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.
