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Hamiltonians for the quantised Volterra hierarchy

Carpentier, Sylvain, Mikhailov, Alexander V, Wang, Jing Ping (2024) Hamiltonians for the quantised Volterra hierarchy. Nonlinearity, 37 (9). Article Number 095033. ISSN 1361-6544. (doi:10.1088/1361-6544/ad68b8) (KAR id:106890)

Abstract

This paper builds upon our recent work, published in Carpentier et al (2022 Lett. Math. Phys. 112 94), where we established that the integrable Volterra lattice on a free associative algebra and the whole hierarchy of its symmetries admit a quantisation dependent on a parameter ω. We also uncovered an intriguing aspect: all odd-degree symmetries of the hierarchy admit an alternative, non-deformation quantisation, resulting in a non-commutative algebra for any choice of the quantisation parameter ω. In this study, we demonstrate that each equation within the quantum Volterra hierarchy can be expressed in the Heisenberg form. We provide explicit expressions for all quantum Hamiltonians and establish their commutativity. In the classical limit, these quantum Hamiltonians yield explicit expressions for the classical ones of the commutative Volterra hierarchy. Furthermore, we present Heisenberg equations and their Hamiltonians in the case of non-deformation quantisation. Finally, we discuss commuting first integrals, central elements of the quantum algebra, and the integrability problem for periodic reductions of the Volterra lattice in the context of both quantisations.

Item Type: Article
DOI/Identification number: 10.1088/1361-6544/ad68b8
Additional information: For the purpose of open access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission.
Uncontrolled keywords: quantisation, quantum algebra, 37K10, Hamiltonians, quantisation ideal, periodic Volterra system, non-Abelian euqtions, Volterra hierarchy, 81R12
Subjects: Q Science > QC Physics
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Funders: Engineering and Physical Sciences Research Council (https://ror.org/0439y7842)
SWORD Depositor: JISC Publications Router
Depositing User: JISC Publications Router
Date Deposited: 16 Aug 2024 13:57 UTC
Last Modified: 05 Nov 2024 13:12 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/106890 (The current URI for this page, for reference purposes)

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