Abstract
We study the family of causal double product integrals
\prod_{a
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.