The theory of strict deformation quantization of the two sphere $S^2\subset\mathbb{R}^3$ is used to prove the existence of the classical limit of mean-field quantum spin chains, whose ensuing Hamiltonians are denoted by $H_N$ and where $N$ indicates the number of sites. Indeed, since the fibers $A_{1/N}=M_{N+1}(\mathbb{C})$ and $A_0=C(S^2)$ form a continuous bundle of $C^*$-algebras over the base space $I=\{0\}\cup 1/\mathbb{N}^*\subset[0,1]$, one can define a strict deformation quantization of $A_0$ where quantization is specified by certain quantization maps $Q_{1/N}: \tilde{A}_0 \rightarrow A_{1/N}$, with $\tilde{A}_0$ a dense Poisson subalgebra of $A_0$. Given now a sequence of such $H_N$, we show that under some assumptions a sequence of eigenvectors $\psi_N$ of $H_N$ has a classical limit in the sense that $\omega_0(f):=\lim_{N\to\infty}\langle\psi_N,Q_{1/N}(f)\psi_N\rangle$ exists as a state on $A_0$ given by $\omega_0(f)=\frac{1}{n}\sum_{i=1}^nf(\Omega_i)$, where $n$ is some natural number. We give an application regarding spontaneous symmetry breaking (SSB) and moreover we show that the spectrum of such a mean-field quantum spin system converges to the range of some polynomial in three real variables restricted to the sphere $S^2$.

中文翻译：

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平均场量子自旋系统的经典极限

两个球体的严格变形量化理论$S^2\subset\mathbb{R}^3$被用来证明平均场量子自旋链的经典极限的存在，其随后的哈密顿量用$H_N表示$，其中 $N$ 表示站点的数量。事实上，由于纤维 $A_{1/N}=M_{N+1}(\mathbb{C})$ 和 $A_0=C(S^2)$ 形成了一个连续的 $C^*$-代数丛在基空间 $I=\{0\}\cup 1/\mathbb{N}^*\subset[0,1]$ 上，可以定义 $A_0$ 的严格变形量化，其中量化由特定量化指定映射 $Q_{1/N}：\tilde{A}_0 \rightarrow A_{1/N}$，其中 $\tilde{A}_0$ 是 $A_0$ 的密集泊松子代数。现在给定这样的 $H_N$ 序列，我们表明在某些假设下，$H_N$ 的特征向量 $\psi_N$ 序列在 $\omega_0(f) 的意义上具有经典极限：=\lim_{N\to\infty}\langle\psi_N,Q_{1/N}(f)\psi_N\rangle$ 作为 $A_0$ 上的状态存在，由 $\omega_0(f)=\frac{1 }{n}\sum_{i=1}^nf(\Omega_i)$，其中 $n$ 是某个自然数。我们给出了一个关于自发对称性破缺 (SSB) 的应用，而且我们证明了这样一个平均场量子自旋系统的频谱收敛到限制在球体 $S^2$ 的三个实变量中的一些多项式的范围。