群表示论的物理方法(Ⅱ) 置换群亚标准基和酉群Gelfand基

本文中分析了态空间置换群f表示论中的矛盾。指出了原因在于当态指标有重复时,f的单个群元算符没有确定作用。根据文献[1]给出了这种情形下f的亚标准基的定义、标志和求法,并证明了f亚标准基同时就是酉群的Gelfand基。     

光偏振态的群论描述:转动矩阵的生成元

应用偏振光描述中的变换矩阵与群论的对应关系[1]和相应的计算理论,讨论了与偏振光学系统中的Jones矩阵、Mueller矩阵相对应的SU(2)群、SO(3)群和Lorentz群的生成元问题,给出了用单位矩阵、Pauli自旋矩阵和稀疏矩阵分别作为无耗偏振光学系统中SU(2)群元(Jones矩阵)和SO(3)群元(Mueller矩阵)生成元以及部分损耗偏振光学系统中的幺模群(Jones矩阵)和Lorentz群(Mueller矩阵)生成元的具体形式;矩阵计算理论说明这些群元的生成元表示可以简化偏振光学系统的计算。     

群表示论的物理方法(Ⅱ)——置换群亚标准基和酉群Gelfand基

本文中分析了态空间置换群g_f表示论中的矛盾。指出了原因在于当态指标有重复时,g_f的单个群元算符没有确定作用。根据文献[1]给出了这种情形下g_f的亚标准基的定义、标志和求法,并证明了g_f亚标准基同时就是酉群的Gelfand基。     

构造dN谱项函数的一个新方法

本文利用群链U(5)(?)SO(5)(?)SO(3)(?)G的各级下降系数讨论了原子谱项函数的构造方法,计算了d~N谱项函数所需要的全部下降系数。此外,还给出了静电作用微扰哈密顿以酉群生成元表达的简洁形式。     

新环形库仑势的束缚连续跃迁矩阵元

利用新环形库仑势的归一化的束缚态径向波函数和按“k/2π标度”归一化的散射态径向波函数,本文给出了新环形库仑势的任意幂次的束缚连续跃迁矩阵元的通项表达式.为了简化高幂次的束缚连续跃迁矩阵元的计算,我们还推导出了不同幂次的束缚连续跃迁矩阵元之间所满足的递推关系,并提出了计算径向波函数微商的矩阵元的计算办法.本文结果可广泛的用于原子与分子的散射问题特别是环形分子的散射问题之中.     

基于量子图态的量子秘密共享

本文基于量子图态的几何结构特征,利用生成矩阵分割法,提出了一种量子秘密共享方案.利用量子图态基本物理性质中的稳定子实现信息转移的模式、秘密信息的可扩展性以及新型的组恢复协议,为安全的秘密共享协议提供了多重保障.更重要的是,方案针对生成矩阵的循环周期问题和因某些元素不存在本原元而不能构造生成矩阵的问题提出了有效的解决方案.在该方案中,利用经典信息与量子信息的对应关系提取经典信息,分发者根据矩阵分割理论获得子秘密集,然后将子秘密通过酉操作编码到量子图态中,并分发给参与者,最后依据该文提出的组恢复协议及图态相关理论得到秘密信息.理论分析表明,该方案具有较好的安全性及信息的可扩展性,适用于量子网络通信中的秘密共享,保护秘密数据并防止泄露.     

八面体晶场强场理论的U群方法

本文利用二次量子化算子的对易性找到了一条适用于八面体场中d电子强场态分类的群链,构成了对称匹配波函数,讨论了矩阵元的计算方法。 关键词：     

谐振子任意次幂坐标算符矩阵元的一种计算方法

利用相干态和正规乘积对谐振子任意次幂坐标算符X^1矩阵元进行了讨论,导出了计算X^1矩阵元的一般公式,其结果与《大学物理》发表的两篇文章一致,为处理谐振子的微扰问题提供了一种新的方法。     

谐振子系统坐标算符矩阵元的简单计算方法

详细讨论了在粒子数表象和相干态表象中如何简捷地计算谐振子系统坐标算符矩阵元的问题,给出了具体的计算过程并对文献中的相关处理方法和过程进行了评述.     

Tb^3＋离子的能级之间约化矩阵元计算

本文利用中间耦合波函数,系统计算了Tb~(3 )离子的22个壮态之间U~K张量的约化矩阵元,为基础理论研究提供基本数据。     

TRANSFORMATION COEFFICIENTS BETWEEN SO(6) AND SU(4) GELFAND-ZETLIN STATES

In this note, we have given a method of calculating the transformation coefficients between SO(6) and SU(4) Gelfand Zetlin states.     

Koszul duality

This paper tries to describe a natural framework for the canonical equivalence between derived categories of graded modules over symmetric and exterior algebra, which has been established by J.N. Bernstein, I.M. Gelfand and S.I. Gelfand.     

Gelfand-Tzetlin coordinates for the unitary supergroup

The Gelfand-Tzetlin method provides explicit coordinates on the parameter space of the unitary groupU(k) which make direct evaluations of group integrals possible. It is closely related to the Gelfand construction of finite-dimensional irreducible representations. We generalize the Gelfand-Tzetlin method to the unitary supergroupU(k ₁/k₂). The coordinates on the parameter space for supergroup integrals and the invariant Haar measure are evaluated. As an example, the supersymmetric Harish-Chandra-Itzykson-Zuber integral is calculated. A generalized Gelfand pattern containing anticommuting variables is introduced which determines the representation.This article was processed by the author using the Latex style filepljour1 from Springer-Verlag.     

AnO(3, 1) approach to Maxwell's equations

Using the principal series representations of the Lorentz group, a method parallel to that of Gelfand and Yaglom is suggested to obtain Maxwell's equations, which dispenses with the arbitrary introduction of a degenerate transformation with respect to which the photon equations are invariant. The method also gives subsidiary conditions which, in conjunction with the masslessness of the particle, yield the Lorentz condition and the correct values of photon polarization.     

A differential structure for quantum mechanics

A method of obtaining a non-commutative analogue of a differential structure from the action of a Lie group on a C^*-algebra is proposed. The addition of this structure to the usual structure of quantum mechanics turns out to be equivalent to the replacement of the Hilbert space by a Gelfand triple (rigged Hilbert space).     

A New Scheme of Integrability for (bi)Hamiltonian PDE

We develop a new method for constructing integrable Hamiltonian hierarchies of Lax type equations, which combines the fractional powers technique of Gelfand and Dickey, and the classical Hamiltonian reduction technique of Drinfeld and Sokolov. The method is based on the notion of an Adler type matrix pseudodifferential operator and the notion of a generalized quasideterminant. We also introduce the notion of a dispersionless Adler type series, which is applied to the study of dispersionless Hamiltonian equations. Non-commutative Hamiltonian equations are discussed in this framework as well.     

Symplectic structures,their Bäcklund transformations and hereditary symmetries

It is shown that compatible symplectic structures lead in a natural way to hereditary symmetries. (We recall that a hereditary symmetry is an operator-valued function which immediately yields a hierarchy of evolution equations, each having infinitely many commuting symmetries all generated by this hereditary symmetry. Furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations). This result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations.In addition we show how symplectic structures transform under Bäcklund transformations. This leads to a method for generating a whole class of symplectic structures from a given one.Several examples and applications are given illustrating the above results. Also the connection of our results with those of Gelfand and Dikii, and of Magri is briefly pointed out.     

A construction of Multidimensional Dubrovin-Novikov Brackets

A method for the construction of classes of examples of multi-dimensional, multi-component Dubrovin-Novikov brackets of hydrodynamic type is given. This is based on an extension of the original construction of Gelfand and Dorfman which gave examples of Novikov algebras in terms of structures defined from com- mutative, associative algebras. Given such an algebra, the construction involves only linear algebra.     

On a Classical Limit of <Emphasis Type="Italic">q</Emphasis>-Deformed Whittaker Functions

Previously, we derive a representation of q-deformed \({\mathfrak{gl}_{\ell+1}}\) -Whittaker function as a sum over Gelfand–Zetlin patterns. This representation provides an analog of the Shintani–Casselman–Shalika formula for q-deformed \({\mathfrak{gl}_{\ell+1}}\) -Whittaker functions. In this note, we provide a derivation of the Givental integral representation of the classical \({\mathfrak{gl}_{\ell+1}}\) -Whittaker function as a limit q → 1 of the sum over the Gelfand–Zetlin patterns representation of the q-deformed \({\mathfrak{gl}_{\ell+1}}\) -Whittaker function. Thus, Givental representation provides an analog the Shintani–Casselman–Shalika formula for classical Whittaker functions.