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

Hudson, Robin L.; Pei, Yuchen

Annales de l’Institut Henri Poincaré D, Tome 5 (2018) no. 4, pp. 467-512.

We study the family of causal double product integrals

$$\prod _{a<x<y<b}\left(1+i\frac{\lambda }{2}(d{P}_{x}d{Q}_y-d{Q}_{x}d{P}_y)+i\frac{\mu }{2}(d{P}_{x}d{P}_y+d{Q}_{x}d{Q}_y)\right),$$

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 [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.

Classification : 81-XX, 05-XX, 06-XX
Mots-clés : causal double product, Lévy's stochastic area, position and momentum Brownian motions, linear extensions, Catalan numbers, Dyck paths