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We study the family of causal double product integrals
where and 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 . The main problem solved in this paper is the explicit evaluation of the continuum limit of the latter, and showing that is a unitary operator. The kernel of 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. 467–512DOI 10.4171/AIHPD/60