Fuzzy hyperspheres via confining potentials and energy cutoffs

Abstract

Abstract We simplify and complete the construction of fully O(D)-equivariant fuzzy spheres S Λ d , for all dimensions d ≡ D − 1 , initiated in Fiore and Pisacane (2018 J. Geom. Phys. 132 423–51). This is based on imposing a suitable energy cutoff on a quantum particle in R D 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 x ‾ i generate the whole algebra of observables on the Hilbert space ; applying polynomials in the x ‾ i to any we recover the whole . The commutators of the x ‾ i are proportional to the angular momentum components, as in Snyder noncommutative spaces. , as carrier space of a reducible representation of O(D), is isomorphic to the space of harmonic homogeneous polynomials of degree Λ in the Cartesian coordinates of (commutative) R D + 1 , which carries an irreducible representation π Λ of O ( D + 1 ) ⊃ O ( D ) . Moreover, is isomorphic to π Λ U s o ( D + 1 ) . We resp. interpret , as fuzzy deformations of the space of (square integrable) functions on S d and of the associated algebra of observables, because they resp. go to as Λ diverges (with ℏ fixed). With suitable ℏ = ℏ ( Λ ) ⟶ Λ → ∞ 0 , in the same limit goes to the (algebra of functions on the) Poisson manifold T ∗ S d ; more formally, yields a fuzzy quantization of a coadjoint orbit of O ( D + 1 ) that goes to the classical phase space T ∗ S d .