多维量子标准形、Moyal星积和环面量子化

Spectral Distances: Results for Moyal Plane and Noncommutative Torus

The spectral distance for noncommutative Moyal planes is considered in the framework of a non compact spectral triple recently proposed as a possible noncommutative analog of non compact Riemannian spin manifold. An explicit formula for the distance between any two elements of a particular class of pure states can be determined. The corresponding result is discussed. The existence of some pure states at infinite distance signals that the topology of the spectral distance on the space of states is not the weak * topology. The case of the noncommutative torus is also considered and a formula for…

多维环面辛线性映射的算术量子唯一遍历性

Arithmetic Quantum Unique Ergodicity for Symplectic Linear Maps of the Multidimensional Torus

我们期待的期望值的量化线性辛映射的多维环面和它们的分布在半经典极限。我们构建的超级疤痕，是稳定的算术对称的系统和本地化周围不变的流形。我们证明了这些超疤痕存在时，只有有各向同性的有理子空间，线性映射下不变。在没有疤痕的情况下，我们计算了去对称化系统的矩阵元的涨落方差，并对它们的极限分布提出了一个猜想。

Entropy of quantum limits for symplectic linear maps of the multidimensional torus

In the case of a linear symplectic map A of the 2d-torus, semiclassical measures are A-invariant probability measures associated to sequences of high energy quantum states. Our main result is an explicit lower bound on the entropy of any semiclassical measure of a given quantizable matrix A in Sp(2d,Z). In particular, our result implies that if A has an eigenvalue outside the unit circle, then a semiclassical measure cannot be carried by a closed orbit of A.

Torus as phase space: Weyl quantization, dequantization and Wigner formalism

The Weyl quantization of classical observables on the torus (as phase space) without regularity assumptions is explicitly computed. The equivalence class of symbols yielding the same Weyl operator is characterized. The Heisenberg equation for the dynamics of general quantum observables is written through the Moyal brackets on the torus and the support of the Wigner transform is characterized. Finally, a dequantization procedure is introduced that applies, for instance, to the Pauli matrices. As a result we obtain the corresponding classical symbols.

接近于KT频率向量的多维环面上流的重整化

Renormalisation of flows on the multidimensional torus close to a KT frequency vector

利用作用在d-环面上的向量场空间上的重整化算子R，证明了等价于常数的向量场子流形的存在性.这个结果是由R的一个不动点w的存在性得到的，它是双曲的.这是对称为Koch型的常数向量场w的某类KT所做的。变换R是使用时间重新标度，一个线性变化的基础加上一个周期性的非线性映射同位素的身份，我们得到了一个“同伦技巧”。

The Multidimensional Berry-Hannay Model

The aim of this paper is to construct the Berry-Hannay model of quantum mechanics on a 2n-dimensional symplectic torus. We construct a simultaneous quantization of the algebra $\cal A$ of functions on the torus and the linear symplectic group $\G = \Sp(\mathrm{2n},\Z)$. In the construction we use the quantum torus $\A$, which is a deformation of $\cal A$, together with a $\G$-action on it. We obtain the quantization via the action of $\G$ on the set of equivalence classes of irreducible representations of $\A$. For $\h \in \Q$ this action has a unique fixed point. This gives a canonical projec…