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

∏ a < x < y < b 1 + i λ 2 ( d P x d Q y - d Q x d P y ) + i μ 2 ( d P x d P y + d Q x d Q y ) ,

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