General O(D)-equivariant fuzzy hyperspheres via confining potentials and energy cutoffs

We summarize our recent construction [1–3] of new fuzzy hyperspheres SΛd of arbitrary dimension d ∈ N covariant under the full orthogonal group O(D), D = d+1. We impose 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. Consequently, the commutators of the Cartesian coordinates xi are proportional to the angular momentum components Lij, as in Snyder’s noncommutative spaces. The xi generate the whole algebra of observables AΛ and thus the whole Hilbert space HΛ when applied to any state. HΛ carries a reducible representation of O(D) isomorphic to the space of harmonic homogeneous polynomials of degree Λ in the Cartesian coordinates of (commutative) RD+1; the latter carries an irreducible representation πΛ of O(D+1) ⊃ O(D). Moreover, AΛ is isomorphic to πΛ (Uso(D+1)). We identify the subspace CΛ ⊂ AΛ spanned by fuzzy spherical harmonics. We interpret {HΛ}Λ∈N, {CΛ}Λ∈N as fuzzy deformations of the space Hs ≡ L2(Sd) of square integrable functions and the space C(Sd) of continuous functions on Sd respectively, {AΛ}Λ∈N as fuzzy deformation of the associated algebra As of observables, because they resp. go to Hs,C(Sd), As as Λ diverges (with fixed ℏ). 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. These models might be useful in quantum field theory, quantum gravity or condensed matter physics.