New fuzzy spheres through confining potentials and energy cutoffs

Abstract

We briefly report on our recent construction [1] of new fuzzy spheres S dΛ is built 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 depend on (and diverge with) Λ ∈ N. The commutator of the coordinates depends only on the angular momentum, as in Snyder noncommutative spaces. As Λ → ∞ the Hilbert space dimension diverges, S d Λ → S d , and we recover ordinary quantum mechanics on S d . These models might be useful in quantum field theory, quantum gravity or condensed matter physics.

The x i -eigenvalue problem on some new fuzzy spheres

“…To overcome these shortcomings, in [14,15] we have built new fuzzy spheres {S 1 Λ } Λ∈N and {S 2 Λ } Λ∈N , which are a fully O(2)-covariant fuzzy circle and a fully O(3)-covariant fuzzy 2sphere, respectively; the right-hand side of (1) 1 depends on the angular momentum components and therefore is parity invariant as in Snyder commutation relations [16], see eq. ( 27) 1 below.…”

“…The plan of the paper is as follows. In section 2 we briefly recall the construction procedure [14,15] of these fuzzy spaces and how to diagonalize Toeplitz tridiagonal matrices; in sections 3,4 we study the x i -eigenvalue equation on S 1 Λ , S 2 Λ respectively; in section 5 we compare results on S 2 Λ and FS; in section 6 (the Appendix) we have concentrated lengthy calculations and complex proofs.…”

Twisted submanifolds of $${\mathbb {R}}^n$$

“…Finally, in [35][36][37]39] an alternative approach to introduce noncommutative (fuzzy) embedded submanifolds S in R n was proposed and applied to the spheres; it is obtained projecting the algebra of observables of a quantum particle in R n , in a confining potential with a very sharp minimum on S, to the Hilbert subspace with energy below a certain cutoff. by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.…”

A fuzzy bipolar celestial sphere

“…The first step in order to obtain a non-commutative deformation of the celestial sphere will be to deform the algebra of spherical harmonics (3.20). This essentially boils down to the introduction of fuzzy spherical harmonics [23][24][25][26] which can be thought of as the algebra of functions on a non-commutative space known as the fuzzy sphere [14,[27][28][29][30][31][32][33][34][35][36][37]. This deformation of the algebra of spherical harmonics is concretely realized in terms of a "quantization map" between the commutative algebra of functions on the two-sphere C(S 2 ) and the algebra of N × N complex matrices M N (C),…”