f4403 Geometrization of matter and antimatter through the coadjoint action of a group on its momentum space. 3 : Geometrical description of Dirac's antimatter. A first geometrical interpretation of antimatter
after Feynman and the so-called CPT theorem. (p3)
...Two more sectors have to be explored. In the third, we examine the impact of the ( l = - 1 ; m = - 1 ) element on the momentum and movement.
...( l = - 1 ) reverses the {z i}. According to our geometric definition, this new movement corresponds to antimatter, as it takes place in the second sector of space { z 1 , z 2 , z 3 , z 4 , z 5 , z 6, x, y , z , t }.
( m = - 1 ) gives a PT-symmetry, reverses the signs of ( x, y , z , t )
...But ( l m = + 1 ) keeps the charges unchanged. This is "PT-symmetric antimatter", so that it is a geometric description of antimatter after Feynman.
...But the group belongs to the antichron sector, so that (coadjoint action) the energy and the mass of the particle is reversed.
...A PT-symmetrical object does not fully coincide with Dirac's antimatter, for it changes the sign of the mass. If such particles exist, they can produce full annihilation with positive mass particles.
. **Fig. 6 : ****( **l= -1 ; m = -1 ) **elements transform movement of normal matter into movement of antimatter **(z-Symmetry) of PT-symmetrical object, running backward in time. Geometric description of Feynman's vision of antimatter. Does not fully coincide with Dirac's one : negative mass and negative energy.
The last elements correspond to the sector ( l= 1 ; m = -1 )
( l = 1 ) --- > the movement is still in the matter's sector :
no z-Symmetry.
( m = -1 ) implies a PT-symmetry. The particle runs backward in time.
( l = -1 ) : C-Symmetry. The charges are reversed.
...This is CPT-symmetrical matter, so that it corresponds to a geometrical interpretation of the so-called "CPT theorem", which asserts that the CPT-symmetric of a particle should be identical to that particle. That's not true. This movement corresponds to an antichron movement. The particle goes backward in time, so that (coadjoint action) its mass and energy become negative .
If CPT-symmetrical particles do exist and if they collide with normal particles, complete annihilation occurs.
. **Fig.7 : ( **l = 1 ; m = - 1 ) case. Corresponds to CPT-symmetry. But the coadjoint action gives negative mass and energy. The CPT-symmetric of a particle of matter is a particle of matter, but with negative mass. ...Now, examine the impact on photons' movement and moment. The z-Symmetry has no impact on it : there is no "antiphoton". As all the charges of the photon are zero, it does not change it. It is identical to its antiparticle.
...The coadjoint action of orthochron components modifies the movement and the moment of the photon, but keeps unchanged its energy. See figure 8.
. Fig. 8 : Coadjoint action of orthochron elements on photon's movement and moment. ** **
** ** . **Fig.9 ** : The coadjoint action of antichron elements on photon's movement and moment, reverses the photon energy : it travels backwards in time. ** **
