运动方程和诺特定理是经典场论中的核心概念。然而在实际的量子场论中,需要使用量子运动方程和量子诺特定理来研究一系列与物理观测量相关的关联函数或者矩阵元。经典场论中的运动方程和诺特定理与其量子版本之间具有非常微妙,但却十分...运动方程和诺特定理是经典场论中的核心概念。然而在实际的量子场论中,需要使用量子运动方程和量子诺特定理来研究一系列与物理观测量相关的关联函数或者矩阵元。经典场论中的运动方程和诺特定理与其量子版本之间具有非常微妙,但却十分重要的联系与区别。透析这一点,无论对于前沿工作者,抑或是初学场论的研究生,都具有非常重要的意义。它揭示了经典物理与量子物理之间的本质差别。本文利用路径积分量子化方法,从第一性原理出发,基于泛函计算体系,推导出了量子版本的运动方程与诺特定理,详细探讨了其与经典版本之间的联系与区别。这对于量子物理的前沿计算具有重要的指导和规范作用,也为初学场论的研究生快速进入前沿领域的研究,提供了一条脉络清晰,可操作性很强的道路。Equation of motion and Noether’s theorem are core concepts in classical field theory. However, in practical quantum field theory, it is necessary to use quantum equation of motion and quantum Noether’s theorem to study a series of correlation functions or matrix elements related to physical observables. There is a very subtle yet crucial connection and difference between the classical and quantum versions of the equation of motion and Noether’s theorem. Analyzing this is of great importance not only for researchers at the forefront but also for graduate students new to field theory. It reveals the fundamental differences between classical and quantum physics. This paper utilizes the path integral quantization method, starting from first principles, based on functional calculation, to derive the quantum versions of the equations of motion and Noether’s theorem, and it thoroughly discusses their connections and differences with the classical versions. This serves as an important guide and standard for frontier calculations in quantum physics and also provides a clear and practical path for graduate students new to field theory to quickly engage in forefront research.展开更多
Based on generalized Apell-Chetaev constraint conditions and to take the inherent constrains for singular Lagrangian into account,the generalized canonical equations for a general mechanical system with a singular hig...Based on generalized Apell-Chetaev constraint conditions and to take the inherent constrains for singular Lagrangian into account,the generalized canonical equations for a general mechanical system with a singular higher-order Lagrangian and subsidiary constrains are formulated. The canonical symmetries in phase space for such a system are studied and Noether theorem and its inversion theorem in the generalized canonical formalism have been established.展开更多
In the light of the local Lorentz transformations and the general Noether theorem, a new formulate ofthe general covariant energy-momentum conservation law in f(R) gravity is obtained, which does not depend on thecoor...In the light of the local Lorentz transformations and the general Noether theorem, a new formulate ofthe general covariant energy-momentum conservation law in f(R) gravity is obtained, which does not depend on thecoordinative choice.展开更多
Self-adjoint theorem is introduced to match the corresponding functional of the given differential equations,and then Noether＇s theorem is used to determine the extended conservation laws of the original equations. F...Self-adjoint theorem is introduced to match the corresponding functional of the given differential equations,and then Noether＇s theorem is used to determine the extended conservation laws of the original equations. Finally, as the application of the method, the conservation laws of Drinfel＇d-Sokolov-Wilson equation and Benjamin-Bona-Mahony equation are constructed.展开更多
In this paper we give a new method to investigate Noether symmetries and conservation laws of nonconservative and nonholonomic mechanical systems on time scales , which unifies the Noether's theories of the two ca...In this paper we give a new method to investigate Noether symmetries and conservation laws of nonconservative and nonholonomic mechanical systems on time scales , which unifies the Noether's theories of the two cases for the continuous and the discrete nonconservative and nonholonomic systems. Firstly, the exchanging relationships between the isochronous variation and the delta derivatives as well as the relationships between the isochronous variation and the total variation on time scales are obtained. Secondly, using the exchanging relationships, the Hamilton's principle is presented for nonconservative systems with delta derivatives and then the Lagrange equations of the systems are obtained. Thirdly, based on the quasi-invariance of Hamiltonian action of the systems under the infinitesimal transformations with respect to the time and generalized coordinates, the Noether's theorem and the conservation laws for nonconservative systems on time scales are given. Fourthly, the d'Alembert-Lagrange principle with delta derivatives is presented, and the Lagrange equations of nonholonomic systems with delta derivatives are obtained. In addition, the Noether's theorems and the conservation laws for nonholonomic systems on time scales are also obtained. Lastly, we present a new version of Noether's theorems for discrete systems. Several examples are given to illustrate the application of our results.展开更多
In this work we study the Lagrangian and the conservation laws for a wave equation with a dissipative source. Using semi-inverse method, we show that the equation possesses a nonlocal Lagrangian with an auxiliary func...In this work we study the Lagrangian and the conservation laws for a wave equation with a dissipative source. Using semi-inverse method, we show that the equation possesses a nonlocal Lagrangian with an auxiliary function.As a result, from a modified Noether's theorem and the nonclassical Noether symmetry generators, we construct some conservation laws for this equation, which are different from the ones obtained by Ibragimov's theorem in [Y. Wang and L. Wei, Abstr. App. Anal. 2013(2013) 407908]. The results show that our method work for arbitrary functions f(u)and g(u) rather than special ones.展开更多
文摘运动方程和诺特定理是经典场论中的核心概念。然而在实际的量子场论中,需要使用量子运动方程和量子诺特定理来研究一系列与物理观测量相关的关联函数或者矩阵元。经典场论中的运动方程和诺特定理与其量子版本之间具有非常微妙,但却十分重要的联系与区别。透析这一点,无论对于前沿工作者,抑或是初学场论的研究生,都具有非常重要的意义。它揭示了经典物理与量子物理之间的本质差别。本文利用路径积分量子化方法,从第一性原理出发,基于泛函计算体系,推导出了量子版本的运动方程与诺特定理,详细探讨了其与经典版本之间的联系与区别。这对于量子物理的前沿计算具有重要的指导和规范作用,也为初学场论的研究生快速进入前沿领域的研究,提供了一条脉络清晰,可操作性很强的道路。Equation of motion and Noether’s theorem are core concepts in classical field theory. However, in practical quantum field theory, it is necessary to use quantum equation of motion and quantum Noether’s theorem to study a series of correlation functions or matrix elements related to physical observables. There is a very subtle yet crucial connection and difference between the classical and quantum versions of the equation of motion and Noether’s theorem. Analyzing this is of great importance not only for researchers at the forefront but also for graduate students new to field theory. It reveals the fundamental differences between classical and quantum physics. This paper utilizes the path integral quantization method, starting from first principles, based on functional calculation, to derive the quantum versions of the equations of motion and Noether’s theorem, and it thoroughly discusses their connections and differences with the classical versions. This serves as an important guide and standard for frontier calculations in quantum physics and also provides a clear and practical path for graduate students new to field theory to quickly engage in forefront research.
文摘Based on generalized Apell-Chetaev constraint conditions and to take the inherent constrains for singular Lagrangian into account,the generalized canonical equations for a general mechanical system with a singular higher-order Lagrangian and subsidiary constrains are formulated. The canonical symmetries in phase space for such a system are studied and Noether theorem and its inversion theorem in the generalized canonical formalism have been established.
基金Supported by the National Natural Science Foundation of China under Grant No.10905027
文摘In the light of the local Lorentz transformations and the general Noether theorem, a new formulate ofthe general covariant energy-momentum conservation law in f(R) gravity is obtained, which does not depend on thecoordinative choice.
基金Supported by "Math + X" Fund of Dalian University of Technology, Science Foundation of Dalian University of Technology under Grant No. SFDUT0808the National Key Basic Research Development of China under Grant No. 2004CB318000
文摘Self-adjoint theorem is introduced to match the corresponding functional of the given differential equations,and then Noether＇s theorem is used to determine the extended conservation laws of the original equations. Finally, as the application of the method, the conservation laws of Drinfel＇d-Sokolov-Wilson equation and Benjamin-Bona-Mahony equation are constructed.
基金supported by the National Natural Science Foundations of China (Grant Nos.11072218 and 11272287)the Natural Science Foundations of Zhejiang Province of China (Grant No.Y6110314)
文摘In this paper we give a new method to investigate Noether symmetries and conservation laws of nonconservative and nonholonomic mechanical systems on time scales , which unifies the Noether's theories of the two cases for the continuous and the discrete nonconservative and nonholonomic systems. Firstly, the exchanging relationships between the isochronous variation and the delta derivatives as well as the relationships between the isochronous variation and the total variation on time scales are obtained. Secondly, using the exchanging relationships, the Hamilton's principle is presented for nonconservative systems with delta derivatives and then the Lagrange equations of the systems are obtained. Thirdly, based on the quasi-invariance of Hamiltonian action of the systems under the infinitesimal transformations with respect to the time and generalized coordinates, the Noether's theorem and the conservation laws for nonconservative systems on time scales are given. Fourthly, the d'Alembert-Lagrange principle with delta derivatives is presented, and the Lagrange equations of nonholonomic systems with delta derivatives are obtained. In addition, the Noether's theorems and the conservation laws for nonholonomic systems on time scales are also obtained. Lastly, we present a new version of Noether's theorems for discrete systems. Several examples are given to illustrate the application of our results.
基金Supported by National Natural Science Foundation of China under Grant No.11101111Zhejiang Provincial Natural Science Foundation of China under Grant Nos.LY14A010029 and LY12A01003
文摘In this work we study the Lagrangian and the conservation laws for a wave equation with a dissipative source. Using semi-inverse method, we show that the equation possesses a nonlocal Lagrangian with an auxiliary function.As a result, from a modified Noether's theorem and the nonclassical Noether symmetry generators, we construct some conservation laws for this equation, which are different from the ones obtained by Ibragimov's theorem in [Y. Wang and L. Wei, Abstr. App. Anal. 2013(2013) 407908]. The results show that our method work for arbitrary functions f(u)and g(u) rather than special ones.
