量子系统的整体正则对称性

基于相空间路径积分,分别导出了正规和奇异拉氏量系统在增广相空间中整体对称的Ward恒等式,从而可给出Green函数间的关系.在量子水平上建立了正则整体对称和守恒量之间的关系.一般来说,经典理论中对称性所联系的守恒量,在量子理论中不一定再保持.这里给出的形式其显著优点在于勿需作出生成泛函中对正则动量的路径积分.重新讨论了含Hopf项和Chem-Simons项的非线性σ-模型,在量子水平上对“分数自旋”性质给予了说明.     

位形空间非定域Ward恒等式

从Faddeev Popov(FP)方法对规范理论给出的位形空间生成泛函出发，导出了位形空间非定域变换下的Ward恒等式。应用于非Abel Chern Simons(CS)理论，得到了CS规范场鬼场正规顶角间的Ward恒等，并把此结果与文献［1］做了对比，对规范理论用位形空间路径积分讨论更简便。     

混合互补问题牛顿型算法的二阶收敛性

在凸规划理论中,通过ＫＴ条件,往往将约束最优化问题归结为一个混合互补问题来求解。该文就正则解和一般解两种情形分别给出了求解混合互补问题牛顿型算法的二阶收敛性的充分性条件,并在一定条件下证明了非精确牛顿法和离散牛顿法所具有的二阶收敛性。     

变形色散水波方程的Auto-Backlund变换、多孤子解和有理分式解

利用未知数变换并借助Ｍａｔｈｅｍａｔｉｃａ软件,给出了变形色散水波方程的Ａｕｔｏ－Ｂａｅｃｋｌｕｎｄ变换以及它与热传导方程和线性方程之间的Ｄａｒｂｏｕｘ变换。进而用此变换,获得了变形色散水波方程的多孤子解,有理分式解及其他精确解。这种思路也适用于其它的非线性方程。     

集值Superpramart的上鞅逼近

文中讨论了可积随机集条件期望的若干性质,在此基础上,给出了集值Superpramart的上鞅逼近.同时,证明了集值Superpramart在Kuratowski-Mosco意义下的收敛定理.     

一个方程组的解及相关边值问题

该文考虑方程组 t(u_x - v_y - w_t ) + w = 0 (H) u_y = -v_x, u_t = -w_x, v_t = w_y 的解f = u + iv + jt ∈C²,找到(H)可解的一个充要条件,并讨论相关边值问题解的存在性和积分表示,推广了1992年H. Leutwiler［1］的结果.     

有界对称域上的BMOp空间的乘子

该文得到在Bergman度量下刻划有界p-次平均振动函数(BMO^p)空间的乘子的几个等价条件,包含了文献［３］中的BMO²空间的结果.     

Banach空间二阶非线性奇异微分方程的解

文中讨论了Banach空间中一类右端具有奇异性的二阶非线性微分方程的边值问题,利用不动点方法,获得了该边值问题解的存在性定理.最后还给出了相应的例子.     

规范理论生成泛函的联合不变性与手征反常项

本文提出了手征群下规范理论生成泛函联合不变性的概念,并提出了五维正常化方案来计算Fermi场泛函积分测度在有限的非阿贝尔手征变换下的改变,由此给出手征反常项(即作用量的补充项)的显示形式,并讨论了它的物理意义.     

发展基于RIESZ表示算子的最小二乘混合有限元方法

针对一类广义抽象混合结构变分方程,研究了其解的存在唯一性问题;同时发展基于Riesz表示算子的有限元方法.     

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.     

From Slavnov-Taylor identities to the ZJ equation

In their work devoted to the proof of the renormalizability of non-Abelian gauge theories in the broken phase, Lee and Zinn-Justin have directly benefited from Slavnov’s important contributions. Later generalized in the form of the BRST symmetry, the Slavnov-Taylor identities eventually have led to a remarkable quadratic equation for the renormalized gauge action (sometimes called the Zinn-Justin (ZJ) equation), allowing a completely general proof of renormalizability of non-Abelian gauge theories.     

Cluster Partition Function and Invariants of 3-Manifolds

The author reviews some recent developments in Chern-Simons theory on a hyperbolic 3-manifold M with complex gauge group G.The author focuses on the case of G =SL(N,C) and M being a knot complement:M =S3 \ k.The main result presented in this note is the cluster partition function,a computational tool that uses cluster algebra techniques to evaluate the Chern-Simons path integral for G =SL(N,C).He also reviews various applications and open questions regarding the cluster partition function and some of its relation with string theory.     

Groups with anti-symmetric mappings

Conclusion The above theorems are sufficient to demonstrate the existence of antisymmetric mappings for all non-Abelian group of order less than 36, except for the group <a, b\a ³ =b ⁸ =e, ab =ba ²> of order 24, and this group can be shown to have one. On the basis of our results, and the fact that we have no conditions on a non-Abelian group that would eliminate any from having anti-symmetric mappings, we conjecture that allnon-Abelian groups have anti-symmetric mappings.     

The theory of non-Abelian tensor fields: Gauge transformations and curvature

We take one more step in formulating the theory of non-Abelian two-tensor fields: we find gauge transformation rules and the curvature tensor for them. To define the theory, we use the surface exponential. We derive a differential equation for the exponential and attempt to formulate its definition as a matrix model. We discuss applications of our construction to the Yang-Baxter equation for integrable models and to string field theory. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 1, pp. 73–91, April, 2006.     

Cohen–Macaulayness of Rees Algebras of Modules

Abstract

The notion of quasi-polynomials is very important in the theory of functional identities. For example, results on quasi-polynomials were tools in the solution of long-standing Herstein's Lie map conjectures. In this paper, we show that functional identities involving quasi-polynomial of degree one have only standard solutions on d-free sets.     

Yang—mills fields with external current

A procedure for quantizing a non-Abelian gauge theory with Lagrangian $$L = - \frac{1}{4}F_{\mu v}^a F_a^{\mu v} + J_a^\mu V_\mu ^a$$ near a nontrivial classical solution is considered. The theories are classified with respect to the external current. The gluon propagator in a model spherically symmetric non-Abelian field is constructed and investigated.     

Dynamical breakdown of parity and chiral invariance in the generalized three-dimensional four-fermion field theory

The generalized three-dimensional four-fermion field theory (1) is investigated. It is shown that this model describes five different phases, including a phase with simultaneous breakdown of both parity and chiral invariance. Interaction of the fermions and the gauge field leads to the dynamical appearance of the Chern-Simons term in only one of the two phases with broken parity.Institute of High Energy Physics, Serpukhov. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 1, pp. 42–50, April. 1993.