The Dirac equation says that the anti-particle is the particle (refer to my first post.) One equation, the relativistic Dirac equation, describes four plane waveform solutions for one 'element.' Two of these plane waveform solutions of the form i(kx-wt) and i(-kx-wt) are for a positive energy solution for the electron. The other two of these plane waveform solutions of the form i(kx+wt) and i(-kx+wt) are for a negative energy solution for the positron. Look at the Dirac equation and plug in these plane waveform solutions and you will see that this is true. Also, the positron plane waveform solutions produce negative energy for the time-independent Schrodinger equation if and only if they are hyperbolic. They are hyperbolic if the imaginary number is removed from the superscript of the exponential part of the plane waveform solution.
This suggests that they are pseudo-spherical as well and from David Hestenes' spacetime algebra solution for the real Dirac equation using geometric algebra, this is elicited from the term (iR) in the plane waveform solution for the positron.
Please refer to David Hestenes' seminal paper 'Mysteries and Insights into the Dirac Equation.' Pseudo-spheres have an imaginary radius for real axes or a real radius for imaginary axes.
In my paper referred to in the previous post, the Unexpected Lorentz Transformation reveals a negative Lorentz factor which relates to negative energy. Note that the derivation relates to the plane waveform solutions for the positron in the Dirac equation.
-x+ct and x+ct are used in the derivation which relates to the waveforms i(kx + wt) and i(-kx+wt). From special relativity, there is nothing prohibiting light waves from propagating backwards in time. Proper time for light waves is zero. From a timelike perspective, light waves appear to propagate forwards in time. From a spacelike perspective, light waves appear to propagate backwards in time.
Matter is spherical and has positive curvature. Anti-matter is pseudo-spherical and has negative curvature. The mathematics and physics is all there as I have just described. I am not saying anything that is outlandish. The real Dirac equation, special relativity, the Lorentz transformation and the time-independent Schrodinger equation all reveal this. Consider too spacelike Minkowski spacetime. Just as timelike Minkowski spacetime has three real space axes and an imaginary time axis, spacelike Minkowski spacetime has three imaginary space axes and a real time axis. From geometric algebra, there are three imaginary space axes and a real time axis as the basis set for the spacetime that have not as yet been given a physical description. The physical description is a spacelike Minkowski spacetime. {x,y,z,it} is orthogonal to {ix,iy,iz,t}. ix and iy and iz describe a pseudo-sphere. real t means forwards in time. imaginary t means backwards in time. Feynman said this as well (that positrons and anti-particles move backwards in time.)