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 :
...vWe extend the precedent group to a four-components
orthochron set. This operation gives a geometrical interpretation of antimatter
after Dirac.
1) Introduction :
...In a former paper [1] we have presented a
description of elementary particles ins a ten-dimensional space, i.e. space-time
(x,y,z,t) plus six additional dimensions :
(1)
{ z1, z2
, z3
, 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 figured the link between the species of moments
and the species of movement, suggesting that :
-
The movement of matter corresponds to { zi > 0 } sector.
-
The movement of antimatter corresponds to { zi < 0 } sector.
-
The movement of photons corresponds to { zi = 0 } plane.
All that must be now justified.
2) Introducing a four components group. Geometrization
of Dirac's antimatter.
...The precedent 16-dimensional
group had two components, correspondong to the two orthochron components of
the Lorentz group, Ln ( neutral component ) and Ls ,
with :
(5)
Lo ( orthochron sub-group ) = Ln
U Ls
Our group was an extension of the orthochron Poincaré
sub-group :
(6)
Go = Gn U Gs
and we wrote it :
(7)
The corresponding coadjoint action was :
(8)
with :
(9)
{ci} = { 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 : {zi} ----> {- zi}
some element should reverse the additional dimensions.
With :
(11)
we can write the precedent group into a more compact form
:
(12)
It contains the neutral element :
(13)
The matrix that reverses the additional dimensions is
be the following orthochron commuter :
(14)
We can duplicate the precedent group through the operation
:
(15)
go x
goc
It is equivalent to write the new four component group,
whose element is :
(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 : {zi} ----> {- zi}
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.
