On a quantum causal stochastic double product integral related to L\'evy area

TL;DR

This paper explicitly evaluates a quantum causal double product integral related to Lévy area, showing it converges to a unitary operator with a kernel expressed via Bessel functions, using lattice path combinatorics.

Contribution

It provides the first explicit evaluation of a continuum limit of a quantum stochastic double product integral, connecting it to Bessel functions and combinatorial path enumeration.

Findings

The continuum limit operator W is unitary.

Kernel of W expressed in terms of Bessel functions.

Enumeration of lattice paths related to Dyck paths and Catalan numbers.

Abstract

We study the family of causal double product integrals \begin{equation*} \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) \end{equation*} 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]. 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.