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

Abstract

We summarize our recent construction [1-3] of new fuzzy hyperspheres 𝑆 𝑑 Λ 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 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. Consequently, the commutators of the Cartesian coordinates 𝑥 𝑖 are proportional to the angular momentum components 𝐿 𝑖 𝑗 , as in Snyder's noncommutative spaces. The 𝑥 𝑖 generate the whole algebra of observables A Λ and thus the whole Hilbert space H Λ when applied to any state. H Λ carries a reducible representation of 𝑂 (𝐷) isomorphic to the space of harmonic homogeneous polynomials of degree Λ in the Cartesian coordinates of (commutative) R 𝐷+1 ; the latter carries an irreducible representation 𝝅 Λ of 𝑂 (𝐷 +1) ⊃ 𝑂 (𝐷). Moreover, A Λ is isomorphic to 𝝅 Λ (𝑈𝑠𝑜(𝐷 +1)). We identify the subspace C Λ ⊂ A Λ spanned by fuzzy spherical harmonics. We interpret {H Λ } Λ∈N , {C Λ } Λ∈N as fuzzy deformations of the space H 𝑠 ≡ L 2 (𝑆 𝑑 ) of square integrable functions and the space 𝐶 (𝑆 𝑑 ) of continuous functions on 𝑆 𝑑 respectively, {A Λ } Λ∈N as fuzzy deformation of the associated algebra A 𝑠 of observables, because they resp. go to H 𝑠 , 𝐶 (𝑆 𝑑 ), A 𝑠 as Λ diverges (with fixed ℏ). With suitable ℏ = ℏ(Λ) Λ→∞ −→ 0, in the same limit A Λ goes to the (algebra of functions on the) Poisson manifold 𝑇 * 𝑆 𝑑 ; more formally, {A Λ } Λ∈N yields a fuzzy quantization of a coadjoint orbit of 𝑂 (𝐷 +1) that goes to the classical phase space 𝑇 * 𝑆 𝑑 . These models might be useful in quantum field theory, quantum gravity or condensed matter physics.