论文标题
平均场量子自旋系统的经典极限
The classical limit of mean-field quantum spin systems
论文作者
论文摘要
对两个球体的严格变形量化理论$ s^2 \ subset \ mathbb {r}^3 $用于证明存在平均场量子旋转链的经典限制，其随后的汉密尔顿人随之而来的汉密尔顿人用$ h_n $表示，而$ n $ n $ n $表示该站点的数量。实际上，由于纤维$ a_ {1/n} = m_ {n+1}（\ mathbb {c}）$和$ a_0 = c（s^2）$形成$ c^*$ - 代数 - 基本空间上的代数$ i = \ i = \ i = \ cup 1/\ cup 1/\ mathb supb n} n} $ a_0 $的严格变形量化，其中某些量化图指定量化$ q_ {1/n}：\ tilde {a} _0 \ rightArrow a_ {1/n} $，带有$ \ tilde {a} _0 $ a} _0 $ a lenge poisson subalgebra of $ a__0 $。现在给定了一系列这样的$ h_n $，我们表明，在某些假设下，$ h_n $的$ψ_n$ $ψ_n$在某种意义上是经典的限制，从某种意义上说：= \ lim_ {n \ to \ fo \ fo \ in \ to \ infty} \ langle} \ langle mente} $ a_0 $由$ω_0（f）= \ frac {1} {n} \ sum_ {i = 1}^nf（ω_i）$给出，其中$ n $是一些自然数字。我们提供了有关自发对称性断裂（SSB）的应用程序，此外，我们表明，这种平均场量子自旋系统的频谱收敛到一些多项式范围，其中三个真实变量仅限于Sphere $ s^2 $。
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 $ψ_N$ of $H_N$ has a classical limit in the sense that $ω_0(f):=\lim_{N\to\infty}\langleψ_N,Q_{1/N}(f)ψ_N\rangle$ exists as a state on $A_0$ given by $ω_0(f)=\frac{1}{n}\sum_{i=1}^nf(Ω_i)$, 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 $S^2$.
