Gauge symmetry as symmetry of matrix coordinates

We propose a new point of view on gauge theories, based on taking the action of symmetry transformations directly on the space coordinates. Via this approach the gauge fields are not introduced at the first step, and they can be interpreted as fluctuations around some classical solutions of the model. The new point of view is connected to the lattice formulation of gauge theories, and the parameter of the non-commutativity of the coordinates appears as the lattice spacing parameter. Through the statements concerning the continuum limit of lattice gauge theories, the suggestion arises that the non-commutative spaces are the natural ones to formulate gauge theories at strong coupling. Via this point of view, a close relation between the large-N limit of gauge theories and string theory can be made manifest. Received: 16 June 2000 / Published online: 8 September 2000     

On the relevance of matrix coordinates for the inside of baryons

It is argued that one natural choice for the coordinates of the constituents of a baryonic state in a SU(N) gauge theory is the choice of hermitian matrices. It is discussed that the relevance of matrix coordinates is supported at least by the restricted form of the color symmetry. Based on the previous investigations in this direction, the consequences of this idea are reviewed. The model has been considered in which it originates in the D0-branes of the string theory. Received: 15 August 2002 / Published online: 15 January 2003 RID="a" ID="a" e-mail: fath@iasbs.ac.ir     

Electrodynamics on matrix space: non-Abelian by coordinates

Faddeev and Niemi have proposed a decomposition of SU(N) Yang–Mills theory in terms of new variables, appropriate for describing the theory in the infrared limit. We extend this method to SO(2N) Yang–Mills theory. We find that the SO(2N) connection decomposes according to irreducible representations of SO(N). The low-energy limit of the decomposed theory is expected to describe soliton-like configurations with nontrivial topological numbers. How the method of decomposition generalizes for SO(2N+1) Yang–Mills theory is also discussed. Received: 22 November 2000 / Published online: 8 June 2001     

Low-lying spectrum of the Y-string three-quark potential using hyper-spherical coordinates

We calculate the energies of three-quark states with definite permutation symmetry (i.e. of SU(6) multiplets) in the N=0, 1, 2 shells, confined by the Y-string three-quark potential. The exact Y-string potential consists of one term, the so-called three-string term, and three angle-dependent two-string terms. Due to this technical complication we treat the problem at three increasingly accurate levels of approximation: (1) the (approximate) three-string potential expanded to first order in trigonometric functions of hyper-spherical angles; (2) the (approximate) three-string potential to all orders in the power expansion in hyper-spherical harmonics, but without taking into account the transition(s) to two-string potentials; (3) the exact minimal-length string potential to all orders in a power expansion in the hyper-spherical harmonics, and taking into account the transition(s) to two-string potentials. We show the general trend of improvement of these approximations: the exact non-perturbative corrections to the total energy are of the order of one per cent, as compared with approximation (2), yet the exact energy differences between the [20,1⁺],[70,2⁺],[56,2⁺],[70,0⁺]-plets are shifted to 2:2:0.9, from the Bowler and Tynemouth separation rule 2:2:1, which is obeyed by approximation (2) at the one per cent level. The precise value of the energy separation of the first radial excitation (“Roper”) [56^′,0⁺]-plet from the [70,1⁻]-plet depends on the approximation, but does not become negative, i.e. the “Roper” remains heavier than the odd-parity [70,1⁻]-plet in all of our approximations.     

Complex coordinates and Kähler potential for the Atiyah-Hitchin metric

In the case of self dual metrics, we detail and put together the Geroch three-dimensional formulation of the Einstein equations and the Boyer-Finley results on Killing vectors to obtain a set of complex coordinates and a Kähler potential for the Atiyah-Hitchin metric.     

An effective scaling frequency factor method for scaling of harmonic vibrational frequencies: The use of redundant primitive coordinates

A modified effective scaling frequency factor (ESFF) method that takes advantage of the potential energy distribution (PED) coefficients calculated in the basis of redundant primitive internal coordinates is presented. This approach is simpler and more flexible than that based on the natural internal coordinates. The sets of optimal scaling factors for routine 9- and 11-parameter ESFF calculations based on B3LYP/6-311G∗∗ force fields are derived from Baker’s training set of 30 molecules (660 frequencies). The calculated root-mean-square (RMS) deviations for all frequencies are 11.42 and 11.44 cm⁻¹ for 9- and 11-parameter scaling, respectively. They are somewhat lower than in the case of ordinary ESFF calculations. The new sets of factors seem to be particularly well suited for scaling of frequencies in the middle region of the vibrational spectra (1000-2500 cm⁻¹) - the RMS values in this range are 8.37 for 9-, and 9.56 cm⁻¹ for 11-parameter scaling. These values are to be compared with 9.20 and 10.29 cm⁻¹, respectively, calculated within the natural coordinates based ESFF formalism.     

Mechanization of operations within an algebraic structure: Application to the transformation of the potential energy function of a linear triatomic molecule from internal coordinates to normal coordinates

In this article a program is described for the automatic computation of the coefficients involved in the expansion, with respect to normal coordinates, of the potential energy function of a linear triatomic molecule, starting from the coefficients of the expansion of the same function with respect to internal coordinates. The two expansions are limited to the terms of degree 6 with respect to the coordinates. This program, which is available upon request to the authors, uses a new method presented in a previous article.     

On the calculation of the transition matrix for an arbitrary potential

In this paper the calculation of the elements of the transition operator in the angular momentum representation is carried out. It is shown that these matrix elements can be expressed in terms of surface integrals over the surrounded surface in the usual case of scattering of a plane wave by an arbitrary short range potential as well as in the case of scattering of a spherical wave by a surrounding potential field.     

Vaidya spacetime in the diagonal coordinates

We have analyzed the transformation from initial coordinates (v, r) of the Vaidya metric with light coordinate v to the most physical diagonal coordinates (t, r). An exact solution has been obtained for the corresponding metric tensor in the case of a linear dependence of the mass function of the Vaidya metric on light coordinate v. In the diagonal coordinates, a narrow region (with a width proportional to the mass growth rate of a black hole) has been detected near the visibility horizon of the Vaidya accreting black hole, in which the metric differs qualitatively from the Schwarzschild metric and cannot be represented as a small perturbation. It has been shown that, in this case, a single set of diagonal coordinates (t, r) is insufficient to cover the entire range of initial coordinates (v, r) outside the visibility horizon; at least three sets of diagonal coordinates are required, the domains of which are separated by singular surfaces on which the metric components have singularities (either g ₀₀ = 0 or g ₀₀ = ∞). The energy–momentum tensor diverges on these surfaces; however, the tidal forces turn out to be finite, which follows from an analysis of the deviation equations for geodesics. Therefore, these singular surfaces are exclusively coordinate singularities that can be referred to as false fire-walls because there are no physical singularities on them. We have also considered the transformation from the initial coordinates to other diagonal coordinates (η, y), in which the solution is obtained in explicit form, and there is no energy–momentum tensor divergence.     

Half-shellT matrix for Coulomb-modified Graz separable potential

We construct a closed form expression for the off-shell Jost function for scattering by the Coulomb-distorted Graz separable potential and express it in the ‘maximal reduced form’. Our result is particularly suitable for numerical computation. We present a case study in support of this and examine the role of Coulomb interaction in thep — p half-shell scattering in the¹ S ₀ channel.     

Instantons in non-Cartesian coordinates

The multi-instanton solutions of ’t Hooft and of Jackiw, Nohl, and Rebbi are generalized to the case of curvilinear coordinates. The resulting formulas are considerably simplified if the transformation of coordinates is supplemented with a gauge transformation. As a result, the gauge field develops a term that has the same form as in Cartesian coordinates and which describes pseudoparticles and a compensating addition of a geometric origin (it is determined by the coordinate frame used). The singularities of the compensating field are irrelevant to physical quantities, but they can affect gauge-dependent quantities.     

Morse 势中矩阵元Xνν′、Pνν′的计算

矩阵元ＸＶＶ’和ＰＶＶ’在有关量子力学问题中有相关重要的应用，本文利用广义拉盖尔多项式的正交性和分部积分的方法，推导Ｍｏｒｓｅ势中ＸＶＶ’、ＰＶＶ’的计算公式。     

The supersymmetric WKB approximation of ring-shaped oscillator potential in r and θ dimensions with spherical polar coordinates

According to calculation of the energy spectrum of ring-shape oscillator potential by using the supersymmetric WKB approximation, it is shown that the energy spectrum of some noncentral separable potentials can be exactly obtained in r and θ dimensions by above method.     

用抛物柱坐标求解两共焦抛物板间的电势和电场

利用抛物柱坐标求解两共焦抛物柱板间的电势和电场分布,并对抛物柱板的电荷密度分布作了讨论.     

Imaginary Weyl coordinates in the Kerr space-time

By introducing imaginary Weyl coordinates the Kerr geometry can be described by the Weyl coordinate system as good as the Boyer-Lindquist system.     

Vibrational coordinates in the electron jump model

The role of quantization of vibrational coordinates in the electron jump model of alkali atom, dimer plus halogen molecule collisions is examined. Despite the bound states of the molecule ions involved, a classical approximation may offer good approximation over a wide collision energy range for the energy distributions of the ions in a single electron jump. Some results are calculated for K + Br₂ and K₂ + Br₂. The theory assumes that the electron jump probability is governed by the electronic matrix element H₁₂ alone. Elementary arguments are offered to suggest that this may be a reasonable approximation.     

用转移矩阵方法来研究发生在势阱内的量子反射

本文利用转移矩阵方法对两种简单的具有吸引尾的势进行了研究.我们发现反射系数在波矢k→0时R=1,反射率与波矢k在k→0时呈线性关系.这说明在发生阱内量子反射时有利于超冷原子生存,因为超冷原子是低速低能原子.此外,我们可以用一个反射时间变化量Δt来判断量子反射的延迟或者超前.