Asymptotic localization of symbol correspondences for spin systems and sequential quantizations of $S^2$
P. A. S. Alcantara, P. de M. Rios
TL;DR
The paper investigates how the Poisson algebra of smooth functions on the 2-sphere emerges from finite-dimensional quantum spin-j systems in the j→∞ limit by studying SU(2)-equivariant symbol correspondences W^j. It introduces a geometric, asymptotic localization criterion based on the symbols of J_3-invariant projectors, and connects this to a broader framework of sequential quantizations that act on well-nested Hilbert spaces. A key result is the splitting of symbol correspondences into mutually exclusive classes (isometric, mapping-positive, positive-dual) and the identification of precise conditions under which classical localization implies Poisson-type limits, as well as the general Edmonds-based criteria for Poisson behavior in the general case. The work further extends to mu-analytic localization, explores concrete quantization constructions, and demonstrates how asymptotic localization governs the convergence of quantized functions to their classical counterparts. Collectively, these findings sharpen the semiclassical understanding of spin systems and provide concrete criteria for when quantum-to-classical transition faithfully recovers Poisson geometry.
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 S^2, 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 the first part of this paper, we establish a more intuitive criterion for when the Poisson algebra of smooth functions on $S^2$ 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 [20] for describing symbol correspondence sequences of (anti-)Poisson type, is now 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, the classical localization condition is stronger than Poisson emergence. We thus also consider some weaker notions of asymptotic localization of projector-symbols. In the second part of this paper, for each sequence of symbol correspondences of (anti-)Poisson type, we define the sequential quantization of a smooth function on $S^2$ and its asymptotic operator acting on a ground Hilbert space. Then, after presenting some concrete examples of these constructions, we obtain some relations between asymptotic localization of a symbol correspondence sequence and the asymptotics of its sequential quantization of smooth functions on $S^2$.