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Explicit construction of a unitary double product integral

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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.

Ograniczanie wyników

1 Banach Center Publications

1 Hudson R.

1 Jones P.

1 2011

Wyniki wyszukiwania

Explicit construction of a unitary double product integral