量子水平的Noether恒等式

基于Green函数的相空间生成泛函,导出了定域变换下的量子正则Noether恒等式;对规范不变系统,导出了位形空间中的量子Noether恒等式.指出在某些情形下由量子Noether恒等式可导致系统的量子守恒律,这种求量子守恒律的方法与量子Noether(第一)定理的程式不同.用于非Abel Chern-Simons(CS)理论,求出了BRS和PBRS守恒荷,这两个守恒荷完全不同.     

Quantum Noether identities for non-local transformationsin higher-order derivatives theories

Based on the phase-space generating functional of the Green function for a system with a regular/singular higher-order Lagrangian, the quantum canonical Noether identities (NIs) under a local and non-local transformation in phase space have been deduced, respectively. For a singular higher-order Lagrangian, one must use an effective canonical action I_eff^P in quantum canonical NIs instead of the classical I^P in classical canonical NIs. The quantum NIs under a local and non-local transformation in configuration space for a gauge-invariant system with a higher-order Lagrangian have also been derived. The above results hold true whether or not the Jacobian of the transformation is equal to unity or not. It has been pointed out that in certain cases the quantum NIs may be converted to conservation laws at the quantum level. This algorithm to derive the quantum conservation laws is significantly different from the quantum first Noether theorem. The applications of our formulation to the Yang-Mills fields and non-Abelian Chern-Simons (CS) theories with higher-order derivatives are given, and the conserved quantities at the quantum level for local and non-local transformations are found, respectively.Received: 12 February 2002, Revised: 16 June 2003, Published online: 25 August 2003Z.-P. Li: Corresponding authorAddress for correspondence: Department of Applied Physics, Beijing Polytechnic University, Beijing 100022, P.R. China     

Approximate Noether symmetries

We present a theory of approximate Noether symmetries, based on the concept of approximate infinitesimal point transformations. The theory is applied to the first-order perturbations of the orbit equations corresponding to Kepler's law and Hooke's law. The effect of the perturbations on the orbits are then discussed.     

On the Noether map

For a wide class of lagrangian systems we show rigorously that the conventional formulation of Noether's theorem provides a bijective map from the set of equivalence classes of Noether's symmetries onto the set of equivalence classes of conserved currents. We further discuss if Noether's theorem is generalized in a significant way by several formulations proposed in this decade.Partially supported by the Junta de Energia Nuclear, Madrid.     

Quantum method of projection

The article is devoted to the method of constructing exact solutions of linear differential equations, which is the quantum analog of the method of projection that A. M. Ol'shanetskii and A. P. Perelomov proposed for integration of finite-dimensional Hamiltonian systems. The method developed in the article is demonstrated on a nontrivial example. Omsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 103–108, September, 1996.     

Perturbation to Noether Symmetry and Noether Adiabatic Invariants of General Discrete Holonomic Systems

The perturbation to Noether symmetry and Noether adiabatic invariants of general discrete holonomic systems are studied. First, the discrete Noether exact invariant induced directly from the Noether symmetry of the system without perturbation is given. Secondly, the concept of discrete high-order adiabatic invariant is presented, the criterion of the perturbation to Noether symmetry is established, and the discrete Noether adiabatic invariant induced directly from the perturbation to Noether symmetry is obtained. Lastly, an example is discussed to illustrate the application of the results.     

Emmy Noether and Her Theorems

Today Noether's principal theorem occupies a prominent place in theoretical physics, though for a long time its significance was largely overlooked. Even now, relatively few physicists realize that Emmy Noether's original paper from 1918 contains two fundamental theorems. Moreover, both theorems are essential for understanding her original motivation, namely to distinguish between proper and improper conservation laws in physics.     

关于Noether对称性的两种理解

介绍了对Lagrange系统Noether对称性的两种理解，一种理解为Lagrange函数的不变性，另一种理解为作用量的不变性.研究表明，这两种理解是不同的.在一些条件下，Lagrange函数的不变性可以成为作用量的不变性，在另一些条件下，作用量的不变性可以成为Lagrange函数的不变性.将Noether对称性理解为作用量的不变性是合理的. 关键词： Lagrange系统 Noether对称性 作用量的不变性 Lagrange函数的不变性     

Noether Symmetry and Noether Conserved Quantity of Nielsen Equation for Dynamical Systems of Relative Motion

Noether symmetry of Nielsen equation and Noether conserved quantitydeduced directly from Noether symmetry for dynamical systems of the relative motion are studied. The definition and criteria of Noether symmetry of a Nielsen equation under the infinitesimal transformations of groups are given. Expression of Noether conserved quantity deduced directly from Noether symmetry of Nielsen equation for the system are obtained. Finally, an example is given to illustrate the application of the results.     

Generalized Noether theorems and applications

We generalize the first and second Noether theorems (Noether identities) to a constrained system in phase space. As an example, the conservation law deriving from Lagrange's formalism cannot be obtained fromH _E via the generalized first Noether theorem (GFNT); Dirac's conjecture regarding secondary first-class constraints (SFCC) is invalid in this example. A preliminary application of the generalized Noether identities (GNI) to nonrelativistic charged particles in an electromagnetic field shows that on the constrained hypersurface in phase space one obtains electric charge conservation. This conservation law is valid whether Dirac's conjecture holds true or not.     

Poincar-Chetaev方程的Noether对称性

建立d’AlembertLagrange原理的PoincarChetaev形式,给出原理在无限小变换下的变形形式,由此得到广义Noether等式以及守恒量的形式.举例说明结果的应用.     

Noether equations and conservation laws

The purpose of the paper is to present a rigorous derivation of the relation between conservation laws and transformations leaving invariant the action integral. The (space-)time development of a physical system is represented by a cross section of a product bundleM. A Lagrange function is defined as a mapping where is the bundle space of the first jet extension ofM. A symmetry transformation is defined as a bundle automorphism ofM, carrying solutions of the Euler-Lagrange equation into solutions of the same equation. An important class of symmetry transformations is that of generalized invariant transformations: they are defined by specifying their action on the Euler-Lagrange equation. The generators of generalized invariant transformations are solutions of a system of linear, homogeneous partial differential equation (Noether equations). The set of all solutions of these equations has a natural structure of Lie algebra. In a simple manner, the Noether equations give rise to differential conservation laws.Supported by Air Force Office of Scientifie Research and Aeronautical Research Laboratories.On leave of absence from the Institute of Theoretical Physics, Warsaw University, Warsaw, Poland.     

Conserved Noether currents in stochastic quantization

A general procedure for constructing Noether conserved currents in the stochastic quantization scheme corresponding to symmetries of the equilibrium theory is proposed. Two different regularizations — the Breit-Gupta-Zaks stochastic time regularization and a new supersymmetric regularization — are employed, and the origin of chiral anomalies is exhibited in this framework.     

Generalized Noether identities for nonlocal transformations

Starting from the transformation properties of an action integral of a system under local and nonlocal transformations, we derive the generalized Noether identities for a variant system under those transformations. The applications of the theory to the Yang-Mills field with higher order Lagrangian is presented under the Coulomb gauge condition, a new conserved PBRS charge is found which differs from the BRS conserved charge, and another conserved charge connected with nonlocal transformation is also obtained.     

Noether symmetries of discrete mechanico-electrical systems

This paper focuses on studying Noether symmetries and conservation laws of the discrete mechanico-electricM systems with the nonconservative and the dissipative forces. Based on the invariance of discrete Hamilton action of the systems under the infinitesimal transformation with respect to the generalized coordinates, the generalized electrical quantities and time, it presents the discrete analogue of variational principle, the discrete analogue of Lagrange-Maxwell equations, the discrete analogue of Noether theorems for Lagrange Maxwell and Lagrange mechanico-electrical systems. Also, the discrete Noether operator identity and the discrete Noether-type conservation laws are obtained for these systems. An actual example is given to illustrate these results.     

Noether Symmetry of Three-Order Lagrangian Equations

Based on the three-order Lagrangian equation, pseudo-Hamilton actoon I^＊ is defined and the three-order Hamilton＇s principle and the conditions are obtained in the paper. Then, the Noether symmetry about three-order Lagrangian equations is deduced. Finally, an example is given to illustrate the application of the result.     

Noether Symmetry of Three-Order Lagrangian Equations

Based on the three-order Lagrangian equation, pseudo-Hamilton action I* is defined and the three-order Hamilton‘s principle and the conditions are obtained in the paper. Then, the Noether symmetry about three-order Lagrangian equations is deduced. Finally, an example is given to illustrate the application of the result.