Explicit construction of a unitary double product integral
Volume 96 / 2011
Abstract
In analogy with earlier work on the forward-backward case, we consider an explicit construction of the forward-forward double stochastic product integral $\def\rra{{\rightarrow\rightarrow}}\prod\limits^\rra (1+dr)$ with generator $dr=\lambda (dA^{\dagger }\otimes dA-dA\otimes dA^{\dagger })$. The method of construction is to approximate the product integral by a discrete double product $\def\rra{{\rightarrow\rightarrow}}\prod\limits^\rra_{(j,k)\in \Bbb{N}_m\times \Bbb{N}_n}\Gamma (R_{m,n}^{(j,k)})=\Gamma \bigl( \prod\limits^\rra_{(j,k)\in \Bbb{N}_m\times \Bbb{N}_n}(R_{m,n}^{(j,k)})\bigr)$ of second quantised rotations $R_{m,n}^{(j,k)}$ in different planes using the embedding of $\Bbb{ C}^m\oplus \Bbb{C}^n$ into $L^2(\Bbb{R})\oplus L^2(\Bbb{R})$ in which the standard orthonormal bases of $\Bbb{C}^m$ and $\Bbb{C}^n$ are mapped to the orthonormal sets consisting of normalised indicator functions of equipartitions of finite subintervals of $\Bbb{R}$. The limits as $m,n\longrightarrow \infty $ 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.
