Generally covariant vs. gauge structure for conformal field theories

We introduce the natural lift of spacetime diffeomorphisms for conformal gravity and discuss the physical equivalence between the natural and gauge natural structure of the theory. Accordingly, we argue that conformal transformations must be introduced as gauge transformations (affecting fields but not spacetime point) and then discuss special structures implied by the splitting of the conformal group.     

Some aspects of conformal field theories on the plane and higher genus Riemann surfaces

We review some aspects of conformal field theories on the plane as well as on higher genus Riemann surfaces.     

Semi-infinite cohomology in conformal field theory and 2D gravity

We discuss various techniques for computing the semi-infinite cohomology of highest weight modules which arise in the BRST quantization of two dimensional field theories. In particular, we concentrate on two such theories - the G/H coset models and 2D gravity coupled to c ≤ 1 conformal matter.     

Conformally invariant field theories in two dimensions and corresponding statistical mechanical models

Belavin, Zamlodochikov and Polyakov have recently proposed a class of conformally invariant field theories in two dimension with exactly determined rational critical indices. We establish a tentative identification of a subset of these theories in terms of the O(n) model and theq-state Potts model in 2-dimensions for appropriaten andq. The results of this work were reported in the conference on “Structural Similarities in Exactly Solved Models” at I.T.P. Santa Barbara, August 1984.     

Arithmetic of Calabi–Yau varieties and rational conformal field theory

It is proposed that certain techniques from arithmetic algebraic geometry provide a framework which is useful to formulate a direct and intrinsic link between the geometry of Calabi–Yau manifolds and the underlying conformal field theory. Specifically, it is pointed out how the algebraic number field determined by the fusion rules of the conformal field theory can be derived from the number theoretic structure of the cohomological Hasse–Weil L-function determined by Artin’s congruent zeta function of the algebraic variety. In this context, a natural number theoretic characterization arises for the quantum dimensions in this geometrically determined algebraic number field.     

Boundary conformal field theory and tunneling of edge quasiparticles in non-Abelian topological states

We explain how (perturbed) boundary conformal field theory allows us to understand the tunneling of edge quasiparticles in non-Abelian topological states. The coupling between a bulk non-Abelian quasiparticle and the edge is due to resonant tunneling to a zero mode on the quasiparticle, which causes the zero mode to hybridize with the edge. This can be reformulated as the flow from one conformally invariant boundary condition to another in an associated critical statistical mechanical model. Tunneling from one edge to another at a point contact can split the system in two, either partially or completely. This can be reformulated in the critical statistical mechanical model as the flow from one type of defect line to another. We illustrate these two phenomena in detail in the context of the ν=5/2 quantum Hall state and the critical Ising model. We briefly discuss the case of Fibonacci anyons and conclude by explaining the general formulation and its physical interpretation.     

An exact solution of the renormalization-group equations for the mean field theory of stable and metastable states

An exact solution of the renormalization-group equations corresponding to the mean field theory of stable and metastable states is given which yields the correct free energies for these states. An unusual feature of this solution is that the renormalized Hamiltonian in the two-phase region becomes a multivalued function of the order parameter for all values of the length rescaling parameter beyond a certain critical value. This is closely related to the multivaluedness of the free energy as a function of magnetic field which characterizes the classical theory of metastable and unstable states. As a consequence of this multivaluedness, the trajectory flow in the space of coupling constants exhibits unusual bifurcation. This leads to difficulties in evaluating the metastable and unstable free energies by a trajectory integral of the spin-independent term, which can be resolved by an extension of the standard formalism.This work was supported by NSF grant #550-346-01 (JDG) and a U.S.-Japan Cooperative Science grant (KK and JDG).     

Parametric smoothness and self-scaling of the statistical properties of a minimal climate model: What beyond the mean field theories?

A quasi-geostrophic intermediate complexity model of the mid-latitude atmospheric circulation is considered, featuring simplified baroclinic conversion and barotropic convergence processes. The model undergoes baroclinic forcing towards a given latitudinal temperature profile controlled by the forced equator-to-pole temperature difference T_E. As T_E increases, a transition takes place from a stationary regime-Hadley equilibrium-to a periodic regime, and eventually to a chaotic regime where evolution takes place on a strange attractor. The attractor dimension, metric entropy, and bounding box volume in phase space have a smooth dependence on T_E, which results in power-law scaling properties. Power-law scalings with respect to T_E are detected also for the statistical properties of global physical observables — the total energy of the system and the averaged zonal wind. The scaling laws, which constitute the main novel result of the present work, can be thought to result from the presence of a statistical process of baroclinic adjustment, which tends to decrease the equator-to-pole temperature difference and determines the properties of the attractor of the system. The self-similarity could be of great help in setting up a theory for the overall statistical properties of the general circulation of the atmosphere and in guiding-on a heuristic basis-both data analysis and realistic simulations, going beyond the unsatisfactory mean field theories and brute force approaches. A leading example for this would be the possibility of estimating the sensitivity of the output of the system with respect to changes in the parameters.     

The QM/MM approach for wavefunctions,energies and response functions within self-consistent field and coupled cluster theories

This paper presents the coupled cluster/molecular mechanics (CC/MM) and self-consistent field/molecular mechanics (SCF/MM) approaches for wavefunctions, energies and response properties. Two physically different theories are derived, the mean-field and the direct-field interaction approaches, together with expressions for the optimization condition of both variational and non-variational wavefunctions and energies. Also derived are the linear response functions at the CC/MM and SCF/MM levels of theory, and the expressions are compared with the vacuum response functions.     

Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations

For singular perturbation problems, the renormalization group (RG) method of Chen, Goldenfeld, and Oono [Phys. Rev. E. 49 (1994) 4502–4511] has been shown to be an effective general approach for deriving reduced or amplitude equations that govern the long time dynamics of the system. It has been applied to a variety of problems traditionally analyzed using disparate methods, including the method of multiple scales, boundary layer theory, the WKBJ method, the Poincaré–Lindstedt method, the method of averaging, and others. In this article, we show how the RG method may be used to generate normal forms for large classes of ordinary differential equations. First, we apply the RG method to systems with autonomous perturbations, and we show that the reduced or amplitude equations generated by the RG method are equivalent to the classical Poincaré–Birkhoff normal forms for these systems up to and including terms of , where is the perturbation parameter. This analysis establishes our approach and generalizes to higher order. Second, we apply the RG method to systems with nonautonomous perturbations, and we show that the reduced or amplitude equations so generated constitute time-asymptotic normal forms, which are based on KBM averages. Moreover, for both classes of problems, we show that the main coordinate changes are equivalent, up to translations between the spaces in which they are defined. In this manner, our results show that the RG method offers a new approach for deriving normal forms for nonautonomous systems, and it offers advantages since one can typically more readily identify resonant terms from naive perturbation expansions than from the nonautonomous vector fields themselves. Finally, we establish how well the solution to the RG equations approximates the solution of the original equations on time scales of .     

场方法的改进及其在积分Riemann-Cartan空间运动方程中的应用

和Hamilton-Jacobi方法类似,Vujanovi?场方法把求解常微分方程组特解的问题转化为寻找一个一阶拟线性偏微分方程(基本偏微分方程)完全解的问题,但Vujanovi?场方法依赖于求出基本偏微分方程的完全解,而这通常是困难的,这就极大地限制了场方法的应用.本文将求解常微分方程组特解的Vujanovi?场方法改进为寻找动力学系统运动方程第一积分的场方法,并将这种方法应用于一阶线性非完整约束系统Riemann-Cartan位形空间运动方程的积分问题中.改进后的场方法指出,只要找到基本偏微分方程的包含m(m≤ n,n为基本偏微分方程中自变量的数目)个任意常数的解,就可以由此找到系统m个第一积分.特殊情况下,如果能够求出基本偏微分方程的完全解(完全解是m=n时的特例),那么就可以由此找到≤系统全部第一积分,从而完全确定系统的运动.Vujanovi?场方法等价于这种特殊情况.     

The mean spherical model in a random external field and the replica method

We provide a quick elementary solution of the mean spherical model in a random external field. This also allows an immediate proof of the self-averaging property of the free energy. We calculate the free energy by means of the replica method, i.e., for any (not necessarily integer) replica numbern, and show that when a phase transition occurs the limits andn 0 are not interchangeable.     

The rank-size scaling law and entropy-maximizing principle

The rank-size regularity known as Zipf’s law is one of the scaling laws and is frequently observed in the natural living world and social institutions. Many scientists have tried to derive the rank-size scaling relation through entropy-maximizing methods, but they have not been entirely successful. By introducing a pivotal constraint condition, I present here a set of new derivations based on the self-similar hierarchy of cities. First, I derive a pair of exponent laws by postulating local entropy maximizing. From the two exponential laws follows a general hierarchical scaling law, which implies the general form of Zipf’s law. Second, I derive a special hierarchical scaling law with the exponent equal to 1 by postulating global entropy maximizing, and this implies the pure form of Zipf’s law. The rank-size scaling law has proven to be one of the special cases of the hierarchical scaling law, and the derivation suggests a certain scaling range with the first or the last data point as an outlier. The entropy maximization of social systems differs from the notion of entropy increase in thermodynamics. For urban systems, entropy maximizing suggests the greatest equilibrium between equity for parts/individuals and efficiency of the whole.     

Isotropisation of flat homogeneous Bianchi type <Emphasis Type="Italic">I</Emphasis> model with a non minimally coupled and massive scalar field

In previous works, we studied the isotropisation of some Bianchi class A models with a minimally coupled scalar field. In this paper we extend these results, in the special case of a Bianchi type I model, to a non minimally coupled scalar field. The Universe isotropisation for the Brans-Dicke and low energy string theories are studied.     

四角对称晶场中4B1(3d3)态离子的磁相互作用及其自旋哈密顿参量研究

基于完全对角化方法，研究了⁴B₁(3d³)态 离子在四角对称晶场中的磁相互作用，分析了自旋哈密顿参量(b⁰₂, g_∥, g_⊥ , Δg)的微观起源.结果表明 ：在被考虑的大部分晶场区域，人们通常考虑的SO(spin-orbit)磁相互作用的贡献最为重要 ；然而，对于零场分裂参量b⁰₂而言，来自其他机理(包 括SS(spin-orbit)，SOO(sp in-other-orbit)，SO-SS-SOO)的贡献在大部分晶场区域超过了20％；在部分晶场区域，其 他机理的贡献甚至超过SO机理的贡献.详细地分析了Macfarlane 零场分裂参量b^{0₂ 近似三阶微扰理论的收敛性，结果表明：该理论在大部分晶场区域收敛性较差.讨论了3d3}态离子第一激发态²E_g分裂的微观起源.并利用 群论方法解 释了在C_4v和C_3v对称晶场中²E_{g< /sub>态分裂的不同机理. 关键词： 4B1(3d³)态离子')" href="#">⁴B1(3d³)态离子 磁相互作用 自旋哈密 顿参量 完全对角化方法(CDM) 微扰理论方法(PTM)}