Connes distance of 2D harmonic oscillators in quantum phase space

We study the Connes distance of quantum states of two-dimensional (2D) harmonic oscillators in phase space. Using the Hilbert-Schmidt operatorial formulation, we construct a boson Fock space and a quantum Hilbert space, and obtain the Dirac operator and a spectral triple corresponding to a four-dimensional (4D) quantum phase space. Based on the ball condition, we obtain some constraint relations about the optimal elements. We construct the corresponding optimal elements and then derive the Connes distance between two arbitrary Fock states of 2D quantum harmonic oscillators. We prove that these two-dimensional distances satisfy the Pythagoras theorem. These results are significant for the study of geometric structures of noncommutative spaces, and it can also help us to study the physical properties of quantum systems in some kinds of noncommutative spaces.     

原子核单粒子共振态的研究进展

共振是自然界普遍存在的有趣现象。在核物理领域,存在许多奇特现象,共振在其中扮演着重要角色。介绍了近期发展的RMF-CSM、RMF-CGF和RMF-CMR方法及其对原子核单粒子共振态的研究。给出了120Sn,31Ne等核的单粒子共振态的能量和宽度及其随质量数、形变的变化规律,分析了19C,31Ne和39Mg等核的晕形成的物理机制和在N=20附近能级反转的原因。特别是新发展的RMF-CMR方法,在描述稳定核和奇特核方面都取得了成功,支持Zr同位素存在巨晕的预言。Resonance is an interesting phenomenon in nature. In nuclear physics, resonance plays an important role in the formation of many exotic phenomena. This paper introduces the recently developed RMF-CSM, RMFCGF, and RMF-CMR methods and their researches on nuclear single-particle resonances. The energies and widths of the single-particle resonant states in 120Sn and 31Ne and their evolution to mass number and deformation are given. In addition, the physical mechanism of the halo formation in 19C, 31Ne and 39Mg and the cause of energy level inversion near N=20 are analyzed. In particular, the newly developed RMF-CMR approach has been successful in describing stable and exotic nuclei and supports the prediction that Zr isotopes exist in a giant halo.     

Modified Form of Wigner Functions for Non-Hamiltonian Systems

Quantization of non-Hamiltonian systems （such as damped systems） often gives rise to complex spectra and corresponding resonant states, therefore a standard form calculating Wigner functions cannot lead to static quasiprobability distribution functions. We show that a modified form of the Wigner functions satisfies a ＊-genvalue equation and can be derived from deformation quantization for such systems.     

Wigner Functions for the Bateman System on Noncommutative Phase Space

We study an important dissipation system, i.e. the Bateman model on noncommutative phase space. Using the method of deformation quantization, we calculate the Exp functions, and then derive the Wigner functions and the corresponding energy spectra.     

Wigner Functions for Non-Hamiltonian Systems on Noncommutative Space

We propose a modified form of Wigner functions for generic non-Hamiltonian systems on noncommutative space and prove that it satisfies the corresponding ＊-genvalue equation. In addition, as an example, we derive exact energy spectra and Wigner functions for a non-Hamiltonian toy model on the noncommutative space.     

基于Wigner函数的一些量子熵的定义

本文分别介绍了冯诺依曼熵、线性熵、Rényi熵和Tsallis熵等几种量子熵,以及它们在相空间中基于Wigner函数的几种不同的定义.通过理论推导和具体的数值计算,证明了文献中利用Wigner函数的绝对值来定义相空间量子熵是不合适的,而本文给出的利用Wigner函数的Moyal星乘积定义的量子熵与通常利用密度矩阵定义的量子熵结果是相一致的.     

Quantum Field Theory with a Minimal Length Induced from Noncommutative Space

From the inspection of noncommutative quantum mechanics, we obtain an approximate equivalent relation for the energy dependence of the Planck constant in the noncommutative space, which means a minimal length of the space. We find that this relation is reasonable and it can inherit the main properties of the noncommutative space. Based on this relation, we derive the modified Klein-Gordon equation and Dirac equation. We investigate the scalar field and φ4 model and then quantum electrodynamics in our theory, and derive the corresponding Feynman rules. These results may be considered as reasonable approximations to those of noncommutative quantum field theory. Our theory also shows a connection between the space with a minimal length and the noncommutative space.