The x i -eigenvalue problem on some new fuzzy spheres

Pisacane

2020

Lett Math Phys

Self Cite

We construct various systems of coherent states (SCS) on the O(D)-equivariant fuzzy spheres S d Λ (d = 1, 2, D = d + 1) constructed in [G. Fiore, F. Pisacane, J. Geom. Phys. 132 (2018), 423-451] and study their localizations in configuration space as well as angular momentum space. These localizations are best expressed through the O(D)-invariant square space and angular momentum uncertainties (∆x) 2 , (∆L) 2 in the ambient Euclidean space R D . We also determine general bounds (e.g. uncertainty relations from commutation relations) for (∆x) 2 , (∆L) 2 , and partly investigate which SCS may saturate these bounds. In particular, we determine O(D)-equivariant systems of optimally localized coherent states, which are the closest quantum states to the classical states (i.e. points) of S d . We compare the results with their analogs on commutative S d . We also show that on S 2 Λ our optimally localized states are better localized than those on the Madore-Hoppe fuzzy sphere with the same cutoff Λ.

Section: O(2)-invariant Ur and Cs Systems On S 1 λsupporting

confidence: 57%

Section: O(2)-invariant Ur and Cs Systems On S 1 λmentioning

confidence: 99%

“…In the appendix we compute (∆L) 2 , (∆x) 2 on the SCS {φ β g } g∈G ; this is the analog of the SCS (28)(29)(30)(31). Again (∆x) 2 is smallest if β = 0.…”

Section: O(3)-invariant Ur and Cs Systems On S 2 λmentioning

confidence: 99%

Section: O(3)-invariant Ur and Cs Systems On S 2 λmentioning

confidence: 99%

Section: Outlook Final Remarks and Conclusionmentioning

confidence: 99%

See 3 more Smart Citations

On localized and coherent states on some new fuzzy spheres

Pisacane

2020

Lett Math Phys

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

Weber

2021

Lett Math Phys

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We propose a general procedure to construct noncommutative deformations of an embedded submanifold M of $${\mathbb {R}}^n$$ R n determined by a set of smooth equations $$f^a(x)=0$$ f a ( x ) = 0 . We use the framework of Drinfel’d twist deformation of differential geometry of Aschieri et al. (Class Quantum Gravity 23:1883, 2006); the commutative pointwise product is replaced by a (generally noncommutative) $$\star $$ ⋆ -product determined by a Drinfel’d twist. The twists we employ are based on the Lie algebra $$\Xi _t$$ Ξ t of vector fields that are tangent to all the submanifolds that are level sets of the $$f^a$$ f a (tangent infinitesimal diffeomorphisms); the twisted Cartan calculus is automatically equivariant under twisted $$\Xi _t$$ Ξ t . We can consistently project a connection from the twisted $${\mathbb {R}}^n$$ R n to the twisted M if the twist is based on a suitable Lie subalgebra $${\mathfrak {e}}\subset \Xi _t$$ e ⊂ Ξ t . If we endow $${\mathbb {R}}^n$$ R n with a metric, then twisting and projecting to the normal and tangent vector fields commute, and we can project the Levi–Civita connection consistently to the twisted M, provided the twist is based on the Lie subalgebra $${\mathfrak {k}}\subset {\mathfrak {e}}$$ k ⊂ e of the Killing vector fields of the metric; a twisted Gauss theorem follows, in particular. Twisted algebraic manifolds can be characterized in terms of generators and $$\star $$ ⋆ -polynomial relations. We present in some detail twisted cylinders embedded in twisted Euclidean $${\mathbb {R}}^3$$ R 3 and twisted hyperboloids embedded in twisted Minkowski $${\mathbb {R}}^3$$ R 3 [these are twisted (anti-)de Sitter spaces $$dS_2,AdS_2$$ d S 2 , A d S 2 ].

Twisted Quadrics and Algebraic Submanifolds in $\mathbb {R}^{n}$

Franco

Weber

2020

Math Phys Anal Geom

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We propose a general procedure to construct noncommutative deformations of an algebraic submanifold M of $\mathbb {R}^{n}$ ℝ n , specializing the procedure [G. Fiore, T. Weber, Twisted submanifolds of$\mathbb {R}^{n}$ ℝ n , arXiv:2003.03854] valid for smooth submanifolds. We use the framework of twisted differential geometry of Aschieri et al. (Class. Quantum Grav. 23, 1883–1911, 2006), whereby the commutative pointwise product is replaced by the ⋆-product determined by a Drinfel’d twist. We actually simultaneously construct noncommutative deformations of all the algebraic submanifolds Mc that are level sets of the fa(x), where fa(x) = 0 are the polynomial equations solved by the points of M, employing twists based on the Lie algebra Ξt of vector fields that are tangent to all the Mc. The twisted Cartan calculus is automatically equivariant under twisted Ξt. If we endow $\mathbb {R}^{n}$ ℝ n with a metric, then twisting and projecting to normal or tangent components commute, projecting the Levi-Civita connection to the twisted M is consistent, and in particular a twisted Gauss theorem holds, provided the twist is based on Killing vector fields. Twisted algebraic quadrics can be characterized in terms of generators and ⋆-polynomial relations. We explicitly work out deformations based on abelian or Jordanian twists of all quadrics in $\mathbb {R}^{3}$ ℝ 3 except ellipsoids, in particular twisted cylinders embedded in twisted Euclidean $\mathbb {R}^{3}$ ℝ 3 and twisted hyperboloids embedded in twisted Minkowski $\mathbb {R}^{3}$ ℝ 3 [the latter are twisted (anti-)de Sitter spaces dS2, AdS2].