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Geometric description
of Dirac's anti-matter.
...We
see thatl = - 1 changes the signs of the ci 's , which
corresponds to a charge conjugation , a C-symmetry.
This groups gives
a geometrical description of anti-matter after Dirac ( positive energy, positive
mass anti-matter ).
...Of
course the C-symmetry does not change the photon, for all its charges are
basically zero. It identifies with its own antiparticle.
Geometric description of Feynmann's anti-matter.
...This
one is supposed to be PT-symmetrical. How to introduce the PT-symmetry in
the group ?
See :
J.P.Petit and P.Midy : " Geometrization of matter and anti-matter through
coadjoint action of a group on its momentum space. 3 : Geometrical description
of Dirac's anti-matter. A first geometrical interpretation of anti-matter
after Feynmann and so-called CPT-theorem". Geometrical Physics B, 3
, 1998.
The subsequent modification
of the group is the following :
(388)
...It
becomes an eight components group, for the orthochron part of Lorentz has
two connex components, so that 2 x 2 x 2 = 8.
It means that we add
the antichron elements :
(389)
Above : we add the antichron
elements to the group.
Below : we add the corresponding
half sector of the momentum space corresponding to negative energy movements.
In a word : we extend
the playing field, which becomes :
(390)
On (388) we see that
( m = - 1 ) elements reverse space-time, achieve
PT-symmetry and correspond to :
(391)
Lst
= - Ln
Lt
= - Ls
We have the following symmetries in the momentum space :
(392)
The calculation of the
coadjoint action of the group (388) on its momentum gives :
(393)
...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.
(394)
Movements of ordinary
matter. Action of orthochron elements of the group, withl = 1**.**
Charges unchanged.
(395)
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 anti-matter.
...The
elements is picked in the grey sector. It corresponds to an "anti-element",
which transform matter into anti-matter : l = - 1 reverse the signs of the
additional dimensions, which is our geometrical definition on antimatter.
Index Dynamic Groups Theory