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Let us move to the second sector (l = -1; m = 1)
(247)
We have z-symmetry. Therefore, our matter is transformed into antimatter, according to the definition we gave earlier. The coadjoint action gives C → -C. There is charge conjugation. Mass and energy remain unchanged. This is antimatter in the sense of Dirac, orthochronous. Charges are reversed, starting with electric charge q.
Let us move to the sector (l = -1; m = -1)
(248)
We have z-symmetry, so matter is transformed into antimatter. Since lm is positive, there is no C-symmetry. Charges remain unchanged. However, there is PT-symmetry. This is what led Feynman to say that ordinary matter (with the same charges), enantiomorphic and time-reversed, would behave like Dirac antimatter (which is C-symmetric). But this overlooks one point. Feynman's antimatter is "antichronous," hence possesses negative mass and energy. In a gravitational field, it should "rise."
Our conclusion:
There is no equivalence between these two types of antimatter.
Let us now consider the last type of motion induced by elements with (l = 1; m = -1). There is no z-symmetry. This motion therefore corresponds to a particle of matter. There is PT-symmetry, due to m = -1.
The coadjoint action, since lm < 0, gives rise to C-symmetry. The object is thus CPT-symmetric.
The "CPT theorem" identifies the CPT-symmetric particle with itself. But we believe this is not true. These CPT-symmetric particles are generated by elements of the group belonging to an antichronous sector. Therefore, the masses and energies of CPT-symmetric particles are negative.
There is no equivalence between the two types of matter.
(249)
In passing, some clarifications regarding photon motions. The orthochronous anticomponents have a coadjoint action on photon motions corresponding to scheme 1 BIS. (246, previous page)
On the other hand, if elements from antichronous sectors act, this will have the effect of reversing the energy of these photons. Scheme 4 BIS, below:
(250)
But under this view, we remain with particles—whether they have non-zero or zero mass—possessing opposite energies, which can still interact. Indeed, we know that anything antichronous comes with E < 0 and m < 0.
According to this model, corresponding to the 1st group of Petit, in summary:
- Only one universe, whose dynamical group is:
(251)
an eight-component group acting on a ten-dimensional space (spacetime plus six additional dimensions).
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We have different symmetries. The z-symmetry (l = -1), affecting all additional dimensions, is taken as the definition of matter-antimatter duality. The PT-symmetry (m = -1).
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The group contains orthochronous and antichronous components, associated with motions having negative energy and mass.
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Analysis of the coadjoint action reveals C-symmetry (inversion of all charges), conditioned by z-symmetry and PT-symmetry: C = l × m.
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There are four fundamental types of motion, hence four types of matter.
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Orthochronous matter: (l = 1; m = 1; C = 1; E > 0)
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Dirac-type orthochronous antimatter: (l = 1; m = 1; C = 1; E > 0)
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CPT-symmetric matter: matter (l = 1; m = -1; C = -1; E < 0): antichronous
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PT-symmetric matter: antimatter (l = -1; m = -1; C = 1; E < 0): antichronous
The proposed solution consists in considering a disconnected momentum space, linked to a disconnected motion space, composed of two sheets, two universes, the quotient space of the proposed group (the second group of Petit) by its orthochronous subgroup.