Fuzzy circle and new fuzzy sphere through confining potentials and energy cutoffs

TL;DR

This paper introduces new fuzzy spheres of dimensions 1 and 2 by confining quantum particles with sharp potential wells, leading to finite-dimensional noncommutative geometries that approximate classical spheres as a parameter diverges.

Contribution

It constructs equivariant fuzzy spheres using energy cutoffs in quantum mechanics, extending previous models to full orthogonal symmetry and providing a convergence framework to classical spheres.

Findings

Finite-dimensional Hilbert spaces with noncommutative coordinates are obtained.

The models exhibit convergence to classical spheres as the cutoff parameter increases.

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

Abstract

Guided by ordinary quantum mechanics we introduce new fuzzy spheres of dimensions d=1,2: we consider an ordinary quantum particle in D=d+1 dimensions subject to a rotation invariant potential well V(r) with a very sharp minimum on a sphere of unit radius. Imposing a sufficiently low energy cutoff to `freeze' the radial excitations makes only a finite-dimensional Hilbert subspace accessible and on it the coordinates noncommutative \`a la Snyder; in fact, on it they generate the whole algebra of observables. The construction is equivariant not only under rotations - as Madore's fuzzy sphere -, but under the full orthogonal group O(D). Making the cutoff and the depth of the well dependent on (and diverging with) a natural number L, and keeping the leading terms in 1/L, we obtain a sequence S^d_L of fuzzy spheres converging (in a suitable sense) to the sphere S^d as L diverges (whereby we recover ordinary quantum mechanics on S^d). These models may be useful in condensed matter problems where particles are confined on a sphere by an (at least approximately) rotation-invariant potential, beside being suggestive of analogous mechanisms in quantum field theory or quantum gravity.