f4205 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 on a 10d-space. Geometrical definition of antimatter. (p5)
4) Suggested geometric definition of antimatter.
...A particle is a species, corresponding to a subset of the momentum space. It corresponds to specific choices in some components of the momentum, the charges :
(51) { q , cB , cL , cm , ct , v }
...A momentum is a movement of a mass-point, governed by a dynamic group. Here an extension of the orthochron Poincaré's subgroup.
...Classically (Dirac's antimatter) one considers that reversing the charge (C-symmetry of charge conjugation) transforms matter into antimatter :
(52) { - q , - cB , - cL , - cm , - ct , - v }
...Then we can classify the particles, through their momentum space, into two subsets, the first containing matter and the second antimatter. Schematically, photons have been represented on the border between the two, for they are identical to antiphotons. See figure 1.
Fig.1** : Classification of particles.**
As we know, each momentum corresponds to a movement. Here we consider movements in a ten-dimensional space, a fibered space-time, as evoked on figure 2.
** ** Fig.2 : Fibered space-time.
As shown in the figure, we suggest that matter-antimatter duality corresponds to a :
(53) z-Symmetry : {z i} ---> { - z i }
...Particles move in { z i> 0 } half-space and antiparticles in the other { z i< 0 } one. Photons move in { z i = 0 } plane. Their movement is not changed by z-Symmetry, so that they are identical to their antiparticle.
...In this paper we deal with an extended 16-dimensional orthochron group. We can figure schematically the coadjoint action of such a group on its moment space and associated movement space. See figures 3, 4 and 5.
**Fig. 3 ** : Movement of matter, in the { z i > 0 } half 10d-space and coadjoint action on the momentum. The link between momentum and movement has been figured.
**Fig. 4 ** : Movement of antimatter, in the { z i < 0 } half 10d-space and coadjoint action on the momentum. The link between momentum and movement has been figured.
Fig. 5** : Movement of photons, in the { z i = 0 } plane** and coadjoint action on the momentum. The link between momentum and movement has been figured.
Conclusion.
...We have extended the orthochron Poincaré subgroup, corresponding to positive energy particles to a 16-dimensional group, acting :
-
On a 16-dimensional momentum space
-
On a 10-dimensional movement space.
...The extension gives the momentum six extra components, which are identified to charges, so that we get a geometric description of usual elementary particles : photon, proton, electron, neutrons , e , m and t neutrinos and their antiparticles.
This provides a classification of particles in terms of momentum's components, defining three basic species :
- Particles - Antiparticles - Photons.
each corresponding to a subset of the ( E > 0 ) momentum space. Then we suggest a basic definition of antimatter, and photons, in terms of peculiar movements in a 10d-space.
{ z i > 0 } corresponding to matter.
{ z i < 0 } corresponding to antimatter.
{ z i = 0 } corresponding to photons.
This is similar to Plato's vision.
...The objects move in a 10-dimensional space, but the inhabitants of the cavern can just see the 4-dimensional (x,y,z,t) shadows of these movements.
References.
[1] J.M.Souriau : Structure des Systèmes Dynamiques, Dunod-France Ed. 1972 and Birkhauser Ed. 1997.
[2] J.M.Souriau : Géométrie et relativité. Ed. Hermann-France, 1964.
[3] P.M.Dirac : "A theory of protons and electrons", Dec. 6th 1929, published in proceedings of Royal Society ( London), 1930 : A **126 **, pp. 360-365
Acknowledgements.
This work was supported by french CNRS and Brevets et Développements Dreyer company, France.
Deposited in sealed envelope at the Academy of Sciences of Paris, 1998.
Copyright french Academy of Science, Paris, 1998.
