Explicit construction of a unitary double product integral

R. L. Hudson; Paul Jones

R. L. Hudson ; Paul Jones

Banach Center Publications, Tome 95 (2011), p. 215-236 / Harvested from The Polish Digital Mathematics Library

Résumé

In analogy with earlier work on the forward-backward case, we consider an explicit construction of the forward-forward double stochastic product integral $\prod^{\to \to} (1 + d r)$ with generator $d r = λ (d A^{†} \otimes d A - d A \otimes d A^{†})$ . The method of construction is to approximate the product integral by a discrete double product $\prod_{(j, k) \in ℕ_{m} \times ℕ ₙ}^{\to \to} Γ (R_{m, n}^{(j, k)}) = Γ (\prod_{(j, k) \in ℕ_{m} \times ℕ ₙ}^{\to \to} (R_{m, n}^{(j, k)}))$ of second quantised rotations $R_{m, n}^{(j, k)}$ in different planes using the embedding of $ℂ^{m} \oplus ℂ ⁿ$ 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.

Publié le : 2011-01-01

Zbl 1271.81091

EUDML-ID : urn:eudml:doc:282100

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     author = {R. L. Hudson and Paul Jones},
     title = {Explicit construction of a unitary double product integral},
     journal = {Banach Center Publications},
     volume = {95},
     year = {2011},
     pages = {215-236},
     zbl = {1271.81091},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc96-0-14}
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R. L. Hudson; Paul Jones. Explicit construction of a unitary double product integral. Banach Center Publications, Tome 95 (2011) pp. 215-236. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc96-0-14/