Beijing Institute of Mathematical Sciences and Applications

Nicolai Reshetikhin , Bart Vlaar , Rui Jie Xu

Alexander Mikhailov

Monday, April 1, 2024 11:30 AM - 1:00 PM

A4-1

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 $\fA$ generated by non-commutative dynamical variables $x_1,x_2,\ldots$, and reduce the problem of quantisation to the problem of studying two-sided quantisation ideals. The dynamical system defines a derivation of the algebra $\p\,:\,\fA\mapsto\fA$. By definition, a two-sided ideal $\cI$ of $\fA$ is said to be a \emph{quantisation ideal} for $(\fA,\p)$ if it satisfies the following two properties: 1. The ideal $\cI$ is $\p$-stable: $\p(\cI)\subset\cI;$ 2. The quotient $\fA/\cI$ admits a basis of normally ordered monomials in the dynamical variables. The multiplication rules in the quantum algebra $\fA/\cI$ 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. quanum 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] Sylvain Carpentier, Alexander V. Mikhailov, and Jing Ping Wang. Hamiltonians for the quantised Volterra hierarchy. arXiv:2312.12077, 2023.(Submitted to Nonlinearity)