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

Projecting a quantum theory onto the Hilbert subspace of states with energies below a cutoff 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 [1,2,3,4,5] of this procedure to spheres Sd⊂RD of radius r=1 (D=d+1>1): making E¯¯¯¯ and the depth of the well depend on (and diverge with) a natural number Λ we obtain new fuzzy spheres SdΛ covariant under the 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 Λ→∞ the Hilbert space dimension diverges, SdΛ→Sd, and we recover ordinary quantum mechanics on Sd. These models might be suggestive for effective models in quantum field theory, quantum gravity or condensed matter physics.