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

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

References:
arXiv:2204.08705