动力学群微分表示的计算:以Lorentz群SO(3,1)为例

The Calculation of Differential Representations of Dynamical Groups: Lorentz Group SO(3,1) as an Example

该文给出了动力学群在群参数空间以及陪集空间上的右、左微分表示和伴随微分表示的符号计算方法.作为例子,计算了Lorentz群SO(3,1)的6-参数和3-参数的右、左及伴随微分表示,这些表示是旋转群SO(3)关于欧拉角和极角的微分表示的相对论性推广.特别,作者给出了伴随微分表示的两种不同的3-参数形式,同时也得到了Wigner小群SO(2,1)和SO(3)的6-参数和3-参数的相应表示.这些表示在相对论性量子陀螺的研究中可得到应用.

The authors present a symbolic computation method for calculating the differential representations for dynamical groups, including the right and the left differential representations, and the adjoint differential representations in the group parameter space and in its roset spaces. As an example, the corresponding representations of Lorentz SO（3,1） group with （i-parameters and 3-parameters are calculated. Especially, two different kinds of 3-parameter adjoint differential representations are obtained. At the same time, the right and the left differential representations, and the adjoint differential representations of Wigner＇s little group SO（2,1） and SO（3） with 6-parameters and 3-parameters are also obtained respectively. These representations are the relativistic generalization of the differential representations of the SO（3） rotation group in the Euler angles and in the polar angles. They may find their applications in the study of relativistic quantum top.