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. (p4)
3) A description of quantum numbers as components
of the moment of an extended group.
The Poincaré group can be extended as many times
one wants. Let us do it six times. Then we get :
(46)
...This two components group
( due to the two components of the orthochron Lorentz group Lo ) acts on a
ten dimensional space :
{ z 1
, z 2
, z 3
, z 4
, z 5
, z 6,
x , y , z , t }
i.e. space time ( x , y , z , t )
plus six additional dimensions
{ z 1
, z 2, z
3 , z 4
, z 5 ,
z 6 }
The momentum becomes :
(47)
Jpe = { c
1, c 2,
c 3, c
4, c 5,
c 6, Jp
}
where Jp represent the classical expression
of the Poincaré group's momentum.
The coadjoint action is :
(48)
All these additional scalars are conserved and we identify
these to the following classical quantum numbers :
(49)
c 1
= q ( electric charge)
c 2
= cB ( baryonic charge)
c 3
= cL ( leptonic charge)
c 4
= cm (
muonic charge)
c 5
= ct (
tauonic charge )
c 6
= v ( gyromagnetic
coefficient)
We give to each first five numbers three possible values
:
{ -1 , 0 , +1 )
The value of the gyromagnetic factor v
depend of the considered particle.
...The momentum space is
supposed to be a continuum, but one assume that discrete values of some components
correspond to real particules, from physics' world. Then we get a description
of elementary particles in terms of group's orbits. We can write the momentum
:
(50)
Jpe = { q , cB
, cL , cm
, ct ,
v , Jp
}
Jj
= { 0 , 0 , 0 , 0 , 0 , 0 , Jp
} : photon
Jp = { 1 , 1 , 0 , 0 , 0 , vp
, Jp } : proton
Jn = { 0 , 1 , 0 , 0 , 0 , vn
, Jp } : neutron
Je = { -1 , 0 , 1 , 0 , 0 , ve
, Jp } : electron
Jne
= { 0 , 0 , 1 , 0 , 0 , vne
, Jp } : electronic neutrino
Jnm
= { 0 , 0 , 0 , 1 , 0 , vnm
, Jp } : m
neutrino
Jnt
= { 0 , 0 , 0 , 0 , 1 , vnt
, Jp } : t neutrino
......
We transform a particle into the corresponding antiparticle
through charge conjugation ( C-symmetry). The charges of the photon are all
zero, so that it identifies with its antiparticle.
