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 12 \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 $SO(3)$.

mathematical physicsanalysis of PDEsdifferential geometry

Audience: researchers in the topic

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Geometry of differential equations seminar