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Projecting a quantum theory onto the Hilbert subspace of states with energiesbelow a cutoff $\overline{E}$ may lead to an effective theory with modifiedobservables, including a noncommutative space(time). Adding a confiningpotential well $V$ with a very sharp minimum on a submanifold $N$ of theoriginal space(time) $M$ may induce a dimensional reduction to a noncommutativequantum theory on $N$. Here in particular we briefly report on our applicationof 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 (anddiverge with) $\Lambda\in\mathbb{N}$ we obtain new fuzzy spheres$S^d_{\Lambda}$ covariant under the {\it full} orthogonal groups $O(D)$; thecommutators of the coordinates depend only on the angular momentum, as inSnyder noncommutative spaces. Focusing on $d=1,2$, we also discuss uncertaintyrelations, localization of states, diagonalization of the space coordinates andconstruction of coherent states. As $\Lambda\to\infty$ the Hilbert spacedimension diverges, $S^d_{\Lambda}\to S^d$, and we recover ordinary quantummechanics on $S^d$. These models might be suggestive for effective models inquantum field theory, quantum gravity or condensed matter physics.

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