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Quantum Probability and Infinite Dimensional Analysis

AN ANALYTIC DOUBLE PRODUCT INTEGRAL

R. L. Hudson

Published Jul 1, 2007• 4 citations
Publisher

Abstract

Abstract. The double product integral → ← ∏ (1+λ(dA † ⊗dA−dA⊗dA †)) is constructed as a family of unitary operators in double Fock space satisfying quantum stochastic differential equations with unbounded operator coefficients. 1. Introduction. A purely algebraic theory of double product integrals such as → ← ∏ (1 + λ(dA ⊗ dA − dA ⊗ dA †)) [4][9] has been developed based on formal power series in the formal

DOI
10.1142/9789812770271_0023

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Abstract. The double product integral → ← ∏ (1+λ(dA † ⊗dA−dA⊗dA †)) is constructed as a family of unitary operators in double Fock space satisfying quantum stochastic differential equations with unbounded operator coefficients. 1. Introduction. A purely algebraic theory of double product integrals such as → ← ∏ (1 + λ(dA ⊗ dA − dA ⊗ dA †)) [4][9] has been developed based on formal power series in the formal
A theory is developed of product integrals of the form ∏a<s<b ∏c<t<d(1+g[h] (ds,dt)). Here [a,b[ and [c,d[ are disjoint finite subintervals of R+, and g[h] is a formal power series in the indeterminate h whose constant term is zero and whose coefficients are elements of L⊗L, where ℒ is the space of basic differentials of a multidimensional quantum stochastic calculus. The product integrals are themselves formal power series in h whose coefficients are finite sums of iterated stochastic integrals against the elements of ℒ. They are symmetrized in such a way that ∏a<s<b ∏c<t<d(1+g[h](ds,dt)) is the image, obtained by applying the representation J[a,b[⊗J[c,d[ to the coefficients, where J[a,b[ is the representation canonically associated with the interval [a,b[, of a formal power series ∏ ∏(1+dg[h]) whose coefficients lie in U⊗U, where 𝒰 is the universal enveloping algebra of the Lie algebra ℒ. It is shown that the naturally conjectured multiplication rule, analogous to the multiplication rule for simple product integrals, holds in the commutative case.
Motivated by the search for solutions of the quantum Yang–Baxter equation, an algebraic theory of quantum stochastic product integrals is developed. The product integrators are formal power series in an indeterminate h whose coefficients are elements of the Lie algebra ℒ labelling the usual integrators of a many-dimensional quantum stochastic calculus. The product integrals are also formal power series in h, whose coefficients are finite iterated additive stochastic integrals which act on the exponential domain in the Fock space of the calculus and which represent elements of the universal enveloping algebra 𝒰 of ℒ. They obey a multiplication rule suggested by the quantum Itô product formula, and are characterized among all such formal power series by a grouplike property.
In this thesis we develop the theory of quantum Wiener integrals on the bosonic Fock space. We study multiple quantum Wiener integrals as an algebra of unbounded operators, investigating its properties, including closedness, common domains and multiplication formulas. We show the applications of the new formalism by providing new proofs to the established theory of quantum stochastic calculus and new conditions for generating quantum stochastic cocycles and quantum stochastic evolutions. The corresponding quasifree case is also studied and the constructions extended to fit in that formalism. We construct the multiple quantum Wiener integral as one operator on a family of operators which we dub operator kernels. This in particular covers the case of quantum stochastic cocycles and evolutions. We show that the family of quantum Wiener integrals forms a WOT-dense algebra of unbounded operators on the bosonic Fock space. We provide more general conditions for an operator kernel to be multiple quantum Wiener integrable, which allows us to treat multiple quantum Wiener integrals as an algebra. We explore the influence of an initial space on the theory. Our setting gives natural conditions for a product of two cocycles (evolutions) to still be a cocycle (an evolution). We apply our theory by solving quantum stochastic differential equations (QSDEs) and by finding more elementary proofs of structure conditions on the generator of a quantum stochastic evolution and of the fundamental estimate in the proof of quantum stochastic Lie–Trotter formula. We also show how our theory unifies and generalises the theory of integral kernels and chaotic representation properties, proving in particular that every Hilbert–Schmidt operator is a quantum Wiener integral.