$O(D)-$等变模糊超球
$O(D)-$equivariant fuzzy hyperspheres
摘要：
本文构造了维数为$d>2$的模糊超球$S^d_Λ$，推广了[G. Fiore，F. Pisacane，J. Geom.Phys.132（2018），423-451]用于$d=1,2$。起始点是$\mathbb{R}^D$中的普通量子粒子，$D:=d+1$，受到旋转不变势阱$V(r)$的影响，该势阱在半径为$r=1$的球上具有非常尖锐的最小值。随后施加一个足够低的能量截止“冻结”径向激发，这使得只有有限维希尔伯特子空间$\mathcal{H}_{Λ,D}$可访问，并在它的坐标非交换{ \it à la Snyder}。此外，坐标算子生成整个代数的观测量$\mathcal{A}_{Λ,D}$，这原来是通过一个合适的不可约矢量表示的$Uso(D+1)$实现。这种构造不仅在$SO(D)$下是等变的，而且在完全正交群$O(D)$下也是等变的，并且使截止和阱的深度以自然数$Λ$增长，结果是模糊球的序列$S^d_Λ$收敛到$S^d$作为$Λ\to\infty$（其中人们在$S^d$上恢复普通量子力学）。
英文摘要：
Fuzzy hyperspheres $S^d_Λ$ of dimension $d>2$ are constructed here generalizing the procedure adopted in [G. Fiore, F. Pisacane, J. Geom. Phys. 132 (2018), 423-451] for $d=1,2$. The starting point is an ordinary quantum particle in $\mathbb{R}^D$, $D:=d+1$, subject to a rotation invariant potential well $V(r)$ with a very sharp minimum on the sphere of radius $r=1$. The subsequent imposition of a sufficiently low energy cutoff `freezes' the radial excitations, this makes only a finite-dimensional Hilbert subspace $\mathcal{H}_{Λ,D}$ accessible and on it the coordinates noncommutative {\it à la Snyder}. In addition, the coordinate operators generate the whole algebra of observables $\mathcal{A}_{Λ,D}$ which turns out to be realizable through a suitable irreducible vector representation of $Uso(D+1)$. This construction is equivariant not only under $SO(D)$, but under the full orthogonal group $O(D)$, and making the cutoff and the depth of the well grow with a natural number $Λ$, the result is a sequence $S^d_Λ$ of fuzzy spheres converging to $S^d$ as $Λ\to\infty$ (where one recovers ordinary quantum mechanics on $S^d$).
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