# Seminar talk, 18 May 2022

Speaker: Andrei Smilga

Title: Noncommutative quantum mechanical systems associated with Lie algebras

Abstract:
We consider quantum mechanics on the noncommutative spaces characterized by the commutation relations

${\displaystyle [x_{a},x_{b}]\ =\ i\theta f_{abc}x_{c}\,,}$

where ${\displaystyle f_{abc}}$ are the structure constants of a Lie algebra. We note that this problem can be reformulated as an ordinary quantum problem in a commuting momentum space. The coordinates are then represented as linear differential operators ${\displaystyle {\hat {x}}_{a}=-i{\hat {D}}_{a}=-iR_{ab}(p)\,\partial /\partial p_{b}}$. Generically, the matrix ${\displaystyle R_{ab}(p)}$ represents a certain infinite series over the deformation parameter ${\displaystyle \theta }$: ${\displaystyle R_{ab}=\delta _{ab}+\ldots }$. The deformed Hamiltonian, ${\displaystyle {\hat {H}}\ =\ -{\frac {1}{2}}{\hat {D}}_{a}^{2}\,,}$ describes the motion along the corresponding group manifolds with the characteristic size of order ${\displaystyle \theta ^{-1}}$. Their metrics are also expressed into certain infinite series in ${\displaystyle \theta }$.

For the algebras ${\displaystyle su(2)}$ and ${\displaystyle u(2)}$, it has been possible to represent the operators ${\displaystyle {\hat {x}}_{a}}$ in a simple finite form. A byproduct of our study are new nonstandard formulas for the metrics on ${\displaystyle SU(2)\equiv S^{3}}$ and on ${\displaystyle SO(3)}$.

Video
References:
arXiv:2204.08705