论文标题
通过限制电势和能量截止
Fuzzy hyperspheres via confining potentials and energy cutoffs
论文作者
论文摘要
我们简化并完成了完全$ o（d）$ - equivariant fuzzy Spheres $ s^d_l $的构建，对于所有维度，$ d \ equiv d-1 $，在[G. Fiore，F。Pisacane，J。Geom。物理。 132（2018），423]。这是基于在$ \ mathbb {r}^d $中对量子粒子施加合适的能量截止，以限制潜在井$ v（r）$，在半径$ r = 1 $的范围内，最低限度很高；在\ mathbb {n} $中$ l \ $ l \的临界值和深度分歧。结果，非交易性的笛卡尔坐标$ \ edline {x}^i $在Hilbert Space $ H_L $上生成了观察值$ a_l $的整个代数; $ h_l $可以在其任何元素中均在$ \ overline {x}^i $中应用多项式恢复。 $ \ Overline {x}^i $的换向器与角动量成分成正比，如Snyder非交通空间。 $ h_l $，作为$ o（d）$的载流空间，是同型均值$ l $ $ l $的谐波均值（交换性）$ \ mathbb {r}^{d+1} $的谐波多项式的空间$ O（D+1）\ supset O（d）$。此外，$ a_l $是同构至$ {\bfπ} _l \ left（uso（d+1）\ right）$。我们解答。解释$ \ {h_l \} _ {l \ in \ Mathbb {n}} $，$ \ {a_l \} _ {l \ in \ Mathbb {n}} $ as fuuzzy as fuugzzy as Space $ h_s $ h_s：= = {= {\ cal cal $ cal on of sarace $ h_s：相关的代数$ a_s $可观察到，因为它们解答。转到$ h_s，a_s $ as $ l $ diverges（带有$ \ hbar $固定）。有了合适的$ \ hbar = \ hbar（l）\ stackrel {l \ to \ infty} {\ longrightArrow} 0 $，在相同的限制$ a_l $中，to（poisson poisson poisson farmord prolold $ termold $ t perions $ t poistold $ t p^*s^*s^d $;更正式地，$ \ {a_l \} _ {l \ in \ mathbb {n}} $产生的$ O（d+1）$的coadjoint轨道的模糊量化量为$ o（d+1）$。
We simplify and complete the construction of fully $O(D)$-equivariant fuzzy spheres $S^d_L$, for all dimensions $d\equiv D-1$, initiated in [G. Fiore, F. Pisacane, J. Geom. Phys. 132 (2018), 423]. This is based on imposing a suitable energy cutoff on a quantum particle in $\mathbb{R}^D$ in a confining potential well $V(r)$ with a very sharp minimum on the sphere of radius $r=1$; the cutoff and the depth of the well diverge with $L\in\mathbb{N}$. As a result, the noncommutative Cartesian coordinates $\overline{x}^i$ generate the whole algebra of observables $A_L$ on the Hilbert space $H_L$; $H_L$ can be recovered applying polynomials in the $\overline{x}^i$ to any of its elements. The commutators of the $\overline{x}^i$ are proportional to the angular momentum components, as in Snyder noncommutative spaces. $H_L$, as carrier space of a reducible representation of $O(D)$, is isomorphic to the space of harmonic homogeneous polynomials of degree $L$ in the Cartesian coordinates of (commutative) $\mathbb{R}^{D+1}$, which carries an irreducible representation ${\bfπ}_L$ of $O(D+1)\supset O(D)$. Moreover, $A_L$ is isomorphic to ${\bfπ}_L\left(Uso(D+1)\right)$. We resp. interpret $\{H_L\}_{L\in\mathbb{N}}$, $\{A_L\}_{L\in\mathbb{N}}$ as fuzzy deformations of the space $H_s:={\cal L}^2(S^d)$ of (square integrable) functions on $S^d$ and of the associated algebra $A_s$ of observables, because they resp. go to $H_s,A_s$ as $L$ diverges (with $\hbar$ fixed). With suitable $\hbar=\hbar(L)\stackrel{L\to\infty}{\longrightarrow} 0$, in the same limit $A_L$ goes to the (algebra of functions on the) Poisson manifold $T^*S^d$; more formally, $\{A_L\}_{L\in\mathbb{N}}$ yields a fuzzy quantization of a coadjoint orbit of $O(D+1)$ that goes to the classical phase space $T^*S^d$.
