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Explicit construction of a unitary double product integral

100%

Hudson R. , Jones P.

Banach Center Publications

2011

tom 96

nr 1

215-236

In analogy with earlier work on the forward-backward case, we consider an explicit construction of the forward-forward double stochastic product integral $∏^{→→}(1 + dr)$ with generator $dr = λ(dA^† ⊗ dA - dA ⊗ dA^†)$. The method of construction is to approximate the product integral by a discrete double product $∏^{→→}_{(j,k)∈ℕ_m×ℕₙ} Γ(R_{m,n}^{(j,k)}) = Γ(∏^{→→}_{(j,k)∈ℕ_m×ℕₙ} (R_{m,n}^{(j,k)}))$ of second quantised rotations $R_{m,n}^{(j,k)}$ in different planes using the embedding of $ℂ^m ⊕ ℂⁿ$ into L²(ℝ) ⊕ L²(ℝ) in which the standard orthonormal bases of $ℂ^m$ and ℂⁿ are mapped to the orthonormal sets consisting of normalised indicator functions of equipartitions of finite subintervals of ℝ. The limits as m,n ⟶ ∞ of such double products of rotations are constructed heuristically by a new method, and are shown rigorously to be unitary operators. Finally it is shown that the second quantisations of these unitary operators do indeed satisfy the quantum stochastic differential equations defining the double product integral.

Deformation coproducts and differential maps

100%

Hudson R. , Pulmannová S.

Studia Mathematica

2008

tom 188

nr 1

1-16

Let 𝒯 be the Itô Hopf algebra over an associative algebra 𝓛 into which the universal enveloping algebra 𝓤 of the commutator Lie algebra 𝓛 is embedded as the subalgebra of symmetric tensors. We show that there is a one-to-one correspondence between deformations Δ[h] of the coproduct in 𝒯 and pairs (d⃗[h],d⃖[h]) of right and left differential maps which are deformations of the differential maps for 𝒯 [Hudson and Pulmannová, J. Math. Phys. 45 (2004)]. Corresponding to the multiplicativity and coassociativity of Δ[h], d⃗[h] and d⃖[h] satisfy the Leibniz-Itô formula and a mutual commutativity condition. Δ[h] is recovered from d⃗[h] and d⃖[h] by a generalised Taylor expansion. As an illustrative example we consider the differential maps corresponding to the quantisation of quasitriangular commutator Lie bialgebras of [Hudson and Pulmannová, Lett. Math. Phys. 72 (2005)].