Geometrization of matter and antimatter through coadjoint action of a group on its momentum space. 4 :
The Twin group. Geometrical description of Dirac's antimatter.
Geometrical interpretations of antimatter after Feynman and the so-called CPT theorem. (p4)
Some comments about the metrics.
All the elements of the group are built from the elements of the complete Lorentz group, which obey :
(7)
(4507)
with
(8)
(4508)
This last matrix is linked to the metric :
(9)
(4509)
so that the two folds have the same signature. If they are described as Minkowski space-times, their metrics are identical. But their arrows of time are opposite.
If one wants to describe the two folds, the two universes, one has to choose one's own arrow of time and spatial orientation.
It is clear that the matter-antimatter duality occurs in both folds. If we call the second fold "twin fold" (A. Sakharov) or "shadow fold" (Green, Schwarz and Salam) or "ghost fold" (the author's choice), the arrow of time in this second fold is opposite (T-symmetry), as predicted by A. Sakharov, and the spatial structures are enantiomorphic (P-symmetry).
In the second fold, the matter is CPT-symmetric with respect to ours. Hence, in that fold, a proton has a negative charge and an electron a positive charge.
Conversely, an anti-electron of that fold, PT-symmetric with respect to ours, has a negative charge, hence an antiproton of the second fold has a positive charge.
To sum up, the second fold is CPT symmetric with respect to ours. As suggested by Andrei Sakharov, we can expect that the violation of the parity principle could be reversed in that fold.
If the absence of antimatter in our fold is a direct consequence of the violation of the parity principle, it is possible that such asymmetry would be reversed in the other fold.
Interacting folds.
All our work in astrophysics and cosmology (see Geometrical Physics A) comes from a system of two coupled field equations :
(10)
S = c ( T - T
- )
(11)
S* =
c ( T
T )
The two minus signs were introduced as an a priori hypothesis. At the end of this work, based on group theory, the explanation emerges. The two folds
must
have opposite arrows of time and
must
be enantiomorphic in order to satisfy constraints coming from the group structure.
Thus, the matter of the other fold, for an observer located in the first, behaves as if it had a negative mass, which comes from the coadjoint action and the T-symmetry.
Conclusion.
Starting from the work of reference [3], we have modified the model in order to avoid encounters between positive and negative mass particles. The solution was to build two ten-dimensional folds (F, F*) as the quotient of the group by its orthochronous subgroup.
Then we get two spaces with opposite arrows of time.
We study the impact of the different components of the group on momentum and motion spaces. It is shown that the matter-antimatter duality occurs in both folds, in both universes.
This work gives a new insight on antimatter, through geometrical tools.
For example, Dirac's antimatter is the antimatter of our own fold.
The matter of the second fold is CPT-symmetric with respect to ours.
The PT-symmetric of a matter particle that belongs to our fold is the antimatter of the other fold.
Matter and antimatter particles of our universe have positive mass and energy.
Matter and antimatter particles of the second fold have negative mass and energy.
Appendix:
Extension of the group.
Consider a group composed of matrices :
(1)
(4513)
A is a square matrix.
B is a column matrix and
O a row matrix, composed of null terms.
Consider the extension :
(2)
(4514)
where
J is the following row sub-matrix :
(3)
(4515)
J being a scalar.
Check that (2) is a group :
(4)
(4516)
(5)
(4517)
(6)
(4518)
Then :
(7)
(4519)
The inverse matrix is :
(8)
(4520)
The element of the Lie algebra is :
(9)
(4521)
Calculate the action of
g 3-1 on the element of the Lie algebra
d g 3
(10)
(4522)
(11)
(4523)
g is a matrix :
(12)
(4524)
so that :
(13)
(4525)
The identification :
(14)
(4526)
gives :
(15)
(4527)
(16)
(4528)
