Confinement,the Wilson loop and the gluon propagator

We show that if the gluon propagator has a highly singular infrared behavior (such as $\text{1}\text{q}{}^4$) in any one gauge then the Wilson loop must fall exponentially to zero with an area law, indicative of confinement. The proof of this claim follows from an inequality derived via a mean value theorem.     

极端条件两相界面流与反应流机理、模型与算法研究进展

清华大学喷雾燃烧与推进实验室长期专注于极高速、强可压和高瞬变等极端条件下的两相流和反应流前沿科学问题,并致力于应用基础研究成果解决航空航天动力与推进系统的关键技术难题。综述了实验室近些年在极端条件下两相流动和含化学反应流动物理机理、数理模型与数值算法等方面的研究进展。首先,介绍了实验室发展的耦合高瞬变相变过程的强可压缩气液两相界面流的数理模型和高精度数值方法,以及针对激波受气液(曲)界面约束情况下,描述非定常激波透射/反射(如波角、波强等物理量关系)的激波动力学分析方法。其次,基于上述模型、算法与分析方法,实验室研究了激波液滴相互作用、高速液滴撞击壁面等一系列问题,解析了上述高瞬变过程中复杂波系与界面的时空演化过程。以被激波或壁面冲击的液滴内流体空化初生为例,揭示了曲界面汇聚膨胀波诱导流体空化的机理,推导了预测空化初生位置的理论公式。最后,介绍了面向发动机燃烧室内的强可压缩两相喷雾反应流动,实验室开发了基于Euler-Lagrange框架的高性能数值仿真软件TURFsim,并成功用于真实复杂几何结构的航空发动机燃烧室和超声速燃烧室的数值模拟。以典型的超声速混合层流动数值模拟为例,总结了...     

Entropy, Superposition and Dynamical Maps

Aspects of quantum entropy and relative quantum entropy are discussed in the Hilbert model. It is shown that finite values of the relative entropy of states implies a superposition relation between the states. The property is studied in case of tensor product of states and for state reductions. A “Schmidt-like” state, derived from the reduced states, is considered. It is shown that its entropy, relative to the product of the reduced states, is not smaller than the entropy of the reduced states. The main existing results concerning the changement of superposition and entropy under dynamical map are recalled in a uniform way. A class of possible dynamical maps, not necessarily linear, is proposed that do not decrease the entropy.     

Randomized Wilson loops,reduced models and the large D expansion

Reduced models are matrix integrals believed to be related to the large N limit of gauge theories. These integrals are known to simplify further when the number of matrices D (corresponding to the number of space–time dimensions in the gauge theory) becomes large. Even though this limit appears to be of little use for computing the standard rectangular Wilson loop (which always singles out two directions out of D), a meaningful large D limit can be defined for a randomized Wilson loop (in which all D directions contribute equally). In this article, a proof-of-concept demonstration of this approach is given for the simplest reduced model (the original Eguchi–Kawai model) and the simplest randomization of the Wilson loop (Brownian sum over random walks). The resulting averaged Wilson loop displays a scale behavior strongly reminiscent of the area law.     

Chaotic Maps, Control Parameter, and Liapunov Exponent

We describe a one-dimensional chaotic map wherethe Liapunov exponent is a smooth function of a controlparameter.     

Zimmermann normal products and the Wilson expansion

The principal part and asymptotic form of the Wilson short distance expansion is derived in the framework of Bogoliubov-Parasiuk-Hepp-Zimmermann renormalized perturbation theory. Use is made of the Zimmermann normal product techniques with all work being performed in the scalar A⁴ model. The expansion of the product of four field operators is given as an explicit example.     

Non-local order in Mott insulators,duality and Wilson loops

It is shown that the Mott insulating and superfluid phases of bosons in an optical lattice may be distinguished by a non-local ‘parity order parameter’ which is directly accessible via single site resolution imaging. In one dimension, the lattice Bose model is dual to a classical interface roughening problem. We use known exact results from the latter to prove that the parity order parameter exhibits long range order in the Mott insulating phase, consistent with recent experiments by Endres et al. [M. Endres, M. Cheneau, T. Fukuhara, C. Weitenberg, P. Schauß, C. Gross, L. Mazza, M.C. Bañuls, L. Pollet, I. Bloch, et al., Science 334 (2011) 200]. In two spatial dimensions, the parity order parameter can be expressed in terms of an equal time Wilson loop of a non-trivial U(1)$U\left(1\right)$ gauge theory in 2+1$2+1$ dimensions which exhibits a transition between a Coulomb and a confining phase. The negative logarithm of the parity order parameter obeys a perimeter law in the Mott insulator and is enhanced by a logarithmic factor in the superfluid.     

Experimental susceptibility of Fe, Fe and Fe and the results of the Wilson theory

The low temperature susceptibility of the local moment system Fe has been measured by the Mössbauer effect. The susceptibility for this system and for the system Fe, Fe and Fe show general agreement with the prediction of the Wilson theory.     

Filtrations, Factorizations and Explicit Formulae for Harmonic Maps

We use filtrations of the Grassmannian model to produce explicit algebraic formulae for all harmonic maps of finite uniton number from a Riemann surface, and so all harmonic maps from the 2-sphere, to the unitary group for a general class of factorizations by unitons. We show how these specialize to give explicit formulae for such harmonic maps to each of the classical compact Lie groups and their inner symmetric spaces—the nonlinear σ-model of particle physics. Our methods also give an explicit Iwasawa decomposition of the algebraic loop group.     

Maps of Matrices and Portrait Maps of the Density Operators of Composite and Noncomposite Systems

We obtain a new inequality for arbitrary Hermitian matrices. We describe particular linear maps called the matrix portrait of arbitrary N × N matrices. The maps are obtained as analogs of partial tracing of density matrices of multipartite qudit systems. The structure of the maps is inspired by “portrait” map of the probability vectors corresponding to the action on the vectors by stochastic matrices containing either unity or zero matrix elements. We obtain new entropic inequalities for arbitrary qudit states including a single qudit and discuss entangled single qudit state. We consider in detail the examples of N = 3 and 4. Also we point out the possible use of entangled states of systems without subsystems (e.g., a single qudit) as a resource for quantum computations.     

Moyal Deformation,Seiberg–Witten Maps,and Integrable Models

A covariant formalism for Moyal deformations of gauge theory and differential equations which determine Seiberg–Witten maps is presented. Replacing the ordinary product of functions by the noncommutative Moyal product, noncommutative versions of integrable models can be constructed. We explore how a Seiberg–Witten map acts in such a framework. As a specific example, we consider a noncommutative extension of the principal chiral model.     

Fourier analysis and Wilson Loops

The quantum average of the Wilson Loop is computed through Fourier analysis of the potentials and functional integration over the coefficients. Simple results are obtained in the abelian case as well as in the N→∞ limit of the nonabelian theory.     

Small-distance-behaviour analysis and Wilson expansions

A previously described method to obtain the asymptotic forms of vertex functions at large momenta is, with the help of Wilson operator product expansion formulas, extended to momenta where the vertex functions of the zero-mass theory underlying the asymptotic forms are infrared singular. To obtain from asymptotic forms information on asymptotic behaviour requires assumptions on the behaviour of the zero-mass theory in the limit of infinite dilatation. One particular set of assumptions is discussed and found to pass a simple consistency test; this set of assumptions leads to power laws, or slight modifications thereof, with coupling-constant-independent exponents. The detailed discussion is given for the ⁴ model.