论文标题
关于与谎言代数相关的非交通量子机械系统的评论
Comments on noncommutative quantum mechanical systems associated with Lie algebras
论文作者
论文摘要
我们考虑以换向关系为特征的非交通空间上的量子力学，$$ [x_a，x_b] \ = \iθf_{abc} x_c} x_c \ ,, $$，其中$ f_ {abc} $是lie代数的结构常数。我们注意到，在通勤势头空间中，可以将此问题重新归类为普通的量子问题。然后将坐标表示为线性差分运算符$ \ hat x_a = -i \ hat d_a = -ie_ {ab}（p）\，\ partial /\ partial /\ partial p_b $。一般而言，矩阵$ e_ {ab}（p）$代表了变形参数$θ$：$ e_ {ab} =δ__{ab} + \ ldots $。变形的汉密尔顿，$ \ hat h = - \ frac 12 \ hat d_a^2 \ ,, $描述了沿相应组歧管的运动，其特征$θ^{ - 1} $的特征大小。他们的指标也以$θ$的形式表示为某些无限系列，其中$ e_ {ab} $具有Vielbeins的含义。对于代数$ su（2）$和$ u（n）$的代数，可以以简单的有限表格表示操作员$ \ hat x_a $。我们研究的副产品是所有领域$ s^n $的指标的新的非标准公式，在相应的投影空间$ rp^n $和$ u（2）$上。
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 $U(2)$.
