arXiv 0901.25372009-01hep-th

庞加莱群上的场与可定向对象的量子描述

Field on Poincare group and quantum description of orientable objects

D. Gitman, A. Shelepin

备注：: 46 pages
发表信息：: Eur.Phys.J.C61:111-139,2009
DOI：: 10.1140/epjc/s10052-009-0954-x

摘要：

提出了一种相对论定向物体的量子力学描述方法。它概括了维格纳的想法，关于处理非相对论定向对象（特别是，一个非相对论转子）的帮助下，两个参考系（空间固定和身体固定）。这种推广的技术实现（例如，在3+1维中）等于引入依赖于庞加莱群$G$的元素的波函数。测试可定向对象和嵌入空间的对称性的变换的完整集合属于群$Π=G\times G$。所有这些变换都可以通过考虑$G$在群$f(x,z)$上的标量函数空间中的广义正则表示来研究，$f(x,z)$依赖于Minkowski空间点$x\in G/Spin(3,1)$以及矩阵$Z\in Spin(3,1)$的元素$z$给出的方向变量。特别地，场$f(x,z)$是通常的自旋张量多分量场的生成函数。在所考虑的理论中，有四种不同类型的旋量，一个可定向的物体用十个量子数来表征。我们研究了相应的相对论波动方程及其对称性。

英文摘要：

We propose an approach to the quantum-mechanical description of relativistic orientable objects. It generalizes Wigner's ideas concerning the treatment of nonrelativistic orientable objects (in particular, a nonrelativistic rotator) with the help of two reference frames (space-fixed and body-fixed). A technical realization of this generalization (for instance, in 3+1 dimensions) amounts to introducing wave functions that depend on elements of the Poincare group $G$. A complete set of transformations that test the symmetries of an orientable object and of the embedding space belongs to the group $Π=G\times G$. All such transformations can be studied by considering a generalized regular representation of $G$ in the space of scalar functions on the group, $f(x,z)$, that depend on the Minkowski space points $x\in G/Spin(3,1)$ as well as on the orientation variables given by the elements $z$ of a matrix $Z\in Spin(3,1)$. In particular, the field $f(x,z)$ is a generating function of usual spin-tensor multicomponent fields. In the theory under consideration, there are four different types of spinors, and an orientable object is characterized by ten quantum numbers. We study the corresponding relativistic wave equations and their symmetry properties.

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