Asymptotic localization of symbol correspondences for spin systems and sequential quantizations of $S^2$

Abstract

Quantum or classical mechanical systems symmetric under $SU(2)$ are called spin systems. A $SU(2)$-equivariant map from $(n+1)$-square matrices to functions on the $2$-sphere, satisfying some basic properties, is called a spin-$j$ symbol correspondence ($n=2j\in\mathbb N$). Given a spin-$j$ symbol correspondence, the matrix algebra induces a twisted $j$-algebra of symbols. In this paper, we establish a new, more intuitive criterion for when the Poisson algebra of smooth functions on the $2$-sphere emerges asymptotically ($n\to\infty$) from the sequence of twisted $j$-algebras. This more geometric criterion, which in many cases is equivalent to the numerical criterion obtained in [Rios&Straume], is given in terms of a classical (asymptotic) localization of symbols of all projectors (quantum pure states) in a certain family. For some important kinds of symbol correspondence sequences, such a classical localization condition is equivalent to asymptotic emergence of the Poisson algebra. But in general, this classical localization condition is stronger than Poisson emergence. We thus also consider some weaker notions of asymptotic localization of projector-symbols. Finally, we obtain some relations between asymptotic localization of a symbol correspondence sequence and its sequential quantizations of the classical spin system, after carefully developing a theory of sequential quantizations of smooth functions on $S^2$ and their asymptotic actions on a ground Hilbert space.Comment: slight edition of expanded version, 56 page