# On localized and coherent states on some new fuzzy spheres

## TL;DR

This paper constructs and analyzes coherent states on new fuzzy spheres, focusing on their localization properties, uncertainty bounds, and comparison with classical and other fuzzy sphere states, revealing improved localization on certain fuzzy spheres.

## Contribution

It introduces new systems of coherent states on equivariant fuzzy spheres, studies their localization and uncertainty properties, and compares their performance with classical and existing fuzzy sphere states.

## Key findings

- Optimal localized states are identified on fuzzy spheres.
- These states better localize than those on Madore-Hoppe fuzzy spheres.
- Uncertainty bounds are established and analyzed for these states.

## Abstract

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 $\Lambda$.