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+#+title: On a causal quantum double product integral related to Lévy
+#+title: stochastic area.
+
+#+date: <2015-07-01>
+
+In [[https://arxiv.org/abs/1506.04294][this paper]] with
+[[http://homepages.lboro.ac.uk/~marh3/][Robin]] we study the family of
+causal double product integrals \[ \prod_{a < x < y < b}\left(1 +
+i{\lambda \over 2}(dP_x dQ_y - dQ_x dP_y) + i {\mu \over 2}(dP_x dP_y +
+dQ_x dQ_y)\right) \]
+
+where $P$ and $Q$ are the mutually noncommuting momentum and position
+Brownian motions of quantum stochastic calculus. The evaluation is
+motivated heuristically by approximating the continuous double product
+by a discrete product in which infinitesimals are replaced by finite
+increments. The latter is in turn approximated by the second
+quantisation of a discrete double product of rotation-like operators in
+different planes due to a result in
+[[http://www.actaphys.uj.edu.pl/findarticle?series=Reg&vol=46&page=1851][(Hudson-Pei2015)]].
+The main problem solved in this paper is the explicit evaluation of the
+continuum limit $W$ of the latter, and showing that $W$ is a unitary
+operator. The kernel of $W$ is written in terms of Bessel functions, and
+the evaluation is achieved by working on a lattice path model and
+enumerating linear extensions of related partial orderings, where the
+enumeration turns out to be heavily related to Dyck paths and
+generalisations of Catalan numbers.