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Consider now a component
( l = - 1 ; m = - 1
)
(396)
(
l = -1 ; m = -1 ) elements transform
the movement of normal matter into the movement of anti-matter (z-Symmetry)
of a PT-symmetrical object, running backward in time. Geometric description
of Feynman's vision of anti-matter. It does not completely coincide with Dirac's
one: negative mass and negative energy.
We obtain Feynman's anti-matter.
But, as pointed out by Souriau in 1973, a PT-symmetrical particle goes backwards
in time. Its mass and its energy are negative.
Remark: in this description,
the fact that the movement corresponds to a positive or negative energy element
does not appear in the evolution space (top-left).
The last elements correspond
to the sector ( l = 1 ; m
= -1 )
( l = 1 ) --> the movement
is still in the matter sector:
no z-Symmetry.
( m
= -1 ) goes with a PT-symmetry. The particle runs backward in time.
( l
= -1 ): C-Symmetry. The charges are reversed.
...This
is CPT-symmetrical matter, so that it corresponds to a geometrical interpretation
of the so-called "CPT theorem", which asserts that the CPT-symmetric of a
particle should be identical to that particle. That is not true. This movement
corresponds to an antichron movement. The particle goes backward in time,
so that (coadjoint action) its mass and energy become negative.
If CPT-symmetrical particles
do exist and if they collide with normal particles, complete annihilation occurs.
(397)
( l = 1 ; m = - 1
) case. Corresponds to CPT-symmetry. But the coadjoint action gives
negative mass and energy. The CPT-symmetric of a particle of matter is a
particle of matter, but with negative mass.
...Now,
examine the impact on the movement and momentum of photons. The z-Symmetry has no impact
on it: there is no "antiphoton". As all the charges of the photon are zero,
it does not change it. It is identical to its antiparticle.
...The
coadjoint action of orthochron components modifies the movement and the momentum
of the photon, but keeps its energy unchanged.
(398)
Coadjoint action
of orthochron elements
on the photon's movement
and momentum.
(399)
The coadjoint action
of antichron elements on the photon's movement and momentum reverses the photon's
energy: it travels backwards in time.
...We
see that reintroducing antichron components in the group yields PT-symmetric
anti-matter and CPT-symmetric matter. Both go backwards in time. Both come
from the action of an antichron element of the group on a normal matter movement.
Anti-matter is nothing but a peculiar movement. The same is true for
CPT-symmetrical matter, which can no longer be identified with normal matter,
as classically asserted by the so-called CPT-theorem, because the mass of a particle
which is the CPT-symmetric of a normal matter particle has negative mass and
negative energy.
Similarly, Feynman's
antimatter has negative mass and negative energy (while Dirac's antimatter
has positive mass and energy).
