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 momentum space. The coordinates are then represented as linear differential operators ${\hat{x}}_a=-i{\hat{D}}_a=-i{R}_{ab}(p)\partial /\partial p_b$. Generically, the matrix $R_{ab}(p)$ represents a certain infinite series over the deformation parameter $\theta$: $R_{ab}={\delta }_{ab}+\dots$. 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 $SO(3)$.

Video
References:
arXiv:2204.08705