arXiv 2204.087052022-04hep-thmath-phmath.GRmath.MP

与李代数相关的非对易量子力学系统的评论

Comments on noncommutative quantum mechanical systems associated with Lie algebras

Andrei Smilga

备注：: 12 pages. Discussion of higher spheres and of higher $N$ in the Gurevich-Saponov model added
DOI：: 10.1016/j.geomphys.2022.104628

摘要：

我们考虑了以对易关系$$ [x_a, x_b] \ =\ iθf_{abc} x_c\,, $$为特征的非对易空间上的量子力学，其中$f_{abc}$是李代数的结构常数.我们注意到，这个问题可以重新表述为一个普通的量子问题在一个交换动量空间。然后将坐标表示为线性微分算子$\hat x_a = -i\hat D_a = -iE_{ab} (p)\, \partial /\partial p_b $。一般地，矩阵$E_{ab}(p)$表示关于变形参数$θ$的某个无穷级数：$E_{ab} = δ_{ab} + \ldots$。变形哈密顿量$\hat H = -\frac 12 \hat D_a^2\,,$描述了沿相应的群流形的运动，其特征尺寸为$θ^{-1}$阶。它们的度量在$θ$中也被表示为某种无穷级数，其中$E_{ab}$具有vielbeins的意义。对于代数$su(2)$和$u(N)$，可以用简单的有限形式表示算子$\hat x_a$。我们的研究的副产品是新的非标准公式的度量上的所有领域$S^n$，相应的投影空间$RP^n$和$U(2)$。

英文摘要：

We consider quantum mechanics on the noncommutative spaces characterized by the commutation relations $$ [x_a, x_b] \ =\ iθf_{abc} x_c\,, $$ 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 $θ$: $E_{ab} = δ_{ab} + \ldots$. 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 $θ^{-1}$. Their metrics are also expressed into certain infinite series in $θ$, 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 $U(2)$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文