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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 $(Deltaboldsymbol{x})^2,(Deltaboldsymbol{L})^2$ in the ambient Euclidean space $mathbb{R}^D$. We also determine general bounds (e.g. uncertainty relations from commutation relations) for $(Deltaboldsymbol{x})^2,(Deltaboldsymbol{L})^2$, 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 $Lambda$.
We study the eigenvalue equation for the Cartesian coordinates observables $x_i$ on the fully $O(2)$-covariant fuzzy circle ${S^1_Lambda}_{Lambdainmathbb{N}}$ ($i=1,2$) and on the fully $O(3)$-covariant fuzzy 2-sphere ${S^2_Lambda}_{Lambdainmathbb{N}
We briefly report our recent construction of new fuzzy spheres of dimensions d=1,2 covariant under the full orthogonal group O(D), D=d+1. They are built by imposing a suitable energy cutoff on a quantum particle in D dimensions subject to a confining
We revise the unireps. of $U(2,2)$ describing conformal particles with continuous mass spectrum from a many-body perspective, which shows massive conformal particles as compounds of two correlated massless particles. The statistics of the compound (b
An algorithm to compute Connes spectral distance, adaptable to the Hilbert-Schmidt operatorial formulation of non-commutative quantum mechanics, was developed earlier by introducing the appropriate spectral triple and used to compute infinitesimal di
We study several formulations of zero-mass relativistic equations, stressing similarities between different frameworks. It is shown that all these massless wave equations have fermionic as well as bosonic solutions.