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

We summarize our recent construction of new fuzzy hyperspheres $S_{\mathrm{\Lambda }}^d$ 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 $\mathrm{\Lambda }\in \mathbb{N}$. Consequently, the commutators of the Cartesian coordinates ${\bar{x}}^i$ are proportional to the angular momentum components $L_{ij}$, as in Snyder’s noncommutative spaces. The ${\bar{x}}^i$ generate the whole algebra of observables ${\mathcal{A}}_{\mathrm{\Lambda }}$ and thus the whole Hilbert space ${\mathcal{H}}_{\mathrm{\Lambda }}$ when applied to any state. ${\mathcal{H}}_{\mathrm{\Lambda }}$ carries a reducible representation of $O(D)$ isomorphic to the space of harmonic homogeneous polynomials of degree $\mathrm{\Lambda }$ in the Cartesian coordinates of (commutative) ${\mathbb{R}}^{D+1}$; the latter carries an irreducible representation ${\pi }_{\mathrm{\Lambda }}$ of $O(D\text{\hspace{0.17em}⁣}+\text{\hspace{0.17em}⁣}1)\supset O(D)$. Moreover, ${\mathcal{A}}_{\mathrm{\Lambda }}$ is isomorphic to ${\pi }_{\mathrm{\Lambda }}(Uso(D\text{\hspace{0.17em}⁣}+\text{\hspace{0.17em}⁣}1))$. We identify the subspace ${\mathcal{C}}_{\mathrm{\Lambda }}\subset {\mathcal{A}}_{\mathrm{\Lambda }}$ spanned by fuzzy spherical harmonics. We interpret $\{{\mathcal{H}}_{\mathrm{\Lambda }}{\}}_{\mathrm{\Lambda }\in \mathbb{N}}$, $\{{\mathcal{C}}_{\mathrm{\Lambda }}{\}}_{\mathrm{\Lambda }\in \mathbb{N}}$ as fuzzy deformations of the space of square integrable functions and the space of continuous functions on $S^d$ respectively, $\{{\mathcal{A}}_{\mathrm{\Lambda }}{\}}_{\mathrm{\Lambda }\in \mathbb{N}}$ as fuzzy deformation of the associated algebra of observables. $\{{\mathcal{A}}_{\mathrm{\Lambda }}{\}}_{\mathrm{\Lambda }\in \mathbb{N}}$ yields a fuzzy quantization of a coadjoint orbit of $O(D\text{\hspace{0.17em}⁣}+\text{\hspace{0.17em}⁣}1)$ that goes to the classical phase space $T^{\ast }{S}^d$. These models might be useful in quantum field theory, quantum gravity or condensed matter physics.