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Banach Center Publications

Quantum stochastic calculus on full Fock space

Michael Skeide

Published Jan 1, 1998• 3 citations
Publisher

Abstract

We present a new version of integration of time-adapted processes with respect to creation, annihilation and conservation processes on the full Fock space. Among the new features, in the first place, there is a new formulation of adaptedness which is both
DOI
10.4064/-43-1-369-379

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We develop a stochastic integration theory with respect to creation, annihilation and gauge operators on the full Fock space. This is done by using a kernel representation for a large class of bounded operators on the full Fock space. It is shown that the kernels form a Banach algebra. Having established the definition of processes and stochastic integrals we go on to prove an Ito formula and use this for examining stochastic evolutions and constructing dilations of special completely positive semigroups. Explicit solutions of the corresponding stochastic differential equations are given.
In Section 21 we have already seen how the classical stochastic processes with independent increments can be realised as suitable linear combinations of the creation, conservation and annihilation operators in the boson Fock space Γ s (ℋ) over a Hilbert space ℋ. This includes, in particular, the Brownian motion and Poisson process. Since a well-developed theory of stochastic integration with respect to these classical processes exists, it is natural to wonder whether it would be possible to introduce quantum probabilistic analogues of stochastic integrals with respect to processes defined by the creation, conservation and annihilation operators. To this end one needs the notions of filtration and adapted processes. This, in turn, depends on the notion of time. We adopt the view that time is an ℝ+-valued observable in a Hilbert space ℋ which, as a spectral measure, has no jumps. The absence of jumps is assumed more as a matter of mathematical convenience than a philosophical or conceptual necessity. We shall now examine how such a notion of time leads to a filtration and the definition of adapted processes. 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.
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.
This chapter begins with introductions of symmetric Fock space Γ(ℍ) and symmetric Guichardet space Φ(ℍ) of a generic complex Hilbert space ℍ. It is shown that these 2 spaces are actually isomorphic and can and will be used interchangeably throughout the chapter. In particular, the simple and yet useful integral-summation formula (see Theorem (3.2.1)) for the Guichardet space is often employed to establish the essential results in quantum stochastic calculus. It has been shown that the symmetric Fock space (and hence the symmetric Guichadet space) provides a plausible mathematical tool for modeling phenomena in quantum optics or quantum electrodynamics. Many results in the quantum physics and quantum probability literature are actually established based on this concrete model space. When the generic complex Hilbert space ℍ = L 2 (ℝ + ; K) (where K is another complex Hilbert space), the class of exponential vectors along with 3 different types of quantum noise processes, namely, the creation, annihilation, and neutral (or guage) processes, can be introduced. It is shown that the subspace generated by the class of exponential vectors is dense in the Fock space. Therefore, it is convenient to verify the properties that hold for Fock space by verifying the same hold for the class of exponential vectors. Parallel to those of Itô integrals with respect to classical Brownian motion and/or Poisson process, the concepts of a quantum stochastic integral of a operator-valued process (as a member of the Fock space) with respect to each of the above mentioned quantum noise processes are constructed. The quantum stochastic calculus, parallel to those of classical Itô calculus, is developed. The theory of quantum stochastic calculus on the symmetric Fock space due originally to Hudson and Parthasarathy [HP84a] has been the topic of investigation by many researchers in the last 20 years. Much of he material presented in this chapter can be found in the books by Meyer [Mey95], Parthasarathy [Par92] and Attal [Att03].