On localized and coherent states on some new fuzzy spheres

We construct various systems of coherent states (SCS) on the O(D)-equivariant fuzzy spheres S^d_L (d = 1; 2, D = d+1) constructed in [G. Fiore, F. Pisacane, J. Geom. Phys. 132 (2018), 423-451] and study their localizations in con guration 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 (Delta x)^2; (Delta 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 Sd. We compare the results with their analogs on commutative Sd. We also show that on S^2_L our optimally localized states are better localized than those on the Madore-Hoppe fuzzy sphere with the same cutoff