f4301 Geometrization of matter and antimatter through coadjoint action of a group on its momentum space. 2 :
Geometrical description of Dirac's antimatter
** Jean-Pierre Petit & Pierre Midy ** Observatoire de Marseille ---
Abstract :
...We extend the previous group to a four-components orthochron set. This operation gives a geometrical interpretation of antimatter after Dirac.
--- ** **
1) Introduction :
...In a previous paper [1], we have presented a description of elementary particles in a ten-dimensional space, i.e. space-time (x,y,z,t) plus six additional dimensions :
(1) **{ **z 1 , z 2 , z 3 , z 4 , z 5 , z 6 }
We presented a 16-dimensions group, an extension of the Poincaré orthochron subgroup, acting on :
-
its 16-dimensions momentum space
-
its 10-dimensional movement space.
The six additional components of the momentum have been identified to the charges of the particles :
(2) { q , cB , cL , cm , ct , v }
so that the momentum becomes :
(3) Jpe = { q , cB , cL , cm , ct , v , Jp } where Jp represent the classical moment, from the orthochron Poincaré sub-group :
(4) Jpo = { E , p , f , **l **}
after J.M.Souriau [1].
We have established the link between the species of moments and the species of movement, suggesting that :
-
The movement of matter corresponds to { z i > 0 } sector.
-
The movement of antimatter corresponds to { z i < 0 } sector.
-
The movement of photons corresponds to { z i = 0 } plane.
All that must be now justified.
2) Introducing a four components group. Geometrization of Dirac's antimatter.
...The previous 16-dimensional group had two components, corresponding to the two orthochron components of the Lorentz group, Ln (neutral component) and Ls, with :
(5) Lo (orthochron subgroup) = Ln U Ls
Our group was an extension of the orthochron Poincaré subgroup :
(6) Go = Gn U Gs
and we wrote it :
(7)
The corresponding coadjoint action was :
(8)
with :
(9) {c i} = { q , cB , cL , cm , ct , v }
...In such a group no element transforms the movement of a matter mass-point into the movement of an antimatter mass-point, and vice versa. According to the chosen definition of antimatter, through a :
(10) z - Symmetry : {z i} ----> {- z i}
some element should reverse the additional dimensions. With :
(11)
we can write the previous group into a more compact form :
(12)
It contains the neutral element :
(13)
The matrix that reverses the additional dimensions is the following orthochron commutator :
(14)
We can duplicate the previous group through the operation :
(15) go x goc
It is equivalent to write the new four component group, whose elements are :
(16)
The corresponding coadjoint action is :
(17)
We see that ( l = - 1 ) reverses the charges. In that case the inversion of the additional dimensions :
(18) z - Symmetry : {z i} ----> {- z i}
goes with a :
(19)
C-symmetry (or charge conjugation ) : { q , cB , cL , cm , ct , v } ---> {- q ,- cB ,- cL ,- cm ,- ct , - v }
which corresponds to Dirac's description of antimatter [4], so that the present paper represents a geometrization of antimatter after Dirac.
