Space with spinor structure and analytical properties of the solutions of Klein-Fock and Schrodinger equations in cylindric parabolic coordinates

TL;DR

This paper explores the implications of extending physical space to include spinor structure using cylindrical parabolic coordinates, constructing solutions to Klein-Fock and Schrödinger equations, and analyzing their properties in vector and spinor spaces.

Contribution

It introduces a novel extension of space with spinor structure in cylindrical parabolic coordinates and constructs solutions to fundamental quantum equations within this framework.

Findings

Solutions form four types, with two being single-valued in vector space and two having discontinuities.

All solutions are orthogonal in the extended spinor domain.

Selection rules differ significantly between vector and spinor coordinate spaces.

Abstract

Possible quantum mechanical corollaries of changing the vectorial geometrical model of the physical space, extending it twice, in order to describe its spinor structure (in other terminology and emphasis it is known as the Hopf's bundle) are investigated. The extending procedure is realized in cylindrical parabolic coordinates. It is done through expansion twice as much of the domain G so that instead of the half plane (u,v>0) now the entire plane (u,v) should be used accompanied with new identification rules over the boundary points. Solutions of the Klein-Fock and Schrodinger equations are constructed in terms of parabolic cylinder functions. Four types of solutions are possible: \Psi_{++}, \Psi_{--} ; \Psi_{+-}, \Psi_{-+}. The first two \Psi_{++}, \Psi_{--} provide us with single-valued functions of the vectorial space points, whereas last two \Psi_{+-},\Psi_{-+} have discontinuities in the frame of vectorial space and therefore they must be rejected in this model. All four types of functions are continuous ones being regarded in the spinor space. It is established that all solutions \Psi_{++}, \Psi_{--}, \Psi_{+-}, \Psi_{-+} are orthogonal to each other provided that integration is done over extended domain parameterizing the spinor space. Simple selection rules for matric elements of the vector and spinor coordinates, (x,y) and (u,v), respectively, are derived. Selection rules for (u,v) are substantially different in vector and spinor spaces.