Energy cutoff, effective theories, noncommutativity, fuzzyness: the case of $O(D)$-covariant fuzzy spheres

Fiore, Gaetano; Pisacane, Francesco

doi:10.22323/1.376.0208

Cited by 3 publications

(7 citation statements)

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“…Many aspects of these new fuzzy spheres deserve investigations: e.g. space uncertainties, optimally localized states and coherent states also for d > 2, as done in [28,29,24] for d = 1, 2; a distance between optimally localized states (as done e.g. in [30] for the FS); extending the construction to particles with spin; QFT based on our fuzzy spheres; application of our fuzzy spheres to problems in quantum gravity, or condensed matter physics; etc.…”

Section: Outlook Discussion and Conclusionmentioning

confidence: 99%

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New fuzzy spheres through confining potentials and energy cutoffs

Fiore

Pisacane

2018

Proceedings of Corfu Summer Institute 2017 "Schools and Workshops on Elementary Particle Physics and Gravity" — PoS(CORFU2017)

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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 . These models might be useful in quantum field theory, quantum gravity or condensed matter physics.

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Section: Outlook Discussion and Conclusionmentioning

confidence: 99%

“…We recall the procedure starting from the general underlying philosophy [24,22,25]. Consider a quantum theory T with Hilbert space H, algebra of observables on H (or with a domain dense in H) A ≡ Lin(H), Hamiltonian H ∈ A.…”

Section: Introductionmentioning

confidence: 99%

New fuzzy spheres through confining potentials and energy cutoffs

Fiore

Pisacane

2018

Proceedings of Corfu Summer Institute 2017 "Schools and Workshops on Elementary Particle Physics and Gravity" — PoS(CORFU2017)

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“…In fact, by (26), (24) normalizing ψ m 0,l amounts to 1 = ´∞ 0 dr r d f 2 l (r) = ´∞ 0 dr g 2 l (r) M 2 l ´∞ −∞ dr e − √ kl(r− rl) 2 ; here (due to the shift 0 → −∞ of the left integration extreme) means equality up to terms of the order 1/ √ 2ke − √ 2k , which has zero asymptotic expansion in 1/ √ 2k, see [22].…”

Section: The Projector P Lmentioning

confidence: 99%

“…If ϕ has the form ϕ(r, θ) = Θ j (θ)φ(r), with Θ j ∈ V j D , then by (24) and (26) this simplifies to…”

Section: The Projector P Lmentioning

confidence: 99%

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Fuzzy hyperspheres via confining potentials and energy cutoffs

Fiore

2023

J. Phys. A: Math. Theor.

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We simplify and complete the construction of fully O(D)-equivariant fuzzy spheres S^d_Λ, for all dimensions d ≡ D − 1, initiated in [G. Fiore, F. Pisacane, J. Geom. Phys. 132 (2018), 423-451]. 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 A_Λ on the Hilbert space H_Λ; applying polynomials in the x^i to any ψ ∈ H_Λ we recover the whole H_Λ. The commutators of the xi are proportional to the angular momentum components, as in Snyder noncommutative spaces. H_Λ, 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, A_Λ is isomorphic to π_Λ (Uso(D+1)). We resp. interpret {H_Λ}_{Λ∈N}, {A_Λ}_{Λ∈N} as fuzzy deformations of the space H_s := L^2(S^d) of (square integrable) functions on S^d and of the associated algebra A_s of observables, because they resp. go to H_s, A_s as Λ diverges (with ℏ fixed). With suitable ℏ = ℏ(Λ)→ 0 as Λ→∞, 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.

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

Fiore

2023

Proceedings of Corfu Summer Institute 2022 "School and Workshops on Elementary Particle Physics and Gravity" — PoS(CORFU2022)

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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.

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Energy cutoff, effective theories, noncommutativity, fuzzyness: the case of $O(D)$-covariant fuzzy spheres

Cited by 3 publications

References 35 publications

New fuzzy spheres through confining potentials and energy cutoffs

New fuzzy spheres through confining potentials and energy cutoffs

Fuzzy hyperspheres via confining potentials and energy cutoffs

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

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