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

Published on 2015-07-01

In this paper with 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 (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.