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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 prevented physicists from using antichron elements, the theory cannot deal with PT and CPT symmetries.
We present an 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 Feynman and the so-called CPT-theorem". Geometrical Physics B, 4, 1998.
...In order to prevent collisions between positive and negative energy particles, we split the evolution space into two folds, which form the quotient of the group by its orthochronous subgroup. We obtain a twin geometry.
We introduce a fold index f = ± 1
f = +1 corresponds to the fold F
f = -1 corresponds to the fold F*.
The Twin-group is:
(400)
...It is still an eight-component 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 of four sectors, but negative energy sectors correspond to particle movements in the F* fold.
(401)
The subsequent symmetries are:
(402)
We can now define the new "playing field".
(403)
The playing field: a two-fold (F and F) space, associated with a two-sector momentum space (E > 0 and E < 0).*
(404)
Movements of ordinary matter. Action of orthochronous elements of the group, with l = 1. Charges unchanged.
Coadjoint action of an (l = -1; m = 1) element of the group on the momentum associated with the movement of normal matter: the new movement corresponds to Dirac's anti-matter.
...On the figure, the line M1 represents the movement of normal, orthochronous matter. We represent straight lines because our group does not take into account force fields, such as gravitational or electromagnetic fields. It only describes the behavior of isolated particles, charged mass-points.
We choose an element in the grey area, corresponding to an (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 lm = -1, the charges are reversed. But as time is not reversed, the energy and mass of the particle remain positive.
This is a geometric description of (orthochronous) anti-matter after Dirac.
...Two more sectors have to be explored. In the third, we examine the impact of an (l = -1; m = -1) element on the momentum and movement.
(l = -1) reverses the {zi}. According to our geometric definition, this new movement corresponds to anti-matter, for it takes place in the second sector of space {z1, z2, z3, z4, z5, z6, x, y, z, t}.
(m = -1) gives a PT-symmetry, reversing the signs of (x, y, z, t).
But (lm = +1) keeps the charges unchanged.
This is "PT-symmetric anti-matter", so it is a geometric description of anti-matter after Feynman.
The movement takes place in the second space sector, in the fold F*.
(406)
(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. Does not completely identify with Dirac's: 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 particle runs backward in time.
(l = -1): C-Symmetry. The charges are reversed.
...This is CPT-symmetrical matter, so 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 antichronous movement. The particle goes backward in time, so that (via coadjoint action) its mass and energy become negative.
...The movement of a particle which is the CPT-symmetric 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 particle of matter, but with negative mass.
