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

arXiv comment41 pages, 1 figure. Version 3: some misprints are corrected

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Cited by in corpus (2)

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