Conformal invariance in quantum gravity

Trace anomalies in a conformal invariant theory do not arise when its conformal invariance in four dimensions is extended to an arbitrary number n of space-time dimensions: the theory can be made finite in any order of perturbation theory by conformal invariant counterterms in n dimensions. Such an extension of conformal invariance is possible provided one works in the framework of spontaneously broken conformal invariance. This is shown explicitly by working out several examples at the one-loop level and by examining the Ward identities which lead to a general proof.We speculate upon possible consequences of these results on the nature of gravitation and other fundamental interactions.     

Vacuum effects in quantum field theory

We discuss some consequences of applying a procedure developed in a previous paper, to implement the conformal and general invariance of the functional integral. It is shown that spontaneous breaking of those symmetries is unavoidable; we consider in particular the linear σ-model, in the conformal limit, and the Einstein gravitation and derive some simple relations among the vacuum expectation value, a phenomenological cut-off and the Lagrangian constants.     

General relations of heavy quark-antiquark potentials induced by reparameterization invariance

A set of general relations between the spin-independent and spin-dependent potentials of heavy quark and anti-quark interactions are derived from reparameterization invariance in the Heavy Quark Effective Theory. It covers the Gromes relation and includes some new interesting relations which are useful in understanding the spin-independent and spin-dependent relativistic corrections to the leading order nonrelativistic potential.Work supported by the National Natural Science Foundation of China     

Broken conformal invariance and spectrum of anomalous dimensions in QCD

Employing the operator algebra of the conformal group and the conformal Ward identities, we derive the constraints for the anomalies of dilatation and special conformal transformations of the local twist-2 operators in Quantum Chromodynamics. We calculate these anomalies in the leading order of perturbation theory in the minimal subtraction scheme. From the conformal consistency relation we derive then the off-diagonal part of the anomalous dimension matrix of the conformally covariant operators in the two-loop approximation of the coupling constant in terms of these quantities. We deduce corresponding off-diagonal parts of the Efremov-Radyushkin-Brodsky-Lepage kernels responsible for the evolution of the exclusive distribution amplitudes and non-forward parton distributions in the next-to-leading order in the flavour singlet channel for the chiral-even parity-odd and -even sectors as well as for the chiral-odd one. We also give the analytical solution of the corresponding evolution equations exploiting the conformal partial wave expansion.     

Projective invariance and the fifth coordinate

The paper considers the consequences of the principle of parametric (projective) invariance in the context of the general variational problem for stationary curves in a four-dimensional space-time. The necessity is shown of introducing a parameter along the trajectory as a fifth coordinate. The condition of cylindricality along it then acquires an obvious significance. The projective invariants are calculated for the trajectories of charged test particles in a gravielectromagnetic field. It is shown that one of these coincides with the density of the electromagnetic-field Lagrangian.     

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.     

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.     

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.     

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.     

Discrete symmetrical perturbation and variational algorithm of disturbed Lagrangian systems

We investigate the perturbation to discrete conformal invariance and the adiabatic invariants of Lagrangian systems. A variational algorithm is proposed for a system subjected to the perturbation quantities. The discrete determining equations of the perturbations to conformal invariance are established. For perturbed Lagrangian systems, the condition of the existence of adiabatic invariant is derived from the discrete perturbation to conformal invariance. The numerical simulations demonstrate that the variational algorithm has the higher precision and the longer time stability than the standard numerical method.     

Conformal invariance and Hojman conserved quantities for holonomic systems with quasi-coordinates

We propose a new concept of the conformal invariance and the conserved quantities for holonomic systems with quasi-coordinates. A one-parameter infinitesimal transformation group and its infinitesimal transformation vector of generators for holonomic systems with quasi-coordinates are described in detail. The conformal factor in the determining equations of the Lie symmetry is found. The necessary and sufficient conditions of conformal invariance, which are simultaneously of Lie symmetry, are given. The conformal invariance may lead to corresponding Hojman conserved quantities when the conformal invariance satisfies some conditions. Finally, an illustration example is introduced to demonstrate the application of the result.     

Conformal invariance and conserved quantities of Appell systems under second-class Mei symmetry

In this paper we introduce the new concept of the conformal invariance and the conserved quantities for Appell systems under second-class Mei symmetry. The one-parameter infinitesimal transformation group and infinitesimal transformation vector of generator are described in detail. The conformal factor in the determining equations under second-class Mei symmetry is found. The relationship between Appell system’s conformal invariance and Mei symmetry are discussed. And Appell system’s conformal invariance under second-class Mei symmetry may lead to corresponding Hojman conserved quantities when the conformal invariance satisfies some conditions. Lastly, an example is provided to illustrate the application of the result.     

Lorentz invariance and origin of symmetries

We reconsider the role of Lorentz invariance in the dynamical generation of the observed internal symmetries. We argue that, generally, Lorentz invariance can be imposed only in the sense that all Lorentz noninvariant effects caused by the spontaneous breakdown of Lorentz symmetry are physically unobservable. The application of this principle to the most general relativistically invariant Lagrangian, with arbitrary couplings for all the fields involved, leads to the appearance of a symmetry and, what is more, to the massless vector fields gauging this symmetry in both Abelian and non-Abelian cases. In contrast, purely global symmetries are generated only as accidental consequences of the gauge symmetry.     

Reconstruction of non-forward evolution kernels

We develop a framework for the reconstruction of the non-forward kernels which govern the evolution of twist-two distribution amplitudes and off-forward parton distributions beyond leading order. It is based on the knowledge of the special conformal symmetry breaking part induced by the one-loop anomaly and conformal terms generated by forward next-to-leading order splitting functions, and thus avoids an explicit two-loop calculation. We demonstrate the formalism by applying it to the chiral odd and flavour singlet parity odd sectors.     

Conformal invariance and conserved quantity of Hamilton systems

This paper studies conformal invariance and conserved quantities of Hamilton system. The definition and the determining equation of conformal invariance for Hamilton system are provided. The relationship between the conformal invariance and the Lie symmetry are discussed, and the necessary and sufficient condition that the conformal invariance would be the Lie symmetry of the system under the infinitesimal one-parameter transformation group is deduced. It gives the conserved quantities of the system and an example for illustration.     

Reparameterization invariance of NRQED self-energy corrections and improved theory for excited D states in hydrogenlike systems

Canonically, the quantum electrodynamic radiative corrections in bound systems have been evaluated in photon energy regularization, i.e., using a noncovariant overlapping parameter that separates the high-energy relativistic scales of the virtual quanta from the nonrelativistic domain. Here, we calculate the higher-order corrections to the one-photon self-energy calculation with three different overlapping parameters (photon energy, photon mass and dimensional regularization) and demonstrate the reparameterization invariance of nonrelativistic quantum electrodynamics (NRQED) using this particular example. We also present new techniques for the calculation of the low-energy part of this correction, which lead to results for the Lamb shift of highly excited states that are important for high-precision spectroscopy.     

Hierarchical order of Galilei and Lorentz invariance in the structure of matter

The structure of matter shows a hierarchical order: (1) from Lorentz invariance in high-energy physics; (2) to Galilei invariance in the low-energy nonrelativistic limit of high-energy physics; and (3) again to Lorentz invariance in condensed matter physics (where the velocity of sound takes the place of the velocity of light). The hierarchical order can be continued downward further to: (4) non-relativistic (velocity small compared to the velocity of sound) condensed matter excitons, obeying Galilei invariance; and (5) to excitonic matter obeying Lorentz invariance with an excitonic matter sound velocity. It was previously conjectured that Lorentz invariance of high-energy physics is preceded by Galilei invariance at the Planck scale. Still further, the conjectured Galilei invariance at the Planck scale may be the result of an underlying five-dimensional non-Euclidean conform invariant metric structure, with three spatial and two time dimensions, compactified onto three spatial and one time dimension.     

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.     

一般完整系统Mei对称性的共形不变性与守恒量

研究一般完整系统Mei对称性的共邢不变性与守恒量.引入无限小单参数变换群及其生成元向量,定义一般完整系统动力学方程的Mei对称性共形不变性,借助Euler算子导出Mei对称性共形不变性的相关条件,给出其确定方程.讨论共形不变性与Noether对称性、Lie对称性以及Mei对称性之间的关系.利用规范函数满足的结构方程得到系统相应的守恒量.举例说明结果的应用. 关键词： 一般完整系统 Mei对称性 共形不变性 守恒量     

Conformal invariance and the six-dimensional formalism

Conformal invariance is discussed assuming the equations are well defined in arbitrary coordinate systems. This assumption leads to some constraints on scale dimensions of terms, and constraints on the introduction of ‘conformally invariant massive equations’. The six-dimensional formalism is then discussed, and is generalized to project to all conformally flat spaces. Finally the imbedding of Minkowski space equations is studied.SO(4, 2) breaking is seen to enter due to the presence of a non-invariant scalar field, and a non-invariant vector field. The theorem relating invariance of the six-space equations underSO(4, 2) to the invariance of their corresponding four-space equations under the conformal group is carefully stated and proved.