Fuzzy hyperspheres via confining potentials and energy cutoffs

Gaetano Fiore

Published 2022-11-23Version 1

We simplify and complete the construction of fully $O(D)$-equivariant fuzzy spheres $S^d_\Lambda$, for all dimensions $d\equiv D-1$, initiated in [G. Fiore, F. Pisacane, J. Geom. Phys. 132 (2018), 423-451]. This is based on imposing 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 $\Lambda\in\mathbb{N}$. We show that the irreducible representation ${{\bf\pi}}_\Lambda$ of $O(D\!+\!1)$ on the space of harmonic homogeneous polynomials of degree $\Lambda$ in the Cartesian coordinates of $\mathbb{R}^{D+1}$ (which we express via trace-free completely symmetric projections) is isomorphic to the Hilbert space $\mathcal{H}_{\Lambda}$ of the particle, as a reducible representation of $O(D)\subset O(D\!+\!1)$. Moreover, we show that the algebra of observables is isomorphic to ${{\bf\pi}}_\Lambda\left(Uso(D\!+\!1)\right)$. As $\Lambda$ diverges (commutative limit) so does the dimension of $\mathcal{H}_{\Lambda}$, and we recover ordinary quantum mechanics on the sphere $S^d$; more formally, we have a fuzzy quantization of a coadjoint orbit of $O(D\!+\!1)$ that goes to the classical phase space $T^*S^d$.