论文标题
两个严格的变形量化的散装边界渐近等效性
Bulk-boundary asymptotic equivalence of two strict deformation quantizations
论文作者
论文摘要
存在$ x_k = s（m_k（{\ Mathbb {c}}}）））$的严格变形量化的存在，$ k \ times k \ times k $矩阵的状态空间$ m_k（{\ mathbb {c}}）$，从既可以通过统计的台层）来登陆（与统计的poisson vercord and Manders and Manders and Man and Manders and Man and Man and Manders and Manders and Manders and Manders and Manders and Manders and Mander and and Manders” \ cite {lmv}。实际上，众所周知，由于增加$ k \ times k $矩阵$ m_k（{\ mathbb {c}}}）$的张力能力引起$ c^*$ - 代数的连续捆绑，$ i = \ i = \ {0 \} $ a_ {1/n} = m_k（{\ Mathbb {c}}）^{\ otimes n} $和$ a_0 = c（x_k）$，我们能够定义严格的变形量化，用$ x_k $ x_k $ X la rieffel，用量化的量子$ q_ p Q _}指定。 \ tilde {a} _0 \ rightarrow a_ {1/n} $，带有$ \ tilde {a} _0 $ a a_0 $的密集泊松subalgebra。类似的结果以符号歧管$ s^2 \ subset \ mathbb {r}^3 $而闻名，在这种情况下，在这种情况下，纤维$ a'_'_ {1/n} = m_ {n+1}（\ mathbb {c}） $ a_0'= c（s^2）$形成$ c^*$ - 代数的连续捆绑包在同一基本空间$ i $上，以及（先验不同的）量化量化量化量的量化，量化的量化。在本文中，我们专注于特定情况$ x_2 \ cong b^3 $（即$ \ mathbb {r}^3 $）的单位三球，并表明对于任何功能$ f \ in \ tilde {a} _0 $ $ \ lim_ {n \ to \ infty} ||（q_ {1/n}（f））| _ {\ text {sym}^n（\ mathbb {\ mathbb {c}^2）} - q_ {1/n} $ \ text {sym}^n（\ mathbb {c}^2）$表示$（\ mathbb {c}^2）^{n \ otimes} $的对称子空间。最后，我们提供了有关（量子）居里 - 韦斯模型的应用程序。
The existence of a strict deformation quantization of $X_k=S(M_k({\mathbb{C}}))$, the state space of the $k\times k$ matrices $M_k({\mathbb{C}})$ which is canonically a compact Poisson manifold (with stratified boundary) has recently been proven by both authors and K. Landsman \cite{LMV}. In fact, since increasing tensor powers of the $k\times k$ matrices $M_k({\mathbb{C}})$ are known to give rise to a continuous bundle of $C^*$-algebras over $I=\{0\}\cup 1/\mathbb{N}\subset[0,1]$ with fibers $A_{1/N}=M_k({\mathbb{C}})^{\otimes N}$ and $A_0=C(X_k)$, we were able to define a strict deformation quantization of $X_k$ à la Rieffel, specified by quantization maps $Q_{1/N}: \tilde{A}_0\rightarrow A_{1/N}$, with $\tilde{A}_0$ a dense Poisson subalgebra of $A_0$. A similar result is known for the symplectic manifold $S^2\subset\mathbb{R}^3$, for which in this case the fibers $A'_{1/N}=M_{N+1}(\mathbb{C})\cong B(\text{Sym}^N(\mathbb{C}^2))$ and $A_0'=C(S^2)$ form a continuous bundle of $C^*$-algebras over the same base space $I$, and where quantization is specified by (a priori different) quantization maps $Q_{1/N}': \tilde{A}_0' \rightarrow A_{1/N}'$. In this paper we focus on the particular case $X_2\cong B^3$ (i.e the unit three-ball in $\mathbb{R}^3$) and show that for any function $f\in \tilde{A}_0$ one has $\lim_{N\to\infty}||(Q_{1/N}(f))|_{\text{Sym}^N(\mathbb{C}^2)}-Q_{1/N}'(f|_{_{S^2}})||_N=0$, were $\text{Sym}^N(\mathbb{C}^2)$ denotes the symmetric subspace of $(\mathbb{C}^2)^{N \otimes}$. Finally, we give an application regarding the (quantum) Curie-Weiss model.
