LAUSR

We construct various systems of coherent states (SCS) on the O ( D )-equivariant fuzzy spheres $$S^d_\Lambda $$ S Λ d ( $$d=1,2$$ d = 1 , 2 , $$D=d+1$$ D = d + 1 ) constructed in Fiore and Pisacane (J Geom Phys 132:423–451, 2018) 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 $$(\Delta \varvec{x})^2,(\Delta \varvec{L})^2$$ ( Δ x ) 2 , ( Δ L ) 2 in the ambient Euclidean space $$\mathbb {R}^D$$ R D . We also determine general bounds (e.g., uncertainty relations from commutation relations) for $$(\Delta \varvec{x})^2,(\Delta \varvec{L})^2$$ ( Δ x ) 2 , ( Δ 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 $$S^d$$ S d . We compare the results with their analogs on commutative $$S^d$$ S d . We also show that on $$S^2_\Lambda $$ S Λ 2 our optimally localized states are better localized than those on the Madore–Hoppe fuzzy sphere with the same cutoff $$\Lambda $$ Λ .