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What is the solution
?
...If,
as suggested by J.M.Souriau, God, in his infinite wisdom, did not create negative
mass and energy particles and prevent physicist to use antichron elements,
the theory cannot deal with PT and CPT symmetries.
We present a alternative
solution in :
J.P.Petit and P.Midy : "Geometrization of matter and anti-matter through
coadjoint action of a group on its momentum space. 4 : The Twin group. Geometrical
description of Dirac's anti-matter. Geometrical interpretations of anti-matter
after Feynmann and so-called CPT-theorem". Geometrical Physics B, 4, 1998.
...In
order to prevent collisions between positive an negative energy particles,
we split the evolution space into two folds, which forms the quotient of the
group by its orthochron sub-group. We get a twin geometry.
We introduce a fold indix
f = ± 1
f = +1 corresponds to
the fold F
f = - 1 corresponds to
the fold F*.
The Twin-group is :
(400)
...It
is still a eight components group. We see that ( m
= - 1 ) elements, which correspond to PT-symmetry, go with a fold commutation
:
f -----> - f
...The
momentum space is still composed by four sectors, but negative energy sectors
corresponds to particle's movements in the F* fold.
(401)
The subsequent symetries
are :
(402)
We can now define the
new "playing field".
(403)
The playing field
: a two folds ( F and F***) space, associated to a two sectors
momentum space**
( E >
0 and E < 0 ).
(404)
Movements of ordinary
matter. Action of orthochron elements of the group, with l
= 1**.**
Charges unchanged.
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.
...On
the figure, the line M1 figures the movement of normal, orthochron
matter. We figure straight lines because our group does not take account of
force field, like gravitational or electromagnetic field. It only runs the
behaviour of lonely particles, charged mass-points.
We chose 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 anti-matter.
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 ) anti-matter after Dirac.
...Two
more sectors has to be explored. On the third we examine the impact of (
l = - 1 ; m = - 1 ) element on the momentum and movement.
( l
= - 1 ) reverses the {z i}. According
to our geometric definition this new movement corresponds to anti-matter,
for it takes place in the second sector of space { z 1,z
2, z 3, z 4, z 5, z 6,
x, y , z , t }.
( m
= - 1 ) gives a PT-symmetry, reverses the signs of ( x, y , z , t )
But (
l m = + 1 ) keeps the charges unchanged.
This is "PT-symmetric
anti-matter", so that it is a geometric description of anti-matter after
Feynmann.
The movement takes place in the second space sector, in the fold F*.
(406)
( l
= -1 ; m = -1 ) elements transform
movement of normal matter into movement of anti-matter (z-Symmetry)
of PT-symmetrical object, runing bacward in time. Geometric description
of Feynmann's vision of anti-matter. Does not identify vompletely with Dirac's
one : negative mass and negative energy.
The last elements correspond to the sector ( l= 1 ; m = -1 )
( l
= 1 ) --- > the movement is still in the matter's sector :
no
z-Symmetry.
( m
= -1 ) goes with a PT-symmetry. The particule 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's not true. This movement
corresponds to an antichron movement. The particle goes backward in time,
si that (coadjoint action) its mass and energy become negative .
...The
movement of a particle which is the CPT-symmetrical of a normal particle takes
place in the fold F*.
(407)
( 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 particule of matter, but with negative mass.
Index Dynamic Groups Theory