Аннотация:
We propose to reformulate the problem of quantisation, focussing on quantisation
of a dynamical system
themselves,
rather than of their Poisson structures [1,2,3].
We begin with a dynamical system
defined on a free associative algebra $\mathfrak{A}={\mathbb C}[[\hbar]]\langle x_1,x_2,\ldots
\rangle $ with
non-commutative dynamical variables $x_1,x_2,\ldots $ and a commutative
parameter $\hbar$. It can be obtained as a
lift of a
classical dynamical system to a free algebra preserving
symmetries and/or other valuable properties of the dynamics. The
non-commutative dynamical
system defines a derivation $\partial \,:\, \mathfrak{A}\to\mathfrak{A}$ of the algebra. A
two-sided ideal $\mathfrak{J}\subset\mathfrak{A}$ is said to be a quantisation ideal for
$(\mathfrak{A},\partial)$ if it is $\partial$-stable: $\partial(\mathfrak{J})\subset\mathfrak{J};$
and the quotient ${\mathcal A}_{\hbar}=\mathfrak{A}\diagup\mathfrak{J}$ admits a basis ${\mathcal B}$ of normally
ordered monomials.
The quotient algebra ${\mathcal A}_\hbar$ is then said to be a quantised
algebra with well defined quantum dynamics $\partial \,:\,
{\mathcal A}_\hbar\to{\mathcal A}_\hbar$ [1].
The new approach enables
us to define and present
first examples of non-deformation quantisations of dynamical systems, i.e.
quantum systems that can be presented in the Heisenberg form
$\dot{a}=\frac{i}{\hbar}[H,a]$, while the algebra of observables ${\mathcal A}_\hbar$
remains non-commutative for any choice of “Planck's constant” $\hbar$
[1,3]. The
quantised algebra ${\mathcal A}_\hbar$ can be viewed as a deformation of the
non-commutative algebra ${\mathcal A}_0={\mathcal A}_\hbar\diagup (\hbar{\mathcal A}_\hbar)$.
In the limit $\hbar\to 0$ it yields a Poisson algebra structure and Hamilton
equations which are well defined on $Z({\mathcal A}_0)\times \frac{{\mathcal A}_0}{Z({\mathcal A}_0)}$
and ${\mathcal A}_0$ respectively, where $Z({\mathcal A}_0)$ is the centre of the algebra ${\mathcal A}_0$.
This talk is based on a joint work (yet in preparation) with Pol Vanhaecke.
[1] A.V.Mikhailov. Quantisation ideals of nonabelian integrable systems. Russ. Math. Surv., 75(5):199, 2020.
[2] V.M.Buchstaber and A.V.Mihkailov. {K}d{V} hierarchies and the quantum {N}ovikov equations. arXiv:2109.06357.
[3] S.Carpentier, A.V.Mikhailov and J.P.Wang. Quantisation of the Volterra hierarchy. Lett. Math. Phys., 112:94, 2022.
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