Artin空间中SU(3)规范场的几何解析

研究4维Artin空间中SU(3)规范场的线性化问题.首先对Yang-Mills方程的推导进行了讨论,给出了恰当的Yang-Mills方程的概念,其具有明确的几何意义.其次,构造了一类线性微分变换,称之为Artin空间SU(3)规范场的示性变换.示性变换是应用数学机械化方法确定的.经由示性变换,将非线性的恰当的Yang-Mills方程变为一组线性方程,实现了SU(3)规范场的场方程的线性化.从而证明了,对于恰当的Yang-Mills方程,SU(3)规范场包括8个独立的规范场.     

SU(2)规范场的恰当形式(欧空间)

应用数学机械化方法研究欧氏空间中SU(2)规范场的正规化问题.首先对Yang-Mills方程的表述进行了讨论,给出了一种具有物理和几何意义的YM-方程,称其为恰当的(exact)YM-方程.对于这种恰当的YM-方程,构造了一类线性微分变换,称之为SU(2)规范场的示性变换.经由示性变换,将非线性的恰当的YM-方程变为一组Laplace方程,实现了SU(2)规范场方程的线性化,即场方程的正规化.从而证明了SU(2)规范场存在3个独立的Yang-Mills规范场.     

SU(3)规范场的恰当形式(欧空间)

应用数学机械化方法讨论SU(3)规范场的规范化问题.首先提出一种具有明确几何意义的Yang-Mills方程,称其为恰当的Yang-Mills方程.然后构造了一类线性微分变换,称之为SU(3)规范场的示性变换,它具体给出联络和截面之间的微分关系.经由示性变换,将非线性的恰当的YM-方程变为一组线性Laplace方程,即实现了规范场YM-方程的线性化.从而证明了SU(3)规范场包括8个独立的Yang-Mills规范场.     

容有平行Yang-Mills场的黎曼空间(英文)

如果一个Yang-Mills场(规范群为任意李群)的场强的所有规范导数均为0,则称这个场为平行的Yang-Mills场。平行规范场是微分几何中对称空间的推广,它是Yang-Mills方程的特解。 本文的主要结果是下列两个定理: 定理1 容有非平凡的平行Yang-Mills场的四维黎曼空间必须是Khler流形或半对称空间,这里半对称流形是满足的黎曼流形,其中分别是曲率张量的自对偶部份及反自对偶部份,而“;”表示共变导数。 定理2 半对称空间如果不是对称空间,则必为Khler-Einstein空间或共形半平坦Einstein空间。这里共形半平坦是指Weyl张量的反自对偶部份或自对偶部份为0。 在附录中作者给出了二维黎曼流形上Yang-Mills方程的所有的整体解。     

ON THE RIEMANNIAN SPACES ADMITTING PARALLEL YANG-MILLS FIELDS

如果一个Yang-Mills场(规范群为任意李群)的场强的所有规范导数均为0,则称这个场为平行的Yang-Mills场.平行规范场是微分几何中对称空间的推广,它是Yang-Mills方程的特解. 本文的主要结果是下列两个定理： 定理1 容有非平凡的平行Yang-Mills场的四维黎曼空间必须是Kahler流形或半对称空间.这里半对称流形是满足 \[R_{ijkl}^ - = 0\](或\[R_{ijkl}^ + = 0\]) 的黎曼流形,其中\[R_{ijkl}^ \pm \]分别是曲率张量的自对偶部份及反自对偶部份,而":"表示共变 导数. 定理2 半对称空间如果不是对称空向,则必为Kahler-Einstein空间或共形半平坦Einstein空间.这里共形半平坦是指Weyl张量的反自对偶部份或自对偶部份为0.在附录中作者给出了二维黎曼流形上Yang-Mills方程的所有的整体解.     

关于具有质量的Yang-Mills方程的静态解(英文)

关于Yang-Mills方程的静态解,Deser,S.证明了:当n≠5时,无质量的紧致群Yang-Mills方程不存在满足条件(ⅰ)无奇性(ⅱ)能量有限(ⅲ)当r→∞oo时,场强f_(λμ)→O足够快的静态解.又已知当n=5时,正则静态解确实存在. 对于具实质量的紧致群Yang-Mills方程,作者证明了:当n≠4时,不存在满足条件(ⅰ)无奇性(ⅱ)能量有限(ⅲ)当r→∞时规范势b_λ与场强f_(λμ)→O足够快的静态解.从而发现在n=5,当质量m→O时,对Yang-Mills方程的可解性问题而言,在性质上有一种“不连续性”.物理学家认为这是存在着经典的不连续性的第一个明确的例子,并对包括Higgs场的情况作了推广的研究. 本文进一步证明了上述两个结果中不仅条件(ⅲ)可以取消,而且条件(ⅱ)也可减弱.即能量为无限,但当以r为半径的球体的总能量趋于无限相当慢时定理仍成立.这时经典的“不连续性”也仍成立.由于能量有限与能量无限在物理上有根本的不同,所揭示的现象是值得注意的. 文中又证明,如果取消总能量趋于无限相当慢这个条件,定理的结论就不成立. 这里的证明方法,可用于更一般的情况.例如包括Higgs场的情况,从而[9]中的结果也得到改进.     

N=2 超对称KdV 方程的双线性形研究

利用双线性方法研究$N=2$超对称KdV方程. 通过适当的相关变换, 将$N=2, a=4$和$N=2, a=1$超对称KdV方程转化成双线性形式, 由此构造了相应方程的解. 对于$N=2, a=1$ 超对称KdV方程, 还得到了它的双线性B\"acklund变换和Lax 表示.     

2n维空间中的广义自对偶Yang-Mills方程的达布变换

从带负幂次谱参数的谱问题出发,构造了一类广义自对偶Yang-Mills方程.这类方程包括若干著名的Lax可积方程,如Takasaki情形、Belavin-Zakharov情形、AblowitzChakravarty-Takhtajall情形和Ma情形.进而建立了这类方程的达布变换的精确表达式.     

A Class Matrix Evolution Equations

在文献[5]中，考虑了如下特征值问题 $\[{\varphi _x} = M\varphi ,{\varphi _x} = \frac{{\partial \varphi }}{{\partial x}}\]$ 其中 $\[\varphi = \left( {\begin{array}{*{20}{c}} {{\varphi _1}}\{{\varphi _2}} \end{array}} \right)\]$ (1) $\[M = \left( {\begin{array}{*{20}{c}} { - i\xi }&{q(x,t)}\{r(x,t)}&{i\xi } \end{array}} \right)\]$ (2) 这里假定特征值$\xi$以某种规律随着时间变化而变化。文章中得出了一类发展方程，其中两个特殊情形：r=1,q=u(x,t)分别可以当做推广的KDV方程和推广的MKDV方程，并证明了不仅在KDV方程和MKDV方程之间存在Miura变换，而且在推广的KDV方程和推广的MKDV方程之间也存在Miura变换，又证明了对推广的KDV方程存在Backlund变换。 本文将[5]的结果推广至矩阵情形： 设 $\[M = \left( {\begin{array}{*{20}{c}} { - i\xi }&{Q(x,t)}\{R(x,t)}&{i\xi I} \end{array}} \right)\]$ (3) 这里Q,R为N*N矩阵，I是N*N单位阵，相应的在(1)式中的向量$\varphi$是2N维向量。我们引进矩阵型的Miura变换，并得到了与[5]相平行的结果。     

方程u_(xx)-X~2u_(tt)+pu_t=0的不变变换群和基本解

方程u_(xx)-x~2u_(tt)+pu_t=0的不变变换群是包含四个参数的李变换群。群生成元的李代数是sl(2,R)。在此基础上,求得方程的格林函数G=u_1+u_2,u_1是对称部份,u_2是反对称部份。     

Non-Abelian tensor gauge fields

A recently proposed extension of Yang-Mills theory contains non-Abelian tensor gauge fields. The Lagrangian has quadratic kinetic terms, as well as cubic and quartic terms describing nonlinear interaction of tensor gauge fields with the dimensionless coupling constant. We analyze the particle content of non-Abelian tensor gauge fields. In four-dimensional space-time the rank-2 gauge field describes propagating modes of helicity 2 and 0. We introduce interaction of the non-Abelian tensor gauge field with fermions and demonstrate that the free equation of motion for the spinor-vector field correctly describes the propagation of massless modes of helicity 3/2. We have found a new metric-independent gauge invariant density which is a four-dimensional analog of the Chern-Simons density. The Lagrangian augmented by this Chern-Simons-like invariant describes the massive Yang-Mills boson, providing a gauge invariant mass gap for a four-dimensional gauge field theory.     

Dirac γ-equation, classical gauge fields and Clifford algebra

An equation, we call Dirac γ-equation, is introduced with the help of the mathematical tools connected with the Clifford algebra. This equation can be considered as a generalization of the Dirac equation for the electron. Some features of Dirac γ-equation are investigated (plane waves, currents, canonical forms). Furthermore, on the basis of local gauge in variance regarding unitary group, a system of equations is introduced consisting of Dirac γ-equation and the Yang-Mills or Maxwell equations. This system of equations describes a Dirac’s field interacting with the Yang-Mills or Maxwell gauge field. Characteristics of this system of equations are studied for various gauge groups and the liaison between the new and the standard constructions of classical gauge fields is discussed. This paper is supported by the Russian Foundation for Basic Research, grant 95-10-00433a.     

Arbitrariness of the general solution and symmetries

The computation of a number of arbitrary functions in the general solution is briefly reviewed. The results are used to study normal systems and their symmetry reduction. We discuss the treatment of gauge systems, especially the analysis of gauge fixing conditions. As examples, the Yang-Mills equations with the Lorentz gauge and Einstein's vacuum field equations with harmonic coordinates are considered.     

Lorentz-invariant quantization of the Yang-Mills theory without Gribov ambiguity

The Yang-Mills theory in the covariant renormalizable gauge, which selects a unique representative in the class of gauge equivalent configurations, is discussed.     

Renormalization of the quantum equation of motion for Yang-Mills fields in the background formalism

The aim of this paper is to compare the renormalizations of the infinite parts of the effective action and quantum equation of motion in gauge fields theory with structural group SU (N) in the one-loop approach and to show that they do not coincide. To this end, we use the background formalism, in which a gauge field is decomposed into the sum of a classical (background) field and a quantum field: A = B + gQ. The appearance of an additional factor, which provides the recovery of the equality of renormalizations of equations and action, is explained by comparison with results of the standard renormalization theory of Yang-Mills fields. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 5–12.     

(Anti)self-dual gauge fields in dimensiond≥4

The (anti)self-duality equations for gauge fields in dimensiond=4 and the generalization of these equations ford>4 are considered. The results on solutions of the (anti)self-duality equations ind4 are reviewed. Some new classes of solutions of Yang-Mills equations ind4 for arbitrary gauge fields are described.Moscow State University; Joint Institute for Nuclear Research, Dubna. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 2, pp. 316–342, February, 1993.     

Classification of generalized symmetries of the Yang-Mills fields with a semi-simple structure group

A complete classification of generalized (or local) symmetries of the Yang-Mills equations on four dimensional Minkowski space with a semi-simple structure group is carried out. It is shown that any generalized symmetry, up to a generalized gauge symmetry, agrees with a first order symmetry on solutions of the Yang-Mills equations. Let be the decomposition of the Lie algebra of the structure group into simple ideals. First order symmetries for -valued Yang-Mills fields are found to consist of gauge symmetries, conformal symmetries for -valued Yang-Mills fields, 1?m?n, and their images under a complex structure of .     

Gauge fields and space-time geometry

Mappings between the SU (2) Yang-Mills gauge theory and the gauge gravity theory are discussed. The global aspects are considered in the framework of the fiber-bundle technique, and three different local constructions (depending on the existence of a fixed background metric) are described in detail. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 117, No. 2, pp. 249–262, November, 1998.