum stochastic germs.The corresponding quantum stochastic master equation on the increasing predual L-spaces is derived as a noncommutative and non-Markov generalization of the Zakai equation driven by the dual quantum Lévy process.This is done by a noncommutative analog of the Girsanov transformation which we introduce here in full generality.
We review the basic concepts of quantum probability and revise the classical and quantum stochastic (QS) calculus using the universal Itô B*-algebra approach. A non-commutative generalization of Lévy process is defined in Fock–Guichardet space in terms of the underlying B*-Itô modular algebra. The main notions and results of classical and QS analysis are reformulated in terms of the non-commutative but associative stochastic covariation of quantum integrals defined as sesquilinear forms adapted to a quantum Lévy process in Fock space. The general semi-Markov QS dynamics is defined in terms of a hemigroup of semi-morphic transformations as adjoints to a semiunitary hemigroup resolving a QS Schrödinger equation in Fock space. The corresponding semi-Markov hemigroup of completely positive contractions resolving a quantum master equation with generalized Lindblad generator as a non-commutative analogue of Feller–Kolmogorov equation is derived by taking vacuum conditional expectation in Fock space.
The quantum decomposition of a classical random variable is one of the deep results of quantum probability: it shows that any classical random variable or stochastic process has a built-in non-commutative structure which is intrinsic and canonical, and not artificially put by hands. Up to now the technique to deduce the quantum decomposition has been based on the theory of interacting Fock spaces and on Jacobi's tri-diagonal relation for orthogonal polynomials. Therefore it requires the existence of moments of any order and cannot be applied to random variables without this property. The problem to find an analogue of the quantum decomposition for random variables without finite moments of any order remained open for about fifteen years and nobody had any idea of how such a decomposition could look like. In the present paper we prove that any infinitely divisible random variable has a quantum decomposition canonically associated to its Lévy–Khintchin triple. The analytical formulation of this result is based on Kolmogorov representation of these triples in terms of 1–cocycles (helices) in Hilbert spaces and on the Araki–Woods–Parthasarathy–Schmidt characterization of these representation in terms of Fock spaces. It distinguishes three classes of random variables: (i) with finite second moment; (ii) with finite first moment only; (iii) without any moment. The third class involves a new type of renormalization based on the associated Lévy–Khinchin function.