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.
Jean-Pierre
Petit & Pierre Midy
Observatoire de Marseille
Abstract :
...Through a new four components
non-connex group, acting on a ten dimensional space, composed by (x,y,z,t)
plus six additional dimensions we give a description of particles like photon,
proton, neutron, electrons, neutrinos ( e, m
and t ) and their anti,
through the coadjoint action on the momentum space. Quantum numbers become
components of the moments. Matter and antimatter are interpreted as two different
movements of mass-points in this
{ z 1,
z 2,
z 3,
z 4,
z 5,
z 6,
x , y , z , t } space
matter movement taking place in the {z
i > 0} half space and antimatter in the
remnant {zi
< 0} one.
The z-Symmetry
:
{zi
---> - zi}
which there goes with charge conjugation, becomes the
definition of matter-antimatter duality.
________________________________________________________
1) Introduction.
...As pointed out by J.M.Souriau
in his book [1] the Poincaré group, as a dynamic group for physics,
arises a problem about the sign of the mass.
Everything starts from the Lorentz group L, whose element
L is axiomaticaly defined by :
(1)
where :
(2)
The Lorentz group acts on space-time :
(3)
through the action :
(4)
The matrix G comes from the expression of the Lorentz
metric (with c=1) :
(5)
We know than the Lorentz group is composed by four components
:
Ln is the neutral componant, which contains
the neutral element 1, i.e. the peculiar matrix :
(6)
Ls , the second component, contains the matrix
:
(7)
which reverses space.
Lt , the third component, contains the matrix
:
(8)
which reverses time.
Lst , the fourth component, contains the matrix
:
(9)
which reverses both space and time.
From the Lorentz group one builds the Poincaré
group Gp, whose element is :
(10)
C is a space-time translation :
(11)
...If we use the four components
of the complete Lorentz group L , (10) will be called the complete Poincaré
group. As the Lotentz group, it owns four components :
- Its neutral component :
(12) (4212) 
built with the neutral component Ln of the
Lorentz group L.
- A second component :
(13)
built with the component Ls of the Lorentz
group.
