---
template: oldpost
title: On a causal quantum double product integral related to Lévy stochastic area.
date: 2015-07-01
archive: false
comments: true
---
In [this paper](https://arxiv.org/abs/1506.04294) with [Robin](http://homepages.lboro.ac.uk/~marh3/) 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 [(Hudson-Pei2015)](http://www.actaphys.uj.edu.pl/findarticle?series=Reg&vol=46&page=1851). 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.