Geometry of 3-Spaces with Spinor Structure

TL;DR

This paper explores the mathematical structure of spinors in 3-space, introducing models based on square roots of vectors, and examines their properties using various coordinate systems and domain doubling techniques.

Contribution

It develops a novel mathematical framework for representing 3-space spinor structures through domain doubling and explicit spinor field mappings.

Findings

Different spinor models are constructed using domain doubling.

Explicit mappings between pseudo vector and proper vector spinor models are provided.

The approach applies to various curvilinear coordinate systems.

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.