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Geometric description of Dirac's anti-matter.
...We see that l = –1 changes the signs of the cᵢ, which corresponds to a charge conjugation, a C-symmetry.
This provides a geometric description of anti-matter after Dirac (positive energy, positive mass anti-matter).
...Of course, the C-symmetry does not affect the photon, since all its charges are essentially zero. It identifies with its own antiparticle.
Geometric description of Feynman's anti-matter.
...This one is supposed to be PT-symmetric. How can PT-symmetry be introduced into the group?
See: J.P. Petit and P. Midy: "Geometrization of matter and anti-matter via the coadjoint action of a group on its momentum space. 3: Geometric description of Dirac's anti-matter. First geometric interpretation of anti-matter after Feynman and the so-called CPT theorem". Geometrical Physics B, 3, 1998.
The subsequent modification of the group is as follows:
(388)
...It becomes an eight-component group, since the orthochronous part of the Lorentz group has two connected components, hence 2 × 2 × 2 = 8.
This means that we add the antichronous elements:
(389)
Above: we add the antichronous elements to the group.
Below: we add the corresponding half-sector of momentum space associated with negative-energy motion.
In short: we expand the domain of action, which becomes:
(390)
On (388), we see that elements with (m = –1) reverse spacetime, realize PT-symmetry, and correspond to:
(391) Lst = – Ln Lt = – Ls
We obtain the following symmetries in momentum space:
(392)
The calculation of the coadjoint action of group (388) on its momentum space yields:
(393)
...It then becomes straightforward to examine the effect of each component on momentum and motion. We consider a reference motion and momentum J+1, corresponding to positive-energy matter (the effect on positive-energy photons will be analyzed in a second step). The sector of the group in which the element is chosen will be shaded gray.
Next, motions of ordinary matter.
l = +1, m = +1
l m = +1
Charges remain unchanged. The motion M2 corresponds to orthochronous matter with positive mass (E > 0).
(394)
Motions of ordinary matter. Action of orthochronous group elements, with l = 1. Charges unchanged. (395)
Coadjoint action of a group element (l = –1; m = +1) on the momentum associated with normal matter motion: the new motion corresponds to Dirac's anti-matter.
...The element is selected from the gray sector. It is an "anti-element," transforming matter into anti-matter: l = –1 reverses the signs of the extra dimensions, which constitutes our geometric definition of anti-matter.