f4404 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. (p4)
Conclusion.
** **...We have extended the group by including antichronous elements. We thus recover the geometrical description of Dirac's antimatter. However, the analysis of the coadjoint action of the antichronous elements of the group leads to PT-symmetric and CPT-symmetric movements.
...We observe that PT-symmetry corresponds to the transformation matter -----> antimatter. It confirms Feynman's idea: the PT-transform of a particle of matter is a particle of antimatter. However, the coadjoint action of the antichronous elements inverts mass and energy. Thus, we cannot identify the PT-transform of a particle of matter with its antiparticle, in accordance with Dirac's description: the latter has negative mass and negative energy.
...Similarly, the CPT-transform of a particle of matter is a particle of matter, but with negative mass, since it evolves towards the past.
...The problem remains unsolved. As recommended by J.M. Souriau, we could limit the dynamic group to its orthochronous part, but then PT and CPT-symmetric objects would be forbidden, and symmetries including time reversal would become impossible.
If we keep the antichronous sector, we obtain an universe filled with particles with both positive and negative mass.
Charybde or Scylla?
In the next article, we will propose another solution.
References.
[1] J.P. Petit & P. Midy : Geometrization of matter and antimatter through the coadjoint action of a group on its momentum space. 2 : Geometrical description of Dirac's antimatter. Geometrical Physics B, 2, March 1998.
[2] J.P. Petit & P. Midy : Geometrization of matter and antimatter through the coadjoint action of a group on its momentum space. 1 : Charges as additional scalar components of the momentum of a group acting in a 10-dimensional space. Geometrical definition of antimatter. Geometrical Physics B, 1, March 1998.
[3] J.M. Souriau : Structure des Systèmes Dynamiques, Dunod-France, 1972 and Birkhäuser, 1997.
[4] J.M. Souriau : Géométrie et relativité, Hermann-France, 1964.
[5] P.M. Dirac : « A theory of protons and electrons », 6 December 1929, published in Proceedings of the Royal Society (London), 1930 : A 126, pp. 360–365
[6] R. Feynman : « The reason for antiparticles » in Elementary particles and the laws of physics, Cambridge University Press, 1987.
Thanks.
...This work was supported by the French CNRS and the company Brevets et Développements Dreyer, France.
Deposited in a sealed envelope at the Academy of Sciences of Paris, 1998.
Copyright Academy of Sciences of Paris, 1998.
