Proceedings of Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity" — PoS(CORFU2019) 2020
DOI: 10.22323/1.376.0208
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Energy cutoff, effective theories, noncommutativity, fuzzyness: the case of $O(D)$-covariant fuzzy spheres

Abstract: Projecting a quantum theory onto the Hilbert subspace of states with energies below a cutoff E may lead to an effective theory with modified observables, including a noncommutative space(time). Adding a confining potential well V with a very sharp minimum on a submanifold N of the original space(time) M may induce a dimensional reduction to a noncommutative quantum theory on N. Here in particular we briefly report on our application [1, 2, 3, 4, 5] of this procedure to spheres S d ⊂ R D of radius r = 1 (D = d+… Show more

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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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“…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%
“…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%
“…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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