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# Annales de l’Institut Henri Poincaré D

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**Volume 5, Issue 4, 2018, pp. 467–512**

**DOI: 10.4171/AIHPD/60**

Published online: 2018-07-25

On a causal quantum stochastic double product integral related to Lévy area

Robin L. Hudson^{[1]}and Yuchen Pei

^{[2]}(1) University of Loughborough, UK

(2) KTH - Royal Institute of Technology, Stockholm, Sweden

We study the family of causal double product integrals \begin{equation*} \prod_{a < x < y < b}\Big(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)\Big), \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 [15]. 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 $W - I$ 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.

*Keywords: *causal double product, Lévy's stochastic area, position and momentum Brownian motions, linear extensions, Catalan numbers, Dyck paths

Hudson Robin, Pei Yuchen: On a causal quantum stochastic double product integral related to Lévy area. *Ann. Inst. Henri Poincaré Comb. Phys. Interact.* 5 (2018), 467-512. doi: 10.4171/AIHPD/60