$O(D)-$equivariant fuzzy hyperspheres

Abstract

Fuzzy hyperspheres $S^d_\Lambda$ 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 $\mathcal{H}_{\Lambda,D}$ accessible and on it the coordinates noncommutative {\it \`a la Snyder}. In addition, the coordinate operators generate the whole algebra of observables $\mathcal{A}_{\Lambda,D}$ 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 $\Lambda$, the result is a sequence $S^d_{\Lambda}$ of fuzzy spheres converging to $S^d$ as $\Lambda\to\infty$ (where one recovers ordinary quantum mechanics on $S^d$).