هندسة المادة والمضادة للمادة عبر التأثير التكميلي

f4302 Geometrization of matter and antimatter through coadjoint action of a group on its momentum space. 2 : Geometrical description of Dirac's antimatter (p2)
3) Coadjoint action on momentum space.

In order to make the things clearer we can graphically figure it.

Fig.1** : The four component orthochron extended group.** The (l=1) components form a a sub-group. Below, the momentum space with its three sub-sets, figuring partcles's, antiparticles' and photons' worlds. Associated two-sectors movement space.

...If we choose an element picked from the ( l = 1 ) sub-group we refind the schemas presented in the precedent paper [1].

Examine the impact of the orthochron commuter goc on the moment and associated movement.

**Fig.2 **: Coadjoint action of the orthochron commuter goc

. **Fig.3 **: Coadjoint action of the orthochron commuter goc on the photon : none, for it is its own antiparticle.
Now, introduce two coupled orthochron matrixes :

(20) go and goc x go

**Fig.4 ** : Coadjoint action of the orthochron commuter goc and conjugated orthochron matrixes go and goc x go

Conclusion.

...We start from the precedent paper [1], where we introduced a 16-dimensional group acting on its 16-dimensions momentum space and 10-dimensional movement space. As in [1] we follow the basic idea : antimatter corresponds to a z-Symmetry, to the inversion of the additional variables. We define a matrix, called orthochron commuter, which achieves z-Symmetry. Then we build a group which contains such element. We get a four components group, composed by the elements go of the ( l = 1 ) sub-group, and by conjugated matrixes goc x go , formed through the action of the orthochron commuter goc on this sub-group. The antimatter becomes another movement of matter, driven by coadjoint action of the group.

References.

[1] J.P.Petit & P.Midy : 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. Geometrical Physics B, 1 , march 1998.
[2] J.M.Souriau : Structure des Systèmes Dynamiques, Dunod-France Ed. 1972 and Birkhauser Ed. 1997.
[3] J.M.Souriau : Géométrie et relativité. Ed. Hermann-France, 1964.
[4] 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.