On localized and coherent states on some new fuzzy spheres

Abstract

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.

The x i -eigenvalue problem on some new fuzzy spheres

“…Therefore the eigenstate χ with maximal eigenvalue of any coordinate is very localized (almost an optimally localized state; the latter are the closest to "classical" states). Evaluating (4) on the approximation of χ determined in the present work is sufficient to prove [17] the bound…”

“…Item (C) of last theorem allows also us to make a connection between our localized states and the classical ones because the α 1 (Λ; 0)-eigenstate approximates a quantum particle on S 2 concentrated (because of the above equivalence between the α 1 (Λ; 0)-eigenstate and the most localized state of our fuzzy space [17]) on the North pole and rotating around the x 3 -axis; on the other hand, if we consider a classical particle forced to stay on S 2 and in the position (0, 0, 1), then it must be L 3 = (L) 3 = r × p 3 = 0, as for our case.…”

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

“…Finally, we mention that in [30,31,50] an alternative approach to introduce NC (more precisely, fuzzy) submanifolds S ⊂ R n has been proposed and applied to spheres, projecting the algebra of observables of a quantum particle in R n , subject to a confining potential with a very sharp minimum on S, to the Hilbert subspace with energy below a certain cutoff.…”

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

“…Λ are constructed and their main features, including uncertainty relations, localization of states, diagonalization of the space coordinates and construction of coherent states [3,4] for d = 1, 2.…”

“…As a T consider now quantum mechanics (QM) of a zero spin particle on R D with a Hamiltonian H(x x x, p p p). If H is the subspace with energies E ≤ E then its dimension is approximately the phase space volume of the classical region B E determined by the inequality H 3 We can obtain a NC, fuzzy approximation of QM on a submanifold N of R D adding a 'dimensional reduction' mechanism, more precisely a V (x x x) with a sharp minimum on N. 4 In the rest of the paper we report on our application [1,2,3,4,5] of the mechanism for N equal to the d = (D−1)-dimensional sphere S d of radius r = 1 (r 2 := x x x 2 is the square distance from the origin) and on the study of the resulting fuzzy spheres for d = 1, 2 [1,2,3,4]; the lower right corner of fig. 1 shows the corresponding region R (a thin spherical shell of radius 1) in the d = 1 case.…”