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# Copyright (C) 2013-2021 Yuchen Pei.
# Permission is granted to copy, distribute and/or modify this
# document under the terms of the GNU Free Documentation License,
# Version 1.3 or any later version published by the Free Software
# Foundation; with no Invariant Sections, no Front-Cover Texts, and
# no Back-Cover Texts. You should have received a copy of the GNU
# Free Documentation License. If not, see <https://www.gnu.org/licenses/>.
# This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA.
#+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.
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