Physics of Superluminal Communication and Estakhr Relativistic Omega Factor

Superluminal communication is a process by which one might send information at FTL (Faster Than Light). I try to developed this idea in detail and with mathematical rigor. The velocity of particle (information) is represented by the group velocity $v_g$. if $v_g\ge c$ then $\gamma=\frac{-i}{\sqrt{\frac{v_g^2}{c^2}-1}}=-i\Omega$ that which means $\gamma$ (Lorentz factor) is an imaginary number (at $v_g\ge c$), that can be written as a real number multiplied by imaginary unit $i$, which is defined by its property $i^2=-1$. and this is $\Omega=\frac{1}{\sqrt{\frac{v_g^2}{c^2}-1}}$ Estakhr Omega factor. then kinetic energy of FTL particle is Complex number $k=E-E_o=-E_o(i\Omega+1)$. we still use Lorentz Symmetry, $\gamma^2-\gamma^2\beta^2=1$ which means faster than light is particle-like, $i^2\Omega^2-i^2\Omega^2\beta^2=\Omega^2\beta^2-\Omega^2=1$. The phase velocity can be found from $v_{ph}=\frac{c^2}{v_g}$, this shows that the phase velocity of FTL particle is less than the speed of light $v_{ph}=\frac{c^2}{v_g\ge c}\le c$. which means that speed of material particles can exceed $c$ but finally, the product of the group and phase velocities is equal to $c^2$, in general: if $v_g\le c$ then $v_{ph}\ge c$, if $v_g\ge c$ then $v_{ph}\le c$, if $v_g=c$ then $v_{ph}=c$ i.e., $v_{g}v_{ph}=c^2$.