Comments on 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 ${\hat{x}}_a=-i{\hat{D}}_a=-i{E}_{ab}(p)\partial /\partial p_b$. Generically, the matrix $E_{ab}(p)$ represents a certain infinite series over the deformation parameter $\theta$: $E_{ab}={\delta }_{ab}+\dots$. 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 $\theta$, with $E_{ab}$ having the meaning of vielbeins. For the algebras $su(2)$ and $u(N)$, 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 all the spheres $S^n$, on the corresponding projective spaces $R{P}^n$ and on $U(2)$.

12 pages. Discussion of higher spheres and of higher $N$ in the Gurevich-Saponov model added

References in corpus (2)

• Magnetic fields in noncommutative quantum mechanics
• Recurrence relations for symplectic realization of (quasi)-Poisson structures