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 .

中文翻译：

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关于一些新的模糊球体上的局部和相干状态

我们在 O ( D ) 等变模糊球体 $$S^d_\Lambda $$ S Λ d ( $$d=1,2$$ d = 1 , 2 , $$ D=d+1$$ D = d + 1 ) 构建于 Fiore 和 Pisacane (J Geom Phys 132:423–451, 2018) 并研究它们在配置空间和角动量空间中的定位。这些定位最好通过 O ( D ) 不变的平方空间和角动量不确定性 $$(\Delta \varvec{x})^2,(\Delta \varvec{L})^2$$ ( Δ x ) 2 , ( Δ L ) 2 在环境欧几里得空间 $$\mathbb {R}^D$$ RD 中。我们还为 $$(\Delta \varvec{x})^2,(\Delta \varvec{L})^2$$ ( Δ x ) 2 , ( Δ L ) 2 ，并部分调查哪些 SCS 可能使这些边界饱和。特别是，我们确定了最优局部相干态的 O ( D ) 等变系统，它们是最接近 $$S^d$$S d 的经典态（即点）的量子态。我们将结果与其在可交换 $$S^d$$S d 上的类似物进行比较。我们还表明，在 $$S^2_\Lambda $$ S Λ 2 上，我们的最优定位状态比具有相同截止值 $$\Lambda $$ Λ 的 Madore-Hoppe 模糊球上的状态更好。