Geometry of 3-Spaces with Spinor Structure

Abstract

A special approach to examine spinor structure of 3-space is proposed. It is based on the use of the concept of a spatial spinor defined through taking the square root of a real-valued 3-vector. Two sorts of spatial spinor according to P-orientation of an initial 3-space are introduced: properly vector or pseudo vector one. These spinors, \eta and \xi, turned out to be different functions of Cartesian coordinates. To have a spinor space model, you ought to use a doubling vector space. The main idea is to develop some mathematical technique to work with such extended models. Two sorts of spatial spinors are examined with the use of curvilinear coordinates (y_{1},y_{2},y_{3}): cylindrical parabolic, spherical and parabolic ones. Transition from vector to spinor models is achieved by doubling initial parameterizing domain G(y_{1},y_{2},y_{3}) \Longrightarrow \tilde{G}(y_{1},y_{2},y_{3}) with new identification rules on the boundaries. Different spinor space models are built on explicitly different spinor fields \xi(y) and \eta(y). Explicit form of the mapping spinor field \eta(y) of pseudo vector model into spinor \xi(y) of properly vector one is given, it contains explicitly complex conjugation.