【作者】 张莹；

【导师】 李子平；

【作者基本信息】 北京工业大学 ， 理论物理， 2005， 硕士

【摘要】 本文回顾了约束Hamilton 系统的多种量子化方案, 着重叙述了Faddeev-Senjanovic(FS)路径积分量子化方案。基于有限自由度系统相空间Green函数的生成泛函,文中导出了正规/奇异Lagrange 量系统在整体变换下不变的量子正则Noether 定理,并将此量子对称性用于Emden 方程,指出经典对称所联系的守恒量在量子理论中不再保持;用于电子-声子相互作用系统,说明一些经典守恒量在量子水平下仍旧保持;导出了有限自由度系统在定域变换下不变的量子正则Noether 恒等式;导出整体变换下规范场在量子水平下的变换性质方程,用于非Abel Chern-Simons(CS)场,求出了量子BRST 荷,讨论了量子水平下场的共形对称性。Poincaré-Cartan(PC)积分不变量在经典力学和场论中占重要地位,在经典理论中由于它和系统的运动方程等价,可视为动力学的一个基本原理。本文从相空间Green 函数的生成泛函出发,考虑系统的在增广相空间中的变换性质,沿量子系统的运动轨线,导出了普遍情况下场论中正规/奇异Lagrange 量系统的量子PC 积分不变量并推广到了高阶微商系统。证明了当场变量变换的Jacobi 行列式不为1 时,仍可导出量子PC 积分不变,这与量子Noether(第一)定理是不同的。指出了在量子水平下该不变量与量子正则方程等价,从而把经典水平下的PC 积分不变量推广到了量子水平。并讨论了量子PC 积分不变量与正则方程、正则变换和Hamilton-Jacobi 方程之间的联系。由于任意子在凝聚态方面的应用占重要地位而引起人们广泛关注,在场论水平可以用CS 理论来描述任意子的分数自旋和分数统计性质。本文对含CS 项与极化子耦合的模型进行了(FS 路径积分)量子化,研究了其量子对称性,利用量子Noether(第一)定理,得到了量子水平下的守恒量和分数自旋性质。并对含CS 项的O(3)非线性σ模型的Abel 理论与非Abel 理论分别进行了量子水平下对称性的研究,同样得到分数自旋性质。在非Abel CS 理论中系统的量子守恒角动量与经典Noether定理导出的结果不同之处在于还必须考虑鬼粒子对系统角动量的贡献,不能简单的认为经典理论中的结论在量子理论中仍保持有效。更多还原

【Abstract】 Several quantization formalisms for constrained Hamiltonian system are reviewed in this thesis. The Faddeev-Senjanovic(FS) path-integral quantization formalism is a mainly one. Based on the phase-space generating functional of Green function for a regular/singular Lagrangian with finite degrees of freedom, the canonical Noether theorem at the quantum level is derived. Appling to the Emden’s equation, the results show that the conserved quantity at the classical level may not hold true at the quantum level. Appling to a system of interacting electron-phonon, the conserved quantity at the classical level still hold true at the quantum level. Quantal canonical Noether identities under the local transformation for a system with finite degree freedom is derived. The equations of transformation properties for global transformation in gauge field theory at the quantum level are also derived. Appling to non-Abelian Chern-Simons(CS) field, the quantal BRST charge is derived and the properties of conformal symmetries are also discussed in this system. The Poincaré-Cartan integral invariant (PCII) plays an important role in classical mechanics and field theories. It can be treated as a fundamental principle of dynamic in classical theories. In this dissertation, based on the invariance of phase space generating function of Green function, considering the transformation property in the extended phase space, along the quantal motion trajectories, the quantal PCII in field theory for a system with a regular/singular Lagrangian are derived and the results are also generalized to higher-order system. For these cases in which the Jacobian of the transformation does not equal to unity, the quantal PCII can be still derived. These cases are different from the quantal first Noether theorem. It is proved that the PCII is equivalent to the quantal canonical equations, thus the classical PCII is extended to the quantum level. The quantal PCII connected with canonical equations and canonical transformation and Hamilton-Jacobi equations are also discussed. Anyons have attracted much attention due to their possible relevance to更多还原