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 $S^d_\Lambda$ ($d=1,2$, $D=d\!+\!1$) 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 $(\Delta\boldsymbol{x})^2,(\Delta\boldsymbol{L})^2$ in the ambient Euclidean space $\mathbb{R}^D$. We also determine general bounds (e.g. uncertainty relations from commutation relations) for $(\Delta\boldsymbol{x})^2,(\Delta\boldsymbol{L})^2$, 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 $S^2_\Lambda$ our optimally localized states are better localized than those on the Madore-Hoppe fuzzy sphere with the same cutoff $\Lambda$.