Conformal invariance and a kind of Hojman conserved quantity of the Nambu system

Conformal invariance and a kind of Hojman conserved quantity of the Nambu system under infinitesimal transfor-mations are studied.The definition and the determining equation of conformal invariance of the system are presented.The necessary and sufficient condition under which the conformal invariance of the system would have Lie symmetry un-der infinitesimal transformations is derived.Then,the condition of existence and a kind of Hojman conserved quantity are obtained.Finally,an example is given to illustrate the application of the results.     

Conformal invariance and conserved quantity of third-order Lagrange equations for non-conserved mechanical systems

This paper studies conformal invariance and conserved quantity of third-order Lagrange equations for non-conserved mechanical systems. Third-order Lagrange equations, the definition and a determining equation of conformal invariance of the system are presented. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The necessary and sufficient condition that conformal invariance of the system would have Lie symmetry under single-parameter infinitesimal transformations is obtained. The corresponding conserved quantity of conformal invariance is derived with the aid of a structure equation. Lastly, an example is given to illustrate the application of the results.     

Generalized Conformal Symmetries and Its Application of Hamilton Systems

In this paper the generalized conformal symmetries and conserved quantities by Lie point transformations of Hamilton systems are studied. The necessary and sufficient conditions of conformal symmetry by the action of infinitesimal Lie point transformations which are simultaneous Lie symmetry are given. This kind type determining equations of conformal symmetry of mechanical systems are studied. The Hojman conserved quantities of the Hamilton systems under infinitesimal special transformations are obtained. The relations between conformal symmetries and the Lie symmetries are derived for Hamilton systems. Finally, as application of the conformal symmetries, an illustration example is introduced.     

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.     

Renormalization group and scale invariance in terms of asymptotic fields

The expansion of the massive renormalized field operator in terms of asymptotic fields is studied. We derive the renormalization group equation for the renormalized field operator. We obtain the renormalized scale transformations with Callan-Symanzik corrections as generated by canonical scale transformations of asymptotic fields.     

Conformal Invariance and Conserved Quantity for?the?Nonholonomic System of Chetaev’s Type

In this paper the definition of conformal invariance and determining equation for the holonomic system which correspond to a nonholonomic system of Chetaev’s type are provided. Conformal factor expression is deduced through relationship between a system’s conformal invariance and Lie symmetry. The necessary and sufficient condition that the system’s conformal invariance would be Lie symmetry under transformations by the infinitesimal one-parameter transformation group is obtained. The conformal invariance of weak and strong Lie symmetry for the nonholonomic system of Chetaev’s type is given using restriction equations and additional restriction equations. And the system’s corresponding conserved quantity is derived with the aid of a structure equation that gauge function satisfied. Lastly, an example is taken to illustrate the application of the result.     

The Algebraic Structure of the Thirring Model

We summarize the representation theory of the group SU(1,1) as needed for the massless Thirring model. Representations of the current operator algebra are given taking account of conformal covariance. The conformal covariance transformation behaviour of the Thirring field is investigated. The Haag-Araki-Kastler observable algebra of the Thirring field is reconstructed from the Wightman theory of this model.     

Conformal invariance of quantum fields in two-dimensional space-time

Our investigations of conformal invariance are based on the theory of analytic representations of the conformal group and its universal covering group. With its help the action of the conformal group on free massless fields, Greenberg fields, Wick products of these fields, and the Thirring fields is studied. In this context we find an infinite set of new operator solutions for the Thirring model that are all equivalent to each other. Explicit constructions of the nonlocal special conformal transformations of all these fields are given.     

Conformal invariance and Hojman conserved quantities of canonical Hamilton systems

This paper discusses the conformal invariance by infinitesimal transformations of canonical Hamilton systems. The necessary and sufficient conditions of conformal invariance being Lie symmetrical simultaneously by the action of infinitesimal transformations are given. The determining equations of the conformal invariance are gained. Then the Hojman conserved quantities of conformal invariance by special infinitesimal transformations are obtained. Finally an illustrative example is given to verify the results.     

Lagrange系统的共形不变性与Hojman守恒量

研究了一般完整Lagrange系统在无限小变换下的共形不变性,推导出共形不变性的确定方程,并且找到在特殊无限小变换下的共形不变性并且是Lie对称性的共形因子,接下来导出Lagrange系统的运动微分方程共形不变时的Hojman守恒量,并给出应用算例. 关键词： Lagrange系统 共形不变性 Hojman守恒量 确定方程     

Conformal Invariance and a New Type of Conserved Quantities of MechanicalSystems with Variable Mass in Phase Space

Conformal invariance and a new type of conserved quantities of mechanical systems with variable mass in phase space are studied. Firstly, the definition and determining equation of conformal invariance are presented. The relationship between the conformal invariance and the Lie symmetry is given, and the necessary and sufficient condition that the conformal invariance would be the Lie symmetry under the infinitesimal transformations is provided.Secondly, a new type of conserved quantities of the conformal invariance are obtained by using the Lie symmetry of the system. Lastly, an example is given to illustrate the application of the results.     

广义Hamilton系统的共形不变性与Hojman守恒量

研究了广义Hamilton系统在无限小变换下的共形不变性,推导出共形不变性的确定方程,找到在无限小变换下的共形不变性并且是Lie对称性的共形因子,最后导出广义Hamilton系统的运动微分方程共形不变时的Hojman守恒量,并给出应用算例. 关键词： 广义Hamilton系统 共形不变性 Hojman守恒量 确定方程     

Invariant Forms of the Lorentz Group

We study trilinear and multilinear invariant forms for the homogeneous Lorentz group. The residues of these trilinear forms generate particular trilinear forms themselves. They appear also if we sum Taylor expansions partially into a series of expressions each of which is covariant under infinitesimal Lorentz transformations. Multilinear invariant forms are submitted to harmonic analysis in different channels. They are thus expressed by invariant functions. Invariant functions for different channels are related by integral equations involving 6χ-symbols, 9χ-symbols etc. as “crossing kernels”. It is shown by construction that all invariant functions and nχ-symbols can be represented as finite sums of Barnes type integrals. As example we analyze explicitly the four-point Schwinger function of the massless Euclidean Thirring field with arbitrary spin and dimension.     

Conformal invariance and conserved quantities of non-conservative Lagrange systems by point transformations

This paper studies the conformal invariance by infinitesimal point transformations of non-conservative Lagrange systems. It gives the necessary and sufficient conditions of conformal invariance by the action of infinitesimal point transformations being Lie symmetric simultaneously. Then the Noether conserved quantities of conformal invariance are obtained. Finally an illustrative example is given to verify the results.     

Conformal Invariance and Noether Conserved Quantities of First-Order Lagrange Systems

In this paper the conformal invariance by infinitesimal transformations of first-order Lagrange systems is discussed in detail. The necessary and sufficient conditions of conformal invariance by the action of infinitesimal transformations being Lie symmetry simultaneously are given. Then we get the conserved quantities of conformal invariance by the infinitesimal transformations. Finally an example is given to illustrate the application of the results.     

Conformal invariance and conserved quantities of dynamical system of relative motion

This paper discusses in detail the conformal invariance by infinitesimal transformations of a dynamical system of relative motion. The necessary and sufficient conditions of conformal invariance and Lie symmetry are given simultaneously by the action of infinitesimal transformations. Then it obtains the conserved quantities of conformal invariance by the infinitesimal transformations. Finally an example is given to illustrate the application of the results.     

Conformal invariance and Hojman conservedquantities of first order Lagrange systems

In this paper the conformal invariance by infinitesimal transformations of first order Lagrange systems is discussed in detail. The necessary and sufficient conditions of conformal invariance and Lie symmetry simultaneously by the action of infinitesimal transformations are given. Then it gets the Hojman conserved quantities of conformal invariance by the infinitesimal transformations. Finally an example is given to illustrate the application of the results.     

Recent developments in conformal invariant quantum field theory

A review of the recent results concerning the kinematics of conformal fields, the analysis of dynamical equations and dynamical derivation of the operator product expansion is given.The classification and transformational properties of fields which are transformed according to the representations of the universal covering group of the conformal group have been considered. A derivation of the partial wave expansion of Wightman functions is given. The analytical continuation to the Euclidean domain of coordinates is discussed. As shown, in the Euclidean space the partial wave expansion can be applied either to one-particle irreducible vertices or to the Green functions, depending on the dimensions of the fields.The structure of Green functions, which contain a conserved current and the energy-momentum tensor, has been studied. Their partial wave expansion has been obtained. A solution of the Ward identity has been found. Special cases are discussed.The program of the construction of exact solution of dynamical equations is discussed. It is shown, that integral dynamical equations for vertices (or Green's functions) can be diagonalized by means of the partial wave expansion. The general solution of these equations is obtained. The equations of motion for renormalized fields are considered. The way to define the product of renormalized fields at coinciding points (arising on the right-hand side) is discussed. A recipe for calculating this product is presented. It is shown, that this recipe necessarily follows from the renormalized equations.The role of bare term and of canonical commutation relations (for unrenormalized fields) is discussed in connection with the problem of the field product determination at coinciding points. As a result an exact relation between fundamental field dimensions is found for various three-linear interactions (section 16 and Appendix 6). The problem of closing the infinite system of dynamical equations is discussed.Al above said results are demonstrated using Thirring model as an example. A new approach to its solving is developed.The program od closing the infinite system of dynamical equations is discussed. The Thirring model is considered as an example. A new approach to the solution of this model is discussed.Methods are developed for the approximate calculation of dimensions and coupling constants in the 3-vertex and 5-vertex approximations. The dimensions are calculated in the γ_?³ theory in 6-dimensional space.The problem of calculating the critical indices in statistics (3-dimensional Euclidean space) is considered. The calculation of the dimension is carried out in the framework of the γ_?⁴ model. The value of the dimension and the critical indices thus obtained coincide with the experimental ones.     

Small distance behaviour in field theory and power counting

For infinitesimal changes of vertex functions under infinitesimal variation of all renormalized parameters, linear combinations are found such that the net infinitesimal changes of all vertex functions are negligible relative to those functions themselves at large momenta in all orders of renormalized perturbation theory. The resulting linear first order partial differential equations for the asymptotic forms of the vertex functions are, in quantum electrodynamics, solved in terms of one universal function of one variable and one function of one variable for each vertex function whereby, in contrast to the renormalization group treatment of this problem, the universal function is obtained from nonasymptotic considerations. A relation to the breaking of scale invariance in renormalizable theories is described.