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. (p2)
- A third component :
(14)
built with the component Lt of the Lorentz
group.
- and a fourth one :
(15)
built with the component Lst of the Lorentz
group.
A group acts on its momentum space [1]. Call Jp
the momentum space associated to the Poincaré group.
...Each peculiar moment Jp 
Jp, is a peculiar movement of the relativistic mass point, described
by this group. On may compute the caodjoint action of the group on the momentum
[1].
The momentum is a set of 10 components (equal to the dimension
of the group). These components are :
(16)
Jp= { E , px , py
, pz , fx , fy , fz , sx
, sy , sz } = { E , p , f , s }
Eis the energy.
p is the impulsion vector :
(17)
f is the passage vector [1].
(18)
s is an antisymmetric (3,3) matrix, whose independant
components are
(19)
{ sx , sy , sz
}
The momentum can be arranged into a matrix form [1], with :
(20)
and :
(21)
Introduce the impulsion-Energy four-vector :
(22)
(23)
or :
(24)
Then the coadjoint action of the Poincaré group
can be written into a matrix form :
(25) )
More explicitely :
(26)
...It is interesting to study
the impact of the different components of the complete Poincaré group
on the components of its momentum. We can concentrate on peculiar matrixes
:
(27)
A is the associated Lorentz matrix.
The coadjoint action gives :
(28)
(29)
and is the neutral component of the complet Poincaré
group.
The corresponding coadjoint action is :
E --> E ; p --> p ; f ---> f
; s ----> s
which reverses space. The corresponding coadjoint action
is :
E --> E ; p --> - p ; f ---> -
f ; s ----> s
which reverses time. The corresponding coadjoint action
is :
E --> - E ; p --> p ; f ---> - f
; s ----> s
which reverses both space and time. The corresponding coadjoint
action is :
E --> - E ; p --> - p ; f ---> f
; s ----> s
As pointed out by J.M.Souriau [1] , The two components
go with the inversion of the energy E ----> - E , so that
it implies the inversion of the mass m ---> - m
Define the following sets of matrixes :
(30)
.
