Increasing tensor powers of the [Formula: see text] matrices [Formula: see text] are known to give rise to a continuous bundle of [Formula: see text]-algebras over [Formula: see text] with fibers [Formula: see text] and [Formula: see text], where [Formula: see text], the state space of [Formula: see text], which is canonically a compact Poisson manifold (with stratified boundary). Our first result is the existence of a strict deformation quantization of [Formula: see text] à la Rieffel, defined by perfectly natural quantization maps [Formula: see text] (where [Formula: see text] is an equally natural dense Poisson subalgebra of [Formula: see text]).We apply this quantization formalism to the Curie–Weiss model (an exemplary quantum spin with long-range forces) in the parameter domain where its [Formula: see text] symmetry is spontaneously broken in the thermodynamic limit [Formula: see text]. If this limit is taken with respect to the macroscopic observables of the model (as opposed to the quasi-local observables), it yields a classical theory with phase space [Formula: see text] (i.e. the unit three-ball in [Formula: see text]). Our quantization map then enables us to take the classical limit of the sequence of (unique) algebraic vector states induced by the ground state eigenvectors [Formula: see text] of this model as [Formula: see text], in which the sequence converges to a probability measure [Formula: see text] on the associated classical phase space [Formula: see text]. This measure is a symmetric convex sum of two Dirac measures related by the underlying [Formula: see text]-symmetry of the model, and as such the classical limit exhibits spontaneous symmetry breaking, too. Our proof of convergence is heavily based on Perelomov-style coherent spin states and at some stage it relies on (quite strong) numerical evidence. Hence the proof is not completely analytic, but somewhat hybrid.

中文翻译：

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Mk(ℂ) 状态空间的严格变形量化及其在居里-魏斯模型中的应用

[公式：见文本] 矩阵 [公式：见文本] 的张量幂的增加已知会产生 [公式：见文本]-代数在 [公式：见文本] 上的连续束 [公式：见文本] ] 和 [Formula: see text]，其中 [Formula: see text]，[Formula: see text] 的状态空间，典型地是紧致 Poisson 流形（具有分层边界）。我们的第一个结果是 [公式：见文本] à la Rieffel 的严格变形量化的存在，由完美自然的量化映射 [公式：见文本] 定义（其中 [公式：见文本] 是同样自然的密集泊松子代数[公式：见正文]）。我们将这种量化形式应用于 Curie-Weiss 模型（具有长程力的示例性量子自旋）在参数域中，其中 [公式：见文本]对称性在热力学极限[公式：见文本]中自发破坏。如果这个限制是针对模型的宏观可观测量（与准局部可观测量相反），它会产生一个具有相空间的经典理论[公式：见正文]（即[公式：见正文]）。然后，我们的量化图使我们能够将由该模型的基态特征向量[公式：见文本]诱导的（唯一）代数向量状态序列的经典极限作为[公式：见文本]，其中序列收敛到在相关的经典相空间[公式：见文本]上的概率测量[公式：见文本]。该度量是两个狄拉克度量的对称凸和，由模型的基础[公式：参见文本]-对称性相关，因此，经典极限也表现出自发的对称性破缺。我们的收敛证明很大程度上基于 Perelomov 风格的相干自旋态，并且在某些阶段它依赖于（非常强大的）数值证据。因此，证明不是完全分析的，而是有点混合的。