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

### Robin L. Hudson

University of Loughborough, UK### Yuchen Pei

KTH - Royal Institute of Technology, Stockholm, Sweden

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## Abstract

We study the family of causal double product integrals

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.

## Cite this article

Robin L. Hudson, Yuchen Pei, On a causal quantum stochastic double product integral related to Lévy area. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 5 (2018), no. 4, pp. 467–512

DOI 10.4171/AIHPD/60