تعميم المادة والمضادة للمادة بفعل مجموعة مُتَعَامِدَة

f4204 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.