Fuzzy hyperspheres via confining potentials and energy cutoffs

We simplify and complete the construction of fully O(D)-equivariant fuzzy spheres SdΛ, for all dimensions d ≡ D − 1, initiated in [G. Fiore, F. Pisacane, J. Geom. Phys. 132 (2018), 423-451]. This is based on imposing a suitable energy cutoff on a quantum particle in RD subject to a confining potential well V (r) with a very sharp minimum on the sphere of radius r = 1; the cutoff and the depth of the well diverge with Λ ∈ N. As a result, the noncommutative Cartesian coordinates xi generate the whole algebra of observables AΛ on the Hilbert space HΛ; applying polynomials in the xi to any ψ ∈ HΛ we recover the whole HΛ. The commutators of the xi are proportional to the angular momentum components, as in Snyder noncommutative spaces. HΛ, as carrier space of a reducible representation of O(D), is isomorphic to the space of homogeneous polynomials of degree Λ in the Cartesian coordinates of (commutative) RD+1, which carries an irreducible representation πΛ of O(D+1) ⊃ O(D). Moreover, AΛ is isomorphic to πΛ (Uso(D+1)). We resp. interpret {HΛ}Λ∈N, {AΛ}Λ∈N as fuzzy deformations of the space Hs := L2(Sd) of (square integrable) functions on Sd and of the associated algebra As of observables, because they resp. go to Hs,As as Λ diverges (with ℏ fixed). With suitable ℏ = ℏ(Λ) Λ→∞ −→ 0, in the same limit AΛ goes to the (algebra of functions on the) Poisson manifold T∗Sd; more formally, {AΛ}Λ∈N yields a fuzzy quantization of a coadjoint orbit of O(D+1) that goes to the classical phase space T∗Sd.