New fuzzy spheres through confining potentials and energy cutoffs

TL;DR

This paper introduces new fuzzy sphere models of dimensions 1 and 2, constructed via energy cutoffs on a quantum particle in a confining potential, with potential applications in quantum field theory, gravity, and condensed matter physics.

Contribution

It presents a novel construction of fuzzy spheres covariant under O(D) using energy cutoffs and confining potentials, extending the framework of noncommutative geometry.

Findings

Fuzzy spheres of dimensions 1 and 2 are constructed with covariant properties.

The models recover ordinary quantum mechanics on spheres as the cutoff diverges.

Potential applications in quantum field theory, quantum gravity, and condensed matter physics.

Abstract

We briefly report our recent construction of new fuzzy spheres of dimensions d=1,2 covariant under the full orthogonal group O(D), D=d+1. They are built by imposing a suitable energy cutoff on a quantum particle in D dimensions subject to a confining potential well V(r) with a very sharp minimum on the sphere of radius r=1; furthermore, the cutoff and the depth of the well depend on (and diverge with) a natural number L. The commutator of the coordinates depends only on the angular momentum, as in Snyder noncommutative spaces. When L diverges, the Hilbert space dimension diverges, too; S^d_L converges to S^d, and we recover ordinary quantum mechanics on S^d. These models might be useful in quantum field theory, quantum gravity or condensed matter physics.