Fuzzy hyperspheres via confining potentials and energy cutoffs

Abstract

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 ${\bf\pi}_L$ of $O(D+1)\supset O(D)$. Moreover, $A_L$ is isomorphic to ${\bf\pi}_L\left(Uso(D+1)\right)$. 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 $T^*S^d$.