View source for Mikhailov A. Quantisation of free associative dynamical systems (abstract)

{{MeetingTalk
| speaker = Alexander Mikhailov
| title = Quantisation of free associative dynamical systems

''Bi-quantisation of stationary KdV hierarchy and Novikov's equations and non-deformation quantisation of the Volterra sub-hierarchy.''
| abstract = Traditional quantisation theories start with classical Hamiltonian systems with variables taking values in commutative algebras and then study their non-commutative deformations, such that the commutators of observables tend to the corresponding Poisson brackets as the (Planck) constant of deformation goes to zero. I am proposing to depart from dynamical systems defined on a free associative algebra <math>\mathfrak{A}</math>. In this approach the quantisation problem is reduced to description of two-sided ideals <math>\mathfrak{J}\subset\mathfrak{A}</math> satisfying two conditions: the ideals have to be invariant with respect to the dynamics of the system and to define a complete set of commutation relations in the quotient algebras <math>\mathfrak{A}_\mathfrak{J}=\mathfrak{A}\diagup\mathfrak{J}</math>.

To illustrate this approach I'll consider the quantisation problem for <math>N</math>-th Novikov equations and the corresponding finite KdV hierarchy. I will show that stationary KdV equations and Novikov's equations admit two compatible quantisations, i.e. two distinct commutation relations between the variables, such that a linear combination of the corresponding commutators i s also a valid quantisation rule leading to the Heisenberg form of quantum equations. The picture is very similar to the bi-Hamiltonian structure in the case of classical integrable equations.

I'll discuss quantisation of the Bogoyavlensky family of integrable <math>N</math>--chains:

<div style='text-align: center;'><math>\frac{du_n}{dt}=\sum_{k=1}^N (u_{n+k}u_n-u_n u_{n-k}),\qquad n\in\mathbb{Z},\qquad\qquad\qquad\qquad(1)</math></div>

quantisation of their symmetries and modifications. In particular, I will show t hat odd degree symmetries of the Volterra chain (<math>N=1</math> in (1)) admit two quantisations, one of them corresponds to known quantisation of the Volterra chain, and another one is new and not deformational.

The talk is partially based on:
* AVM, ''Quantisation ideals of nonabelian integrable systems'', {{arXiv|2009.01838}}, 2020 (Published in Russ. Math. Surv. v.75:5, pp 199-200, 2020).
* V.M. Buchstaber and AVM, ''KdV hierarchies and quantum Novikov's equations'', {{arXiv|2109.06357}}, 2021.
| video = https://video.gdeq.org/AMV-conf-20211213-Alexander_Mikhailov.mp4
| slides = 
| references = 
| event = [[Diffieties, Cohomological Physics, and Other Animals]], 13-17 December 2021, Moscow.<br>''Alexandre Vinogradov Memorial Conference.''
| 79YY-MM-DD = 7978-87-86
}}