Separation of spin and charge in the continuum Schrödinger equation

I describe here the attempt to introduce spin-charge separation in Schrodinger equation. The construction we present here gives a decomposed Schrodinger spinor that has one problem: Its absolute value can only have value between 0 and ${1}{2}$. The problem we solve is to expand and generalize this construction so that one can have a Schrodinger spinor with absolute value that are arbitrary non-negative numbers. It may be that one has to introduce a set of different decompositions to cover all nonnegative values, that is to introduce patches over $\mathbb{R}_{+}^{3}$ so that in each patch one has a different representation. It seems that the decomposition has a direct relation to so called entangled states that have been discussed very much in connection of e.g. quantum computing, and we would like to find this relation and discuss it in detail.

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