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

TL;DR

This paper constructs new covariant fuzzy spheres by projecting quantum theories with energy cutoffs, leading to noncommutative geometries that recover classical spheres as the cutoff diverges, with potential applications in quantum physics.

Contribution

It introduces a method to generate $O(D)$-covariant fuzzy spheres from energy cutoff projections, extending noncommutative geometry models on spheres.

Findings

New fuzzy spheres $S^d_{ ext{Lambda}}$ covariant under $O(D)$ are constructed.

Coordinate commutators depend only on angular momentum, similar to Snyder spaces.

Classical spheres are recovered as the cutoff parameter goes to infinity.

Abstract

Projecting a quantum theory onto the Hilbert subspace of states with energies below a cutoff $\overline{E}$ may lead to an effective theory with modified observables, including a noncommutative space(time). Adding a confining potential well $V$ with a very sharp minimum on a submanifold $N$ of the original space(time) $M$ may induce a dimensional reduction to a noncommutative quantum theory on $N$. Here in particular we briefly report on our application of this procedure to spheres $S^d\subset\mathbb{R}^D$ of radius $r=1$ ($D=d\!+\!1>1$): making $\overline{E}$ and the depth of the well depend on (and diverge with) $\Lambda\in\mathbb{N}$ we obtain new fuzzy spheres $S^d_{\Lambda}$ covariant under the {\it full} orthogonal groups $O(D)$; the commutators of the coordinates depend only on the angular momentum, as in Snyder noncommutative spaces. Focusing on $d=1,2$, we also discuss uncertainty relations, localization of states, diagonalization of the space coordinates and construction of coherent states. As $\Lambda\to\infty$ the Hilbert space dimension diverges, $S^d_{\Lambda}\to S^d$, and we recover ordinary quantum mechanics on $S^d$. These models might be suggestive for effective models in quantum field theory, quantum gravity or condensed matter physics.