$O(D)-$equivariant fuzzy hyperspheres

Abstract

Fuzzy hyperspheres $S_{\Lambda }^d$ 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 à 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 $S_{\Lambda }^d$ of fuzzy spheres converging to $S^d$ as $\Lambda \to \infty$ (where one recovers ordinary quantum mechanics on $S^d$).

71 pages, 2 figures

Cited by in corpus (1)

• Energy cutoff, effective theories, noncommutativity, fuzzyness: the case of O(D)-covariant fuzzy spheres