Fuzzy circle and new fuzzy sphere through confining potentials and energy cutoffs

Abstract

Guided by ordinary quantum mechanics we introduce new fuzzy spheres S d Λ of dimensions d = 1, 2: we consider an ordinary quantum particle in D = d+1 dimensions subject to a rotation invariant potential well V (r) with a very sharp minimum on a sphere of unit radius. Imposing a sufficiently low energy cutoff to 'freeze' the radial excitations makes only a finite-dimensional Hilbert subspace accessible and on it the coordinates noncommutativeà la Snyder; in fact, on it they generate the whole algebra of observables. The construction is equivariant not only under rotations -as Madore's fuzzy sphere -, but under the full orthogonal group O(D). Making the cutoff and the depth of the well dependent on (and diverging with) a natural number Λ, and keeping the leading terms in 1/Λ, we obtain a sequence S d Λ of fuzzy spheres converging to the sphere S d in the limit Λ → ∞ (whereby we recover ordinary quantum mechanics on S d ). These models may be useful in condensed matter problems where particles are confined on a sphere by an (at least approximately) rotation-invariant potential, beside being suggestive of analogous mechanisms in quantum field theory or quantum geometry. arXiv:1709.04807v3 [math-ph]

New fuzzy spheres through confining potentials and energy cutoffs

“…The decomposition of C Λ in irreducible representations of O(3) reads C Λ = 2Λ l=0 V l . In [1] we have shown that for all φ ∈ H s x i φ → (x i /r)φ ; more generally, settingf…”

“…Here we present new fuzzy approximations [1] of quantum mechanics (QM) on S d (d = 1, 2) overcoming these two problems: We start with an ordinary zero-spin quantum particle in R D configuration space (D = d+1) with Hamiltonian…”

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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 .

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On the quantum structure of space-time, gravity, and higher spin in matrix models

“…Covariant higher-dimensional fuzzy spaces were studied e.g. in [2,4,6,54,[88][89][90][91][92][93][94][95], which are similar in spirit to Snyder space [41,96], see also [47] for a somewhat related ansatz. In particular, the relation of fuzzy S 4 to fuzzy CP 2 was pointed out in [5,97,98], which is analogous to the bundle structure discussed in section 5.…”

On localized and coherent states on some new fuzzy spheres

“…the angular momentum components L i map the generic V l into itself, while the x i map it into V l−1 ⊕ V l+1 . Moreover, relations (1) In [12] we have introduced some new fuzzy approximations of S 1 , S 2 -more precisely a fully O(2)-equivariant fuzzy circle {S 1 Λ } Λ∈N and a fully O(3)-equivariant fuzzy 2-sphere {S 2 Λ } Λ∈N ; the latter is free of the mentioned shortcomings. To construct S d Λ (d = 1, 2) we have first projected the Hilbert space L 2 (R D ) of a zero-spin quantum particle in R D (D := d+1) onto the finite-dimensional subspace H Λ spanned by all the ψ fulfilling…”