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.     

Noether恒等式的推广及其应用

从系统的作用量在普遍的定域和非定域变换下的性质出发,导出了含非定城变换的广义Noether恒等式.将其用于高阶微商杨—Mills场论,求出了有别于BRS荷的新PBRS守恒荷和非定域变换下的新守恒荷.     

非定域变换的广义Ward恒等式

基于高阶微商奇异拉氏量系统相空间Green函数的生成泛函,导出了该系统在定域和非定域变换下的广义正则Ward恒等式.对规范不变系统,从位形空间生成泛函出发,导出了该系统在定域、非定域和整体变换下的广义Ward恒等式.用于高阶微商非Abel(Chern-Simons CS)理论,无需作出生成泛函中对正则动量的路径积分,即可导出正规顶角的某些关系.此外还给出了BRS变换下的Ward-Takahashi恒等式.     

奇异拉氏量系统相空间路径积分中的整体对称性

基于奇异拉氏量系统Green函数的相空间生成泛函,导出了相空间中整体变换下的Ward恒等式和整体对称下的量子守恒律.一般它有别于经典Noether守恒律.用于杨-Mills理论,导出了BRS变换下的Ward-Takahashi恒等式和BRS守恒律;用于非Abel-Chern-Simons理论,导出了系统的量子角动量,它有别于经典角动量在于计及了鬼粒子对角动量的贡献.     

高阶微商规范不变系统的整体对称性和量子守恒律

分别从Faddeev–Popov(FP)和Faddeev–Senjanovic(FS)路径积分量子化方法对高阶微商规范不变系统导致的位形空间和相空间生成泛函出发,导出规范系统在量子水平下的守恒律,用于高阶Maxwell非AbelChern–Simons(CS)理论.得到了高阶Maxwell非AbelCS理论与标量场耦合系统的量子BRS守恒荷和量子守恒角动量,无论从位形空间或相空间的生成泛函出发,其结果是相同的.并对CS理论中的分数自旋性质给予了讨论.     

动力系统的正则量子对称性质

分别从正规和奇异拉氏量系统的相空间生成泛函出发,导出了增广相空间中整体对称下的正则形式Ward恒等式.考虑对应的定域交换,得到了量子水平的守恒荷,给出了正则形式的量子Noether定理.讨论了在核子和π介子相互作用中的初步应用.     

广义Noether恒等式与守恒荷

考虑非不变作用理系统在无限连续群下的变换性质,导致了广义Noether恒等式,由此可导出系统的强守恒律和弱守恒律,给出与此相联系的重质量杨-Mills场守恒的PBRS荷,它有别于守恒的BRS荷.讨论了变更性系统的Dirac约束.     

高阶微商系统Dirac猜想的一个反例

由扩展正则作用量导出了高阶微商奇异Lagrange量系统的扩展正则Noether恒等式.从广义约束Hamilton系统相空间中对称性分析，给出高阶微商系统Dirac猜想的一个反例. 用正则Noether定理、 正则Noether恒等式和扩展正则Noether恒等式说明在此反例中Dirac猜想失效, 讨论中没有将约束线性化. 关键词： 高阶微商系统 约束Hamilton系统 正则对称性 Dirac猜想     

高介微商规范不变系统的整体对称性和量子守恒律

分别从Faddeev-Popov(FP)和Faddeev-Senjanovic（FS）路径积分量子化方法对高阶微商规范不变系统导致的位形空间和相空间生成泛函出发，导出规范系统在量子水平下的守恒律，用于高阶Maxwell非Abel Chern-Simons（CS）理论，得到了高阶Maxwell非AbelCS理论与标量场耦合系统的量子BRS守恒荷和量子守恒角动量，无论从闰形空间或相空间的生成泛函出发，其结果是相同的，并对CS理论中的分数自旋性质给予了讨论。     

规范理论中的量子守恒荷

从Faddeev-Popov(F-P)方法对规范理论导致的位形空间生成泛函出发,导出了规范系统在量子情形下的守恒律,用于非Abel Chern-Simons(CS)理论,得到了CS场与Fermi场耦合系统的量子BRS守恒荷和量子守恒角动量. 对CS理论中的分数目旋性质给予了讨论.     

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     

Quantal Noether Identities and Their Applications

Based on the phase-space generating functional of the Green function for a system with a regular/singular Lagrangian, the quantal canonical Noether identities (NI) under the local and non-local transformation in extended phase have been derived, respectively. The result holds true whether the Jacobian of the transformation is equal to unity or not. Based on the configuration-space generating functional of the gauge-invariant system obtained by using Faddeev-Popov (FP) trick, the quantal NI under the local and non-local transformation in configuration space have been also deduced. It is showed that for a system with a singular Lagriangian one must use the effective action in the quantal NI instead of the classical action in corresponding classical NI. It is pointed out that in certain cases, the quantal NI may be converted into the quantal (weak) conservation laws by using the quantal equations of motion. This algorithm to derive the quantal conservation laws differs from the quantal first Noether theorem. The preliminary applications of this formulation to Yang-Mills (YM) fields and non-Abelian Chern-Simons (CS) theories are given. The quantal conserved quantities for non-local transformation in YM fields are obtained. The conserved BRS and PBRS quantities at the quantum level in non-Abelian CS theories are also found. The property of fractional spin in CS theories is discussed. PACS no11.10. Ef; 11.30.−j 11.15. −q.     

Topological Chern—Simons vortices in the O (3) σ -model

Based on the phase-space path integral (functional integral) for a system with a regular or singular Lagrangian, the generalized Ward identities for phase space generating functional under the global transformation in phase space are derived respectively. The canonical Noether theorem at the quantum level is also established. It is pointed out that the connection between the symmetries and conservation laws in classical theories, in general,is no longer preserved in quantum theories. The advantage of our formulation is that we do not need to carry out the integration over the canonical momenta as usually performed. Applying the present formulation to Yang-Mills theory, the quantal BRS conserved quantity and Ward-Takahashi identity for BRS tranformation are derived; the Ward identities for gaugeghost proper vertices and new quantal conserved quantity are also found. In comparison of quantal conservation laws with those one deriving from configuration-space path integral using the Faddeev-Popov(F-P) trick is discussed. A precise study of path-integral quantisation for a nonlinear sigma model with Hopf and Chern-Simons (CS) terms is reexamined. It has been shown that the angular momentum at the quantum level is equal to classical (Noether ) one. Applying our formulation to non-Abelian CS theory, the quantal conserved angular momentum of this system is obtained which differs from classical one in that one needs to take into account the contribution of angular momenta of ghost fields.     

Symmetries in a Constrained System with a Singular Higher-Order Lagrangian

A simple algorithm to construct the generator of gauge transformation for a constrained canonical system with a singular higher-order Lagrangian in field theories is developed. Based on phase-space generating functional of Green function for such a system, the generalized canonical Ward identities under the non-local transformation have been deduced. For the gauge-invariant system, based on configuration-space generating functional, the generalized Ward identities under the non-local transformation have been also derived.The conservation laws are deduced at the quantum level. The applications of the above results to the gauge invariance massive vector field and non-Abelian Chern–Simons(CS) theories with higher-order derivatives are given, a new form of gauge-ghost proper vertices, and Ward–Takahashi identity under BRS transformation and BRS charge at the quantum level are obtained. In the canonical formulation one does not need to carry out the integration over canonical momenta in phase-space path integral as usually performed.     

The Transformation Properties at the Quantum Level in Field Theories for a Gauge-Invariant System with a Higher-Order Lagrangian

Based on the configuration-space generating functional of the Green functions for the gauge-invariant system in higher-order derivatives theories, the equations of the transformation properties at the quantum level have been derived. It follows that the sufficient conditions are found which implies that there exists the conservation laws and the expressions of the quantal conserved laws are also given. Applying the results to the non-Abelian Chern-Simons higher-order derivatives theories, the quantal BRST conserved charge and other conserved charges are found, the transformation properties of the conformal transformation at the quantum level is discussed, the quantal conserved angular momentum is derived, it is pointed out that fractional spin in this system may be also preserved in quantum theories. But the connection between the symmetries and conservation laws in classical theories are not always preserved in quantum theories.     

The quantal Poincaré-Cartan integral invariantfor singular higher-order Lagrangian in field theories

Based on the phase-space generating functional of the Green function for a system with a regular/singular higher-order Lagrangian, the quantal Poincaré-Cartan integral invariant (QPCII) for the higher-order Lagrangian in field theories is derived. It is shown that this QPCII is equivalent to the quantal canonical equations. For the case in which the Jacobian of the transformation may not be equal to unity, the QPCII can still be derived. This case is different from the quantal first Noether theorem. The relations between QPCII and a canonical transformation and those between QPCII and the Hamilton-Jacobi equation at the quantum level are also discussed.Received: 26 May 2004, Revised: 2 December 2004, Published online: 16 March 2005     

Canonical global symmetry and quantal conservation laws for a system with singular higher order Lagrangian

Starting from the phase-space generating functional of the Green function for a system with singular higher order Lagrangian, the generalized canonical Ward identities under the global symmetry transformation in phase space is deduced. The local transformation connected with this global symmetry transformation is studied, and the quantal conservation laws are obtained for such a system. We give a preliminary application to higher derivative Yang-Mills theory; a generalized quantal BRS conserved quantity is found.     

Canonical Symmetries in the Functional Formalism

Based on the phase-space generating functionalof the Green function, the canonical Ward identities(CWI) under local, nonlocal, and global transformationsin phase space for a system with a regular and singular Lagrangian have been derived. Therelation of global canonical symmetries to conservationlaws at the quantum level is presented. The advantage ofthis formulation is that one does not need to carry out the integration over canonicalmomenta in a phase-space path (functional) integral asin the traditional treatment in configuration space. Ingeneral, the connection between global canonicalsymmetries and conservation laws in classical theories isno longer preserved in quantum theories. Applications ofour formulation to the non-Abelian Chern-Simons (CS)theory are given, and new forms for CS gauge-ghost field proper vertices and the quantal conservedangular momentum of this system are obtained; thisangular momentum differs from the classical one in thatone needs to take into account the contribution of angular momenta of ghost fields.