Spinor Structure of P-Oriented Space, Kustaanheimo-Stifel and Hopf Bundle - Connection between Formalisms

TL;DR

This paper explores the relationships between Hopf's bundle, Kustaanheimo-Stiefel's bundle, and spinor structures in 3-space, providing explicit parametrizations and mappings that connect these formalisms in a geometric context.

Contribution

It introduces explicit parametrizations of spatial spinors using spherical and parabolic coordinates and establishes relations between different spinor models and their mappings.

Findings

Explicit spinor parametrizations in spherical and parabolic coordinates.

Existence of Hopf mappings from 3-sphere to 2-sphere.

Linear transformation relations between different spinor models.

Abstract

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 2\pi-rotation now only 4\pi-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, $\xi$ and $\eta$, 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 \xi and \eta by 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 \xi and \eta respectively, 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: \eta = (\xi - i\sigma ^{2}\xi ^{*})/\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.