General $O(D)$-equivariant fuzzy hyperspheres via confining potentials and energy cutoffs

Abstract

We summarize our recent construction of new fuzzy hyperspheres $S^d_{\Lambda}$ of arbitrary dimension $d$ covariant under the {\it full} orthogonal group $O(D)$, $D=d+1$. We impose a suitable energy cutoff on a quantum particle in $\mathbb{R}^D$ subject to 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 $\Lambda\in\mathbb{N}$. Consequently, the commutators of the Cartesian coordinates $\overline{x}^i$ are proportional to the angular momentum components $L_{ij}$, as in Snyder's noncommutative spaces. The $\overline{x}^i$ generate the whole algebra of observables ${\cal A}_{\Lambda}$ and thus the whole Hilbert space ${\cal H}_{\Lambda}$ when applied to any state. $\mathcal{H}_{\Lambda}$ carries a reducible representation of $O(D)$ isomorphic to the space of harmonic homogeneous polynomials of degree $\Lambda$ in the Cartesian coordinates of (commutative) $\mathbb{R}^{D+1}$; the latter carries an irreducible representation $\pi_\Lambda$ of $O(D\!+\!1)\supset O(D)$. Moreover, ${\cal A}_{\Lambda}$ is isomorphic to $\pi_\Lambda\left(Uso(D\!+\!1)\right)$. We identify the subspace ${\cal C}_\Lambda\subset{\cal A}_{\Lambda}$ spanned by fuzzy spherical harmonics. We interpret $\{{\cal H}_\Lambda\}_{\Lambda\in\mathbb{N}}$, $\{{\cal C}_\Lambda\}_{\Lambda\in\mathbb{N}}$ as fuzzy deformations of the space of square integrable functions and the space of continuous functions on $S^d$ respectively, $\{{\cal A}_\Lambda\}_{\Lambda\in\mathbb{N}}$ as fuzzy deformation of the associated algebra of observables. $\{{\cal A}_\Lambda\}_{\Lambda\in\mathbb{N}}$ yields a fuzzy quantization of a coadjoint orbit of $O(D\!+\!1)$ that goes to the classical phase space $T^*S^d$. These models might be useful in quantum field theory, quantum gravity or condensed matter physics.