The stochastic calculus constructed in [2] for fermion Brownian motion is augmented through the inclusion of stochastic integration with respect to the gauge process. The solutions of certain non-commutative stochastic differential equations are used to construct dilations of contraction semigroups on a Hilbert space \mathfrak h_0 and of uniformly continuous, completely positive semigroups on \boldsymbol B(\mathfrak h_0) . Finally we construct a fermion analogue of the classical Poisson process and investigate some of its properties.
The stochastic calculus constructed in [2] for fermion Brownian motion is augmented through the inclusion of stochastic integration with respect to the gauge process. The solutions of certain non-commutative stochastic differential equations are used to construct dilations of contraction semigroups on a Hilbert space \mathfrak h_0 and of uniformly continuous, completely positive semigroups on \boldsymbol B(\mathfrak h_0) . Finally we construct a fermion analogue of the classical Poisson process and investigate some of its properties.
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.
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].
Given a compensated Poisson process (Xt)t≥0 based on (Ω,F,P), the Wiener-Poisson isomorphism W:F+(L2(R+))→L2(Ω,F,P) is constructed. We restrict the isomorphism to F+(L2[0,1]) and prove some novel properties of the Poisson exponentials E(f):=W(e(f)). A new proof of the result Λt+At+At†=W−1XtW is also given. The analogous results for F+(L2(R+)) are briefly mentioned. The concept of a compensated Poisson process over R+ is generalised to any measure space (M,M,μ) as an isometry I:L2(M,M,μ)→L2(Ω,F,P) satisfying certain properties. For such a generalised Poisson process we recall the construction of the generalised Wiener-Poisson isomorphism, WI:F+(L2(M))→L2(Ω,F,P), using Charlier polynomials. Two alternative constructions of WI are also provided, the first using exponential vectors and then deducing the connection with Charlier polynomials, and the second using the theory of reproducing kernel Hilbert spaces. Given any measure space (M,M,μ), we construct a canonical generalised Poisson process I:L2(M,M,μ)→L2(Δ,B,P), where Δ is the maximal ideal space, with B the completion of its Borel σ-field with respect to P, of a C∗-algebra A⊆B(F+(L2(M))). The Gelfand transform A→B(L2(Δ)) is unitarily implemented by the Wiener-Poisson isomorphism WI:F+(L2(M))→L2(Δ). This construction only uses operator algebra theory and makes no a priori use of Poisson measures. A new Fock space proof of the quantum Ito formula for (Λt+At+At†)0≤t≤1 is given. If (Ft)0≤t≤1 is a real, bounded, predictable process with respect to a compensated Poisson process (Xt)0≤t≤1, we show that if Mt=∫0tFsdXs, then on Elb:=linsp{e(f):f∈Llb2[0,1]}, W−1MtW=∫0tW−1FsW(dΛs+dAs+dAs†), and that (W−1MtW)0≤t≤1 is an essentially self-adjoint quantum semimartingale. We prove, using the classical Ito formula, that if (Jt)0≤t≤1 is a regular self-adjoint quantum semimartingale, then (WMtW−1+Jt)0≤t≤1 is an essentially self-adjoint quantum semimartingale satisfying the quantum Duhamel formula, and hence the quantum Ito formula. The equivalent result for the sum of a Brownian and Poisson martingale, provided that the sum is essentially self-adjoint with core Elb, is also proved.
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.