There are no Goldstone bosons in two dimensions

In four dimensions, it is possible for a scalar field to have a vacuum expectation value that would be forbidden if the vacuum were invariant under some continuous transformation group, even though this group is a symmetry group in the sense that the associated local currents are conserved. This is the Goldstone phenomenon, and Goldstone's theorem states that this phenomenon is always accompanied by the appearance of massless scalar bosons. The purpose of this note is to show that in two dimensions the Goldstone phenomenon can not occur; Goldstone's theorem does not end with two alternatives (either manifest symmetry or Goldstone bosons) but with only one (manifest symmetry).Work supported by the National Science Foundation under Grant No. 30819X and Grant No. 30738X.     

The symmetry group paradox for non-rigid molecules

In many situations, the energy levels for a quantum system, whose Hamiltonian is invariant under a specific symmetry group, are split when the Hamiltonian is replaced by a new one with lower symmetry. In non-rigid molecules (NRM), fast quantum tunnelling processes allow the molecule to change between different geometrical configurations related by permutations of identical nuclei (or with inversion as well), resulting in the splitting of the energy levels for the rigid molecule (RM) case where tunnelling is absent. However, for NRM, there is apparently a paradoxical situation where although the original RM energy levels are associated with a symmetry group isomorphic to the point group for the geometrical configuration, the split NRM energy levels are associated with a symmetry group consisting of all permutations and inversions related to the fast quantum tunnelling processes between configurations, and for which the point group is a subgroup. The resolution of this paradox, where energy level splitting is evidently accompanied by an enlargement of the symmetry group, is the subject of this article.     

Internal structure of the monopoles

The non-abelian monopoles are shown to form linear multiplets under the color reflection symmetry of the original gauge group. This implies that as far as the gauge symmetry is kept unbroken only the singlet combinations of the monopoles may be admitted in the physical sector.     

Conformal invariance and conserved quantities of a general holonomic system with variable mass

Conformal invariance and conserved quantities of a general holonomic system with variable mass are studied. The definition and the determining equation of conformal invariance for a general holonomic system with variable mass are provided. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The relationship between the conformal invariance and the Lie symmetry is discussed, and the necessary and sufficient condition under which the conformal invariance would be the Lie symmetry of the system under an infinitesimal one-parameter transformation group is deduced. The conserved quantities of the system are given. An example is given to illustrate the application of the result.     

Special Lie symmetry and Hojman conserved quantity of Appell equations in a dynamical system of relative motion

Special Lie symmetry and the Hojman conserved quantity for Appell equations in a dynamical system of relative motion are investigated. The definition and the criterion of the special Lie symmetry of Appell equations in a dynamical system of relative motion under infinitesimal group transformation are presented. The expression of the equation for the special Lie symmetry of Appell equations and the Hojman conserved quantity, deduced directly from the special Lie symmetry in a dynamical system of relative motion, are obtained. An example is given to illustrate the application of the results.     

Flux compactifications of string theory on twisted tori

Global aspects of Scherk‐Schwarz dimensional reduction are discussed and it is shown that it can usually be viewed as arising from a compactification on the compact space obtained by identifying a (possibly non‐compact) group manifold 𝒢 under a discrete subgroup Γ, followed by a truncation. This allows a generalisation of Scherk‐Schwarz reductions to string theory or M‐theory as compactifications on 𝒢/Γ, but only in those cases in which there is a suitable discrete subgroup of 𝒢. We analyse such compactifications with flux and investigate the gauge symmetry and its spontaneous breaking. We discuss the covariance under O(d,d), where d is the dimension of the group 𝒢, and the relation to reductions with duality twists. The compactified theories promote a subgroup of the O(d,d) that would arise from a toroidal reduction to a gauge symmetry, and we discuss the interplay between the gauge symmetry and the O(d,d,ℤ T‐duality group, suggesting the role that T‐duality should play in such compactifications.     

Strongly first order phase transitions near an enhanced discrete symmetry point

We propose a group theoretic condition which may be applied to extensions of the Standard Model in order to locate regions of parameter space in which the electroweak phase transition is strongly first order, such that electroweak baryogenesis may be a viable mechanism for generating the baryon asymmetry of the universe. Specifically, we demonstrate that the viable corners of parameter space may be identified by their proximity to an enhanced discrete symmetry point. At this point, the global symmetry group of the theory is extended by a discrete group under which the scalar sector is non-trivially charged, and the discrete symmetry is spontaneously broken such that the discrete symmetry relates degenerate electroweak preserving and breaking vacua. This idea is used to investigate several specific models of the electroweak symmetry breaking sector. The phase transitions identified through this method suggest implications for other relics such as dark matter and gravitational waves.     

Conformal Invariance and Conserved Quantities of General Holonomic Systems

Conformed invariance and conserved quantities of general holonomic systems are studied. A one-parameter infinitesimal transformation group and its infinitesimal transformation vector of generators are described. The definition of conformal invariance and determining equation for the system are provided. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The necessary and suttlcient condition, that conformal invariance of the system would be Lie symmetry, is obtained under the infinitesimal one-parameter transformation group. The corresponding conserved quantity is derived with the aid of a structure equation. Lastly, an example is given to demonstrate the application of the result.     

Is the collapse a phase transition?

It is shown that the process of the collapse during quantum measurements is a consequence of the quantum symmetry built in the formalism: there exists a nontrivial group of symmetry under which all possible states are invariant after the measurement of the given observable. On the basis of this fact, the postulate of collapse is derived with purely group theoretical considerations. These are analogous to the ones used in Landau's problem of phase transitions. Some further analogies of the processes of the collapse and the phase transitions are discussed.     

Hidden quantum group symmetry and integrable perturbations of conformal field theories

The hidden quantum group symmetry in the quantum Sine-Gordon model is found. This symmetry provides the possibility to restrict the operator algebra of the model to subalgebras. It is shown that these subalgebras are massive deformations of minimal conformal field theories.Supported in part by the Department of Energy under Grant DE-FG02-88ER25065     

奇异 Chetaev型非完整系统Nielsen方程的Lie-Mei对称性与守恒量

研究奇异Chetaev型非完整系统Nielsen方程的Lie-Mei对称性, 建立系统Nielsen方程的Lie-Mei对称性方程, 给出系统Nielsen方程强Lie-Mei对称性和弱Lie-Mei对称性的定义, 得到对称性导致的Hojman守恒量和Mei守恒量, 最后给出说明性算例. 关键词： 奇异非完整系统 Nielsen方程 Lie-Mei对称性 守恒量     

一般完整系统Mei对称性的逆问题

动力学逆问题是星际航行学、火箭动力学、规划运动学理论的基本问题. Mei对称性是力学系统的动力学函数在群的无限小变换下仍然满足系统原来的运动微分方程的一种新的不变性. 本文研究广义坐标下一般完整系统的Mei对称性以及与Mei对称性相关的动力学逆问题. 首先, 给出系统动力学正问题的提法和解法. 引入时间和广义坐标的无限小单参数变换群, 得到无限小生成元向量及其一次扩展. 讨论由n个广义坐标确定的一般完整力学系统的运动微分方程, 将其Lagrange函数和非势广义力作无限小变换, 给出系统运动微分方程的Mei对称性定义, 在忽略无限小变换的高阶小量的情况下得到Mei对称性的确定方程, 借助规范函数满足的结构方程导出系统Mei对称性导致的Noether守恒量. 其次, 研究系统Mei对称性的逆问题. Mei对称性的逆问题的提法是: 由已知守恒量来求相应的Mei对称性. 采取的方法是将已知积分当作由Mei对称性导致的Noether守恒量, 由Noether逆定理得到无限小变换的生成元, 再由确定方程来判断所得生成元是否为Mei对称性的. 然后, 讨论生成元变化对各种对称性的影响. 结果表明, 生成元变化对Noether和Lie对称性没有影响, 对Mei 对称性有影响, 但在调整规范函数时, 若满足一定条件, 生成元变化对Mei对称性也可以没有影响. 最后, 举例说明结果的应用.     

Twisted modular forms and Atkin-Lehner symmetries

A new “stringy” symmetry leading to zero cosmological constant to one loop is introduced. The partition function is a modular function which gets multiplied by a certain group character under modular transformations. The elementary fact that the sum over group elements of any nontrivial character is zero gives the desired result. The relationship of this symmetry with Moore's Atkin-Lehner transformations, as well as with Gross and Mende's analysis of anomalous heterotic strings is clarified. General classes of two dimensional models are also investigated.     

Conformal invariance and conserved quantities of general holonomic systems in phase space

This paper studies the conformal invariance and conserved quantities of general holonomic systems in phase space. The definition and the determining equation of conformal invariance for general holonomic systems in phase space are provided. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The relationship between the conformal invariance and the Lie symmetry is discussed, and the necessary and sufficient condition that the conformal invariance would be the Lie symmetry of the system under the infinitesimal single-parameter transformation group is deduced. The conserved quantities of the system are given. An example is given to illustrate the application of the result.     

Natural electroweak breaking from a mirror symmetry

We present "twin Higgs models," simple realizations of the Higgs boson as a pseudo Goldstone boson that protect the weak scale from radiative corrections up to scales of order 5-10 TeV. In the ultraviolet these theories have a discrete symmetry which interchanges each standard model particle with a corresponding particle which transforms under a twin or a mirror standard model gauge group. In addition, the Higgs sector respects an approximate global symmetry. When this global symmetry is broken, the discrete symmetry tightly constrains the form of corrections to the pseudo Goldstone Higgs potential, allowing natural electroweak symmetry breaking. Precision electroweak constraints are satisfied by construction. These models demonstrate that, contrary to the conventional wisdom, stabilizing the weak scale does not require new light particles charged under the standard model gauge groups.     

Acceleration-extended Newton-Hooke symmetry and its dynamical realization

Newton-Hooke group is the nonrelativistic limit of de Sitter (anti-de Sitter) group, which can be enlarged with transformations that describe constant acceleration. We consider a higher order Lagrangian that is quasi-invariant under the acceleration-extended Newton-Hooke symmetry, and obtain the Schrödinger equation quantizing the Hamiltonian corresponding to its first order form. We show that the Schrödinger equation is invariant under the acceleration-extended Newton-Hooke transformations. We also discuss briefly the exotic conformal Newton-Hooke symmetry in 2+1 dimensions.     

Unified Symmetry of Nonholonomic System of Non-Chetaev's Type in Event Space

The unified symmetry of a nonholonomic system of non-Chetaev's type in event space under infinitesimal transformations of group is studied.Firstly,the differential equations of motion of the system are given.Secondly,the definition and the criterion of the unified symmetry for the system are obtained.Thirdly,a new conserved quantity,besides the Noether conserved quantity and the Hojman conserved quantity,is deduced from the unified symmetry of a nonholonomic system of non-Chetaev's type.Finally,an example is given to illustrate the application of the result.     

A type of new conserved quantity deduced from Mei symmetry for Nielsen equations in a holonomic system with unilateral constraints

<正>A type of new conserved quantity deduced from Mei symmetry for Nielsen equations in a holonomic system with unilateral constraints is investigated.Nielsen equations and differential equations of motion for the holonomic mechanical system with unilateral constraints are established.The definition and the criterion of Mei symmetry for Nielsen equations in the holonomic systems with unilateral constraints under the infinitesimal transformations of Lie group are also given.The expressions of the structural equation and a type of new conserved quantity of Mei symmetry for Nielsen equations in the holonomic system with unilateral constraints are obtained.An example is given to illustrate the application of the results.     

Large N expansion for Yukawa-type theories

4 dimensional Yukawa-type theories with an internal SU(N) symmetry group are studied in the large N limit by means of path integral techniques. For a simplified model where the bosonic field transforms as a singket under SU(N), we explicitly solve the gap equation and shiw that the symmetry is spontaneously broken.