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

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.