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On a causal quantum stochastic double product integral related to Levy area
Loughborough Univ, Math Dept, Loughborough LE11 3TU, Leics, England..
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2018 (English)In: Annales de l'Institut Henri Poincare (D) Combinatorics, Physics and their Interactions, ISSN 2308-5827, Vol. 5, no 4, p. 467-512Article in journal (Refereed) Published
Abstract [en]

We study the family of causal double product integrals Pi(a < x < y < b) (1 + i lambda/2 (dP(x)dQ(y) - dQ(x)dP(y)) + i mu/2 (dP(x)dP(y) + dQ(x)dQ(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.

Place, publisher, year, edition, pages
European Mathematical Society Publishing House, 2018. Vol. 5, no 4, p. 467-512
Keywords [en]
Causal double product, Levy's stochastic area, position and momentum Brownian motions, linear extensions, Catalan numbers, Dyck paths
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-244164DOI: 10.4171/AIHPD/60ISI: 000457121800001Scopus ID: 2-s2.0-85061457263OAI: oai:DiVA.org:kth-244164DiVA, id: diva2:1289357
Note

QC 20190218

Available from: 2019-02-18 Created: 2019-02-18 Last updated: 2019-03-26Bibliographically approved

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CiteExportLink to record
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Citation style
  • apa
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