P-定向空间的旋量结构、Kustaanheimo-Stifel和Hopf丛--形式之间的联系
Spinor Structure of P-Oriented Space, Kustaanheimo-Stifel and Hopf Bundle - Connection between Formalisms
摘要：
在工作中的一些关系的三种技术，Hopf的丛，Kustaanheimo-Stiefel的丛，3-空间与旋量结构已被审查。旋量空间被看作是一个与普通向量3-空间相比最小（两倍）扩展的实空间：在这一点上，而不是2π-旋转，现在只有4π-旋转被认为是几何空间中的恒等变换。对于初始未扩展多重矢量或伪矢量的给定P-取向，可以分别构造两个不同的空间旋量，$ξ$和$η$。根据定义，这些旋量为我们提供了扩展空间模型的点，每个旋量与向量空间的点对应$2 \longrightarrow 1。对于这两种模型，给出了旋量η和η的球坐标和抛物坐标的显式参数化，抛物坐标系是最便于定义空间旋量的坐标系。Kustaanheimo-Stiefel的四个实坐标U_{a}和V_{a}分别是复旋量ε和η的实部和虚部，它们服从两个二次约束。因此，在这两种情况下，存在一个从部分3球S_{3}到整个2球S_{2}的Hopf映射。发现了两个空间旋量之间的关系：η=（π- iσ^{2} π ^{*}）/ \sqrt{2}，用Kustaanheimo-Stiefel变量U_{a}和V_{a}表示，它是SO（4.R）的线性变换，不进入SU（2）-旋量生成的子群.
英文摘要：
In the work some relations between three techniques, Hopf's bundle, Kustaanheimo-Stiefel's bundle, 3-space with spinor structure have been examined. The spinor space is viewed as a real space that is minimally (twice as much) extended in comparison with an ordinary vector 3-space: at this instead of 2π-rotation now only 4π-rotation is taken to be the identity transformation in the geometrical space. With respect to a given P-orientation of an initial unextended manyfold, vector or pseudovector one, there may be constructed two different spatial spinors, $ξ$ and $η$, respectively. By definition, those spinors provide us with points of the extended space odels, each spinor is in the correspondence $2 \longrightarrow 1 with points of a vector space. For both models an explicit parametrization of the spinors ξand ηby spherical and parabolic coordinates is given, the parabolic system turns out to be the most convenient for simple defining spacial spinors. Fours of real-valued coordinates by Kustaanheimo-Stiefel, U_{a} and V_{a}, real and imaginary parts of complex spinors ξand ηrespectively, obey two quadratic constraints. So that in both cases, there exists a Hopf's mapping from the part of 3-sphere S_{3} into the entire 2-sphere S_{2}. Relation between two spacial spinor is found: η= (ξ- iσ^{2}ξ^{*})/\sqrt{2}, in terms of Kustaanheimo-Stiefel variables U_{a} and V_{a} it is a linear transformation from SO(4.R), which does not enter its sub-group generated by SU(2)-rotation over spinors.
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