论文标题
$ o（d） - $ ecorivariant模糊大超球
$O(D)-$equivariant fuzzy hyperspheres
论文作者
论文摘要
尺寸$ d> 2 $的模糊hyperpheres $ s^d_λ$在这里构建了[G. Fiore，F。Pisacane，J。Geom。物理。 132（2018），423-451] $ d = 1,2 $。起点是$ \ mathbb {r}^d $，$ d：= d+1 $中的普通量子粒子，受旋转不变电位井$ v（r）$的约束，在半径$ r = 1 $的范围内，最低限度很高。随后强加了足够低的能量临界值“冻结”径向激发，这仅使有限维的Hilbert子空间$ \ MATHCAL {H} _ {λ，D} $可访问，并且在其上均不合并{\ it romutations {\ it ran la snyder}。此外，坐标运算符生成可观察到的整个代数$ \ MATHCAL {a} _ {λ，d} $，它通过$ uso（d+1）$的合适的不可约为矢量表示可以实现。 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 $S^d$).
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 $S^d$).
