Geometrization
of matter and antimatter through
coadjoint
action of a group on its momentum space. 3 :
Geometrical
description of Dirac's antimatter.
A first
geometrical interpretation of antimatter
after Feynmann and
so-called CPT-theorem. (p2)
negative energies sectors.
.
.
Fig.2 : Subsequent symmetries
.
.
Fig.3) : The eight components
group
its momentum and movement spaces.
...It becomes easy to examine
the impact of each component on momentum and movement. We shall consider a
reference movement and momentum J+1
, refering to positive energy matter ( the impact on positive energy photons
will be analysed in a second step ). The sector of the group in which the
element is chose will be grey.
Next, the movements of ordinary matter.
l = +1 m
= +1
l
m = +1
The charges are unchanged. The movement M2
refers to (E>0), positive mass, orthochron matter.
.
Fig.4 : Movements of
ordinary matter.
Action of orthochron elements of the group, with l
= 1**.**
Charges unchanged.
Fig. 5 :
Coadjoint action of a ( l
= -1 ; m = 1 ) element
of the group on the momentum associated to
the movement of normal matter : the new movement
corresponds to Dirac's antimatter.
...On the figure 5 the line
M1 figures the movement of normal, orthochron matter. We figures
straight lines because ou group does not take account of force field, like
gravitational or electromagnétic field. It only runs the behaviour
of lonely particles, charged mass-points.
...We choose an element in
the grey area, corresponding to a ( l
= -1 ; m = 1 ) matrix.
The ( l = - 1 ) value
changes the signs of all the zi. They become negative. The new path is in the second sector,
corresponding to antimatter. As l
m = - 1 the charges are
reversed. But as time is not reversed, the energy and the mass of the particle
remains positive. This is a geometric description of ( orthochron ) antimatter
after Dirac.
