报告时间：3月28日 14:00开始
报 告 人：Alexander V. Mikhailov（英国利兹大学数学学院教授）
报告地点：包玉书9号楼113
报告题目：Quantisation Ideals - a novel approach to the old problem of quantisation
报告摘要：We propose to revisit the problem of quantisation and look at it from an entirely new angle, focusing on the quantisation of dynamical systems themselves, rather than of their Poisson structures. We begin with a dynamical system defined on a free associative algebra 𝔄 generated by non-commutative dynamical variables 
, and reduce the problem of quantisation to the problem of studying two-sided quantisation ideals. The dynamical system defines a derivation of the algebra
∂: 𝔄 ↦ 𝔄. By definition, a two-sided ideal 𝓘 of 𝔄 is said to be a quantisation ideal. for (𝔄 , ∂) if it satisfies the following two properties:
1. The ideal 𝓘 is ∂-stable: ∂( 𝓘) ⊂ 𝓘;
2. The quotient 𝔄/ 𝓘 admits a basis of normally ordered monomials in the dynamical variables.
The multiplication rules in the quantum algebra 𝔄/ 𝓘 are manifestly associative and consistent with the dynamics. We found first examples of bi-quantum systems which are quantum counterparts of bi-Hamiltonian systems in the classical theory. Moreover, the new approach enables us to define and present first examples of non-deformation quantisations of dynamical systems, i.e. quantum systems that cannot be obtained as deformations of a classical dynamical system with commutative variables. In order to apply the novel approach to a classical system we need firstly lift it to a system on a free algebra preserving the most valuable properties, such as symmetries, conservation laws, or Lax integrability. The new approach sheds light on the long-standing problem of operator's ordering. We will use the well-known Volterra hierarchy and stationary KdV equations to illustrate the methodology.
References
[1] A. V. Mikhailov. Quantisation ideals of nonabelian integrable systems. Russ. Math. Surv., 75(5):199, 2020.
[2] V. M. Buchstaber and A. V. Mikhailov. KdV hierarchies and quantum Novikov's equations. Ocnmp:12684 - Open Communications in Nonlinear Mathematical Physics, February 15, 2024, Special Issue in Memory of Decio Levi.
[3] S. Carpentier, A. V. Mikhailov and J. P. Wang. Quantisation of the Volterra hierarchy. Lett. Math. Phys., 112:94, 2022.
[4] S. Carpentier, A. V. Mikhailov, and J. P. Wang. Hamiltonians for the quantised Volterra hierarchy. arXiv:2312.12077, 2023.(Submitted to Nonlinearity)
Introduction. Alexander V. Mikhailov, Professor of the School of Mathematics, University of Leeds, UK. His research activity in integrable systems, especially problems of classification and quantization of integrable systems, noncommutative integrable systems. Obtained a PhD in Theoretical and Mathematical Physics at Landau Institute for Theoretical Physics in 1978 and Doctor of Science since 1987. Life fellow of the Clare Hall College, University of Cambridge. Organizer of multiple conferences and workshops, and published more than 100 papers in international journals with more than 3400 citations.
报告人简介：Alexander V.Mikhailov，英国利兹大学数学学院教授。可积系统的研究活动，特别是可积系统、非对易可积系统分类和量化问题。1978年在朗道理论物理研究所获得理论和数学物理博士学位，1987年全博士学位。剑桥大学克莱尔·霍尔学院终身院士。组织了多个会议和研讨会，并在国际期刊上发表了100多篇论文，引用次数超过3400次。
