f4501 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. . Jean-Pierre Petit and Pierre Midy **Observatory of Marseille ** **France. ** ---
Abstract.
Starting from the work of reference [3], we modify the model in order to avoid encounters between positive and negative mass particles. The solution is to build a two-fold space (F, F*) of eighteen dimensions as the quotient of the group by its orthochronous subgroup.
We then obtain two spaces with opposite arrows of time.
We study the impact of the different components of the group on the momentum and movement spaces. It is shown that the matter-antimatter duality occurs in both folds, in both universes. This work brings a new understanding of antimatter, through geometrical tools. Thus, Dirac's antimatter is the antimatter of our own fold. The matter of the second fold is CPT-symmetric with respect to ours. The PT-symmetric of a matter particle belonging to our fold is the antimatter of the other fold. The matter and antimatter particles of our universe have positive mass and energy. The matter and antimatter particles of the second fold have negative mass and energy.
1) Introduction.
In a previous paper [1], we introduced a geometrical definition of antimatter, through a z-symmetry. Charged mass-points are supposed to move in a ten-dimensional space, divided into two sectors:
{ z i > 0 } : and { z i < 0 }. The first corresponds to the movement of matter, the second to the movement of antimatter.
By the way, photons follow the { z i = 0 } surface.
This resembles Plato's cave. The play takes place in a ten-dimensional theater, and inside a four-dimensional cave called space-time, we observe four-dimensional shadows, four-dimensional movements.
In [1], we introduce a group which is an extension of the orthochronous part of the Poincaré group. It allows to describe the charges of the particles in terms of additional components of their momenta. In the paper [2], this group is duplicated through a z-symmetry, which gives a geometric description of Dirac's antimatter. This one has positive mass and energy.
Next step, paper [3], we decide to include antichronous elements in the group. We then get symmetries including T-symmetry, that is, PT-symmetry and CPT-symmetry. We find that the PT-symmetric of a particle of matter is an antiparticle, as suggested by Feynman. We find that the CPT-symmetric of a particle of matter is also a particle of matter, as stated by the so-called "CPT-theorem". However, from the coadjoint action of the group on the momentum components, we find that these two have negative masses and energies. Therefore, it is no longer possible, as suggested by Feynman, to identify the PT-symmetry and the C-symmetry. Similarly, the CPT-symmetry is different from identity, as it reverses the mass. As pointed out in [3], a solution, suggested by the mathematician J.M. Souriau [4], is to abandon the antichronous part of the Lorentz and Poincaré dynamical groups. But then the PT and CPT symmetries disappear.
In the following, we suggest another solution.
2) Construction of a group acting on a two-fold space.
According to [3], the action of our 16-dimensional group on a ten-dimensional space corresponds to:
(1) (4501)
and the corresponding coadjoint action is:
(2) (4502)
See computational details in the annex.
We construct the two-fold space as the quotient of the group by its orthochronous subgroup. According to (1), a point of the space is defined by:
(3) { z 1 , z 2 , z 3 , z 4 , z 5 , x , y , z , t }
Introduce a fold index f = ± 1
A point M of the first fold, called F, is defined by:
(4) { z 1 , z 2 , z 3 , z 4 , z 5 , z 5, x , y , z , t , f = +1 }
and the conjugated point M*, belonging to the second fold F*, by:
(5) { z 1 , z 2 , z 3 , z 4 , z 5 , z 5 , x , y , z , t , f = -1 }
We can write the new action:
(6) (4506)
The coadjoint action on the momentum space remains unchanged. However, the interpretation of the results is different. Negative energy movements occur in another fold. Positive and negative energy particles cannot meet, as they evolve in distinct ten-dimensional twin spaces. Fig.1 (45f1): Two sectors of the momentum space. Fig.2 **** : Associated symmetries 