Comments on noncommutative quantum mechanical systems associated with Lie algebras

TL;DR

This paper explores quantum mechanics on noncommutative spaces defined by Lie algebra commutation relations, reformulating the problem as a standard quantum problem in momentum space and deriving new metric formulas for spheres and projective spaces.

Contribution

It reformulates noncommutative Lie algebra-based quantum mechanics as a standard problem in momentum space and provides new metric formulas for spheres and related spaces.

Findings

Reformulation of noncommutative quantum mechanics as a differential operator problem.

Explicit finite representations for $su(2)$ and $u(N)$ cases.

New formulas for metrics on spheres $S^n$, $RP^n$, and $U(2)$.

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 = -iE_{ab} (p)\, \partial /\partial p_b $. Generically, the matrix $E_{ab}(p)$ represents a certain infinite series over the deformation parameter $\theta$: $E_{ab} = \delta_{ab} + \ldots$. 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 $\theta$, 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 $RP^n$ and on $U(2)$.