JournalsaihpdVol. 5, No. 4pp. 467–512

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

  • Robin L. Hudson

    University of Loughborough, UK
  • Yuchen Pei

    KTH - Royal Institute of Technology, Stockholm, Sweden
On a causal quantum stochastic double product integral related to Lévy area cover

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Abstract

We study the family of causal double product integrals

a<x<y<b(1+iλ2(dPxdQydQxdPy)+iμ2(dPxdPy+dQxdQy)),\prod_{a < x < y < b}\Big(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)\Big),

where PP and QQ 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 WW of the latter, and showing that WW is a unitary operator. The kernel of WIW - 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.

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. 467–512

DOI 10.4171/AIHPD/60