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What is the solution?
...If, as suggested by J.M. Souriau, God, in His infinite wisdom, had not created particles with negative mass and energy and had not prevented physicists from using antichronous elements, the theory would not be able to handle PT and CPT symmetries.
We present an alternative solution in:
J.P. Petit and P. Midy: "Geometrization of matter and antimatter through the coadjoint action of a group on its momentum space. 4: The Twin Group. Geometrical description of Dirac's antimatter. Geometrical interpretations of antimatter after Feynman and the so-called CPT theorem." Geometrical Physics B, 4, 1998.
...To avoid collisions between positive- and negative-energy particles, we divide the evolution space into two folds, forming the quotient of the group by its orthochronous subgroup. We obtain a twin geometry.
We introduce a fold index f = ±1
f = +1 corresponds to fold F
f = –1 corresponds to fold F*.
The twin group is:
(400)
...It remains an eight-component group. We see that elements with (m = –1), corresponding to PT symmetry, are accompanied by a fold permutation: f → –f.
...The momentum space remains composed of four sectors, but the negative-energy sectors correspond to particle motions within fold F*.
(401)
The following symmetries are:
(402) We can now define the new "playing field." (403)
The playing field: a two-fold space (F and F) associated with a two-sector momentum space (E > 0 and E < 0).*
(404)
Movements of ordinary matter. Action of orthochronous group elements, with l = 1. Charges unchanged.
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 antimatter.
...On the figure, line M1 represents the motion of normal orthochronous matter. We depict straight lines because our group does not account for force fields such as gravitational or electromagnetic fields. It models only the behavior of isolated, charged point masses.
We select an element in the shaded region, corresponding to a matrix (l = –1; m = 1). The value l = –1 reverses the signs of all zᵢ, making them negative. The new trajectory lies in the second sector, corresponding to antimatter. Since l·m = –1, charges are reversed. However, since time is not reversed, the particle's energy and mass remain positive.
This is a geometric description of (orthochronous) antimatter after Dirac.
...Two additional sectors must be explored. In the third sector, we examine the effect of an element (l = –1; m = –1) on momentum and motion.
(l = –1) reverses the {zᵢ}. According to our geometric definition, this new motion corresponds to antimatter, as it occurs in the second sector of space {z₁, z₂, z₃, z₄, z₅, z₆, x, y, z, t}.
(m = –1) produces a PT symmetry, reversing the signs of (x, y, z, t).
But since l·m = +1, charges remain unchanged.
This is "PT-symmetric antimatter," thus providing a geometric description of antimatter after Feynman.
The motion takes place in the second sector of space, within fold F*.
(406)
(l = –1; m = –1) elements transform the motion of normal matter into the motion of antimatter (z-symmetry) of a PT-symmetric object evolving backward in time. Geometric description of Feynman's view of antimatter. Does not fully match Dirac’s: negative mass and negative energy.
The final elements correspond to the sector (l = 1; m = –1).
(l = 1) → the motion remains in the matter sector:
no z-symmetry.
(m = –1) is accompanied by a PT symmetry. The particle evolves backward in time.
(l = –1): C symmetry. Charges are reversed.
...This is CPT-symmetric matter, thus corresponding to a geometric interpretation of the so-called "CPT theorem," which claims that the CPT symmetric counterpart of a particle should be identical to the original particle. This is not true. This motion corresponds to an antichronous motion. The particle evolves backward in time, so (via coadjoint action) its mass and energy become negative.
...The motion of a particle that is the CPT-symmetric counterpart of a normal particle occurs within fold F*.
(407)
(l = 1; m = –1) case. Corresponds to CPT symmetry. But the coadjoint action yields negative mass and energy. The CPT-symmetric counterpart of a matter particle is a matter particle, but with negative mass.