Doubled of Minkowski space and spinor wave function

[bayak]

First of all note that 8-dinensional Finsler space $(t,x,y,z,t^*,x^*,y^*,z^*)$ preserving the metric form
\begin{equation}
S^2 = tt^*-xx^*-yy^*-zz^*,
\end{equation}
actually presents doubled of the Minkowski space.

Then the solution with one-dimensional feature localized on the world line of doubled of Minkowski space $(x, y, z, t,t^*,x^*,y^*,z^*)$, which in the distance from it tends to the vacuum potential, should be considered as particle-like solutions. Moreover, if we are interested in particle-like solutions in which the static part of the line potential features has symmetry of the equator of seven-sphere, and the dynamic characteristics of the line is wound on this equator, then the space which is orthogonal to this line (or rather - a congruence of lines) features can be represented by spinor wave function:
\begin{equation}
\begin{cases}
\psi_1= \frac{z_1}{|z|}e^{iS},\\
\psi_2= \frac{z_2}{|z|}e^{iS},\\
\psi_3= \frac{z_3}{|z|}e^{iS},\\
\psi_4= \frac{z_4}{|z|}e^{iS},
\end{cases}
\end{equation}
where $(z_j = x_{2j- 1} + ix_{2j})_4$ --- a point north pole of the sphere with radius $|z| = \sqrt{z_1\bar{z}_1 + \cdots + z_4\bar{z}_4}$, and $S = k_{x}x + k_{y}y + k_{z}z-k_{t}t$ --- this is the path length features in the space $(t, x, y , z, t^*, x^*, y^*, z^*)$. Note also that if we are interested in particle-like solutions with the symmetry of the pseudo-sphere radius $\sqrt{z_1\bar{z}_1 + z_2\bar{z}_2 - z_3\bar{z}_3-z_4\bar{z}_4}$, but their internal symmetry is broken by the inequality $\sqrt{z_1\bar{z}_1 + z_2\bar{z}_2} \gg\sqrt{z_3\bar{z}_3 + z_4\bar{z}_4}$, we can limit the two-component spinor describing the symmetry on three-dimensional sphere with radius $\sqrt{z_1\bar{z}_1 + z_2\bar{z}_2}$.

Whether it is worth discussing?

 

[micromass]

Please stop using physicsforums to try and discuss this.