忽视定义域苦果吃不尽

定义域作为构成函数的三要素之一,它直接制约着函数的解析式、图像和性质,在解题过程中若忽视定义域这个重要条件,将导致错误.现在对几类题型作扼要的剖析如下.     

Nucleon internal structure: a new set of quark, gluon momentum, angular momentum operators and parton distribution functions

It is unavoidable to deal with the quark and gluon momentum and angular momentum contributions to the nucleon momentum and spin in the study of nucleon internal structure. However we never have the quark and gluon momentum, orbital angular momentum and gluon spin operators which satisfy both the gauge invariance and the canonical momentum and angular momentum commutation relation. The conflicts between the gauge invariance and canonical quantization requirement of these operators are discussed. A new set of quark and gluon momentum, orbital angular momentum and spin operators, which satisfy both the gauge invariance and canonical momentum and angular momentum commutation relation, are proposed. The key point to achieve such a proper decomposition is to separate the gauge field into the pure gauge and the gauge covariant parts. The same conflicts also exist in QED and quantum mechanics and have been solved in the same manner. The impacts of this new decomposition to the nucleon internal structure are discussed.     

局域规范群C~∞(M,G)上不存在有限不变正测度(英文)

研究了对 Faddeev- Popov方法有关键意义的局域规范群上不变测度的存在性问题 .证明了局域规范群 C∞ (M,G) (M为一紧致流形 ,G为一矩阵李群 )上不存在有限的平移不变的正测度. In is proved that on the local gauge group C ∞ (M,G) ( M a compact manifold and G a matrix Lie group) there does not exist a finite positive invariant measure.     

Gauge-Invariant Spin and Orbital Angular Momentum of Laguerre--Gaussian Laser

We adopt a gauge-invariant definition to calculate the spin and orbital angular momenta of a so-called Ith order Laguerre-Gaussian laser. The results reveal that photons on the axis of the beam may carry an orbital angular momentum of （l - 1）h besides lh per photon. For the spin, we obtain a more reasonable expression proportional to the beam intensity instead of the gradient of the intensity as previously derived. We also discuss how to experimentally discriminate the angular momentum expressions given here and those commonly accepted in the literature.     

Gauge invariance and quantization applied to atom and nucleon internal structure

The prevailing theoretical quark and gluon momentum,orbital angular momentum and spin operators,satisfy either gauge invariance or the corresponding canonical commutation relation,but one never has these operators which satisfy both except the quark spin.The conflicts between gauge invariance and the canonical quantization requirement of these operators are discussed.A new set of quark and gluon momentum,orbital angular momentum and spin operators,which satisfy both gauge invariance and canonical momentum and angular momentum commutation relation,are proposed.To achieve such a proper decomposition the key point is to separate the gauge field into the pure gauge and the gauge covariant parts.The same conflicts also exist in QED and quantum mechanics,and have been solved in the same manner.The impacts of this new decomposition to the nucleon internal structure are discussed.