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 things clearer, we can illustrate them graphically.
Fig.1** : The four-component orthochron extended group.** The (l=1) components form a subgroup. Below, the momentum space with its three subsets, representing the worlds of particles, antiparticles and photons. Two-sector movement space associated.
...If we choose an element from the (l = 1) subgroup, we find again the diagrams presented in the previous paper [1].
Examine the effect of the orthochron operator goc on the momentum and the associated movement.
**Fig.2 **: Coadjoint action of the orthochron operator goc
. **Fig.3 **: Coadjoint action of the orthochron operator goc on the photon : none, because it is its own antiparticle.
Now, introduce two coupled orthochron matrices :
(20) go and goc x go
**Fig.4 ** : Coadjoint action of the orthochron operator goc and the conjugated orthochron matrices go and goc x go
Conclusion.
...We start from the previous paper [1], where we introduced a 16-dimensional group acting on its 16-dimensional momentum space and on a 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 the orthochron operator, which achieves z-Symmetry. Then we construct a group containing such an element. We obtain a four-component group, composed of the elements go of the (l = 1) subgroup, and of the conjugated matrices goc x go, formed by the action of the orthochron operator goc on this subgroup. Antimatter then becomes another movement of matter, driven by the 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", December 6th 1929, published in the proceedings of the Royal Society (London), 1930 : A 126, pp. 360-365
Acknowledgements.
This work was supported by the French CNRS and by the company Brevets et Développements Dreyer, France.
Deposited in sealed envelope at the Academy of Sciences of Paris, 1998.
Copyright Academy of Sciences of France, Paris, 1998.
