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 : {zi} ---> { - zi }
...Particles move in { zi> 0 } half-space and antiparticles in the other { zi< 0 } one. Photons move in { zi = 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 { zi > 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 { zi < 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 {
zi = 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.
{ zi > 0 } corresponding to matter.
{ zi < 0 } corresponding to antimatter.
{ zi = 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.
