19-23 June 2023
Faculty of Physics, University of Warsaw
Europe/Warsaw timezone
A novel approach to quantisation of dynamical systems
Presented by Prof. Alexander MIKHAILOV
on
20 Jun 2023
from
09:15
to
09:45
Session:
Non-commutative
Content
We propose to revisit the problem of quantisation and look at it from an
entirely new angle, focussing on quantisation of dynamical systems themself, rather than of their Poisson structures. We begin with a lift of a classical dynamical system to a system on a free associative algebra with noncommutative dynamical variables and reduce the problem of quantisation
to the problem of studying of two-sided quantisation ideals, i.e. the ideals
of the free algebra that define the commutation relations of the dynamical
variables and are invariant with respect to the non-commutative dynamics.
Quantum multiplication rules in the quotient algebra over a quantisation
ideal 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. The new approach also sheds light on the
problem of operator’s ordering.
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. arXiv:2109.06357.
[3] S. Carpentier, A.V. Mikhailov and J.P. Wang. Quantisation of the
Volterra hierarchy. Lett. Math. Phys., 112:94, 2022.
Place
Location: Faculty of Physics, University of Warsaw
Address: Pasteura 5, Warsaw
Room: Lecture hall: 0.06