Explicit construction of a unitary double product integral (Q2905550)

This work is a contribution to the theory of double product integrals in quantum stochastic calculus. These integrals take one of four types, depending on whether each of the two products is taken forwards or backwards. Forwards-backwards integrals have been used to obtain explicit solutions of the quantum Yang-Baxter equation and produce quantisations of quasitriangular Lie bialgebras [\textit{R. L. Hudson} and \textit{S. Pulmannová}, Lett. Math. Phys. 72, No. 3, 211--224 (2005; Zbl 1079.53139)].NEWLINENEWLINEIn the paper under review, the object of interest is the rectangular forward-forward integral NEWLINE\[NEWLINE {}_a^b \prod^{\to\to} {}_s^t \bigl( 1 + \lambda ( \mathrm{d} A^\dagger \otimes \mathrm{d} A - \mathrm{d} A \otimes \mathrm{d} A^\dagger ) \bigr), \tag{\(\star\)} NEWLINE\]NEWLINE where \(0 \leqslant a \leqslant b < \infty\), \(0 \leqslant s \leqslant t < \infty\), \(\lambda\) is a fixed real number and \(\mathrm{d} A\) and \(\mathrm{d} A^\dagger\) are the annihilation and creation integrators of quantum stochastic calculus. It is shown that the integral (\(\star\)) is the second quantisation of a unitary operator, which is obtained as the limit of a discrete double product of rotation matrices.NEWLINENEWLINEFor motivation, it is noted that (\(\star\)) is closely associated with certain causal double product integrals which appear in a quantum version of Lévy area; a paper by the first author on this subject is in preparation.NEWLINENEWLINEFor the entire collection see [Zbl 1248.46002].