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The matter of the second universe possesses a number of properties (relative to ours):
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It is C-symmetric. The protons in this universe are negatively charged.
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It is enantiomorphic (the structures of this matter are, with respect to ours, "mirror images"). Consequence of P-symmetry.
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It is T-symmetric and retrograde, evolving in "reverse time".
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It is E-symmetric: its energy and mass are negative.
Two ghost particles attract each other according to Newton. However, if an interaction between sheets is considered, a particle and a ghost particle repel each other according to "Anti-Newton".
(256)
It remains to analyze the motions associated with the last sector (l = -1; lm = -1).
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We have z-symmetry. This therefore corresponds to antimatter.
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We have T-symmetry, hence E-symmetry. The motion takes place in the second universe, the ghost universe.
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We have PT-symmetry.
This is "Feynman's antimatter", but reinterpreted. The motion occurs in the universe where motions with negative energy take place.
(257)
This group is written, using the previous notation:
(258)
It acts on a ten-dimensional space with two sheets (we introduce a sheet index f = ±1).
The calculation of the coadjoint action yields the same result:
(259) c'i = l m c i (i ranging from 1 to 6)
Once again, we identify the additional scalars c i of the momentum with the charges of the particles. Thus:
(260) C = l m
If C = -1, we have a symmetry (charge inversion).
The proposed matrix encodes all the properties illustrated graphically above.
In summary:
We propose a dynamic group with eight components, acting on a two-sheeted space, which is the quotient of this group by its orthochronous subgroup.
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The group acts on a ten-dimensional space with two sheets, corresponding to sheet index values ±1.
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Various symmetries are present. The z-symmetry (l = -1), affecting all additional dimensions, is taken as the definition of matter-antimatter duality. The PT-symmetry (m = -1). PT-symmetry implies F-symmetry (sheet symmetry), which in turn is synonymous with E-symmetry (symmetry between motions with E > 0 and motions with E < 0).
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The group contains both orthochronous and antichronous components, associated with motions having negative energy and mass.
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The analysis of the coadjoint action reveals the 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, and thus four types of matter.
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Two occur in the orthochronous sheet and correspond to the motions of matter and antimatter in the sense of Dirac, C-symmetric, having the same mass and energy as the matter of which they are the symmetric counterparts.
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The other two occur in the antichronous sheet, where particles with negative energy and mass therefore travel. These are matter particles and antimatter particles. The matter-antimatter duality exists in the second universe.
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Since these two sheets are disjoint, particles with positive energy and particles with negative energy can no longer meet and annihilate.
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The matter of the antichronous universe has negative mass and energy. It is CPT-symmetric relative to ours. This is our interpretation of the "CPT Theorem". A CPT-symmetric particle of a matter particle is not identical to that particle. It is the matter of the other universe, retrograde, enantiomorphic, with negative mass. In that other universe, charges are inverted (C-symmetry), so protons are negatively charged and electrons positively charged.
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The antimatter of the other, antichronous universe is PT-symmetric relative to ours. This is our interpretation of "Feynman's antimatter". It is indeed antimatter, but it is not identical to antimatter in the sense of Dirac. It travels in the second universe, antichronous and enantiomorphic. Its mass and energy are negative. It possesses the same charges as particles in our universe. Thus, an antielectron in the antichronous universe is negatively charged, and an antiproton in that universe is positively charged.
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Since the second universe is P-symmetric relative to ours, structures analogous to those in our universe are enantiomorphic, mirror images.
Note on metrics.
The dynamic groups of the two sheets are constructed from the same initial elements (the orthochronous elements of the Lorentz group). The matrices
(261) L = m Lo with **m = ± 1
present in all matrices of the group satisfy the axiom
(262) with:
(263)
Thus, the sheets F and F* have the same signature ( - - - - + ).
On masses.
We have seen that the sign of mass and energy is directly linked to the direction of time. Any transformation that inverts time also inverts mass m and energy E. This is a relative inversion, with respect to an observer located in a given sheet. Thus, matter and antimatter in the ghost universe, evolving in a sheet F* where the arrow of time is reversed, will behave, with respect to our reference matter, as if they possessed negative mass and energy. Hence the justification for the system of two field equations:
(264) S = c ( T - T* )
(265) S* = c ( T* - T )