هندسة المادة والمضادة للمادة بواسطة التأثير التكميلي

f4205 Geometrization of matter and antimatter through 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 sub-set of the momentum space. It corresponds to peculiar 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 sub-group.

...Classically ( Dirac's antimatter ) one considers that reversing the charge ( C-symmetry of charge conjugation) transforms matter into anti matter

(52) { - q , - cB , - cL , - cm , - ct , - v }

...Then we can classify the particles, through their momentum space, into two sub-sets, the first containing matter and the second anti matter. Schematically, photons have been figured on the borde between the two, for they are identical to antiphotons. See figure 1.

Fig.1** : Classification of particles.**

As we know each momentum correspond 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 of 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 othochron Poincaré sub-group, 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 antis.

This provides a classification of particles in terms of momentum's components, defining three basic species :

• Particles - Antiparticles - Photons.

each corresponding to a sub-set 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.
Déposé sous pli cacheté à l'Académie des Sciences de Paris, 1998.
Copyright french Academy of Science, Paris, 1998.