f4203 Geometrization of matter and antimatter through the coadjoint action of a group on its momentum space. 1 : Charges as additional scalar components of the momentum of a group acting on a 10d-space. Geometrical definition of antimatter. (p3) The complete Poincaré group is :
(31) Gp = Gn U Gs U Gt U Gst
The neutral component Gn is the first sub-group. The orthochron group [1] :
(32) Go = Gn U Gs
is also a sub-group of the Poincaré group.
The antichron part of the group [1] :
(33) Gac = Gt U Gst is not a group. Obviously :
(34) Gp = Go U Gac
...As pointed out in [1] the presence of the elements of Gac = Gt U Gst may produce negative mass particles, as peculiar movements of matter, running backward in time. In his book [1] J.M.Souriau suggests two solutions :
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Either one simply decides that negative masses cannot exist.
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Either the Poincaré group is limited to its orthochron subgroup.
(35) Go = Gn U Gs
2) The central extension of the Poincaré group. (36)
is the central extension of the Poincaré group, built from the orthochron sub-group. The corresponding action is : (37)
z is an additional dimension, a fifth dimension. The dimension of the group becomes 11 and the momentum gets a corresponding extra component :
(38) Jpe = { c , M , P } = { c , Jp }
The coadjoint action gives : (39)
...The physical meaning of this 11th component c was never clearly understood. Through his geometric quantification method, J.M.Souriau shows that the spin must be quantized [1]. Choosing a coordinate system in which the passage becomes zero, and considering only z-motions, the Jp momentum matrix becomes :
(40)
where E is the energy, p the modulus of the vector impulse and s the spin.
Photons correspond to
(41)
with two distinct helicities : right and left (polarization).
Neutrinos correspond to :
(42)
with also two distinct helicities.
Non zero mass particles like proton, electron, neutron, correspond to :
(43)
with : (44)
(45))
...From the extended Poincaré group (36), through the Kostant-Kirilov-Souriau method one can derive [1] the relativistic quantum Klein-Gordon equation. Similarly [1] the non-relativist Bargmann group (1960) gives the non-relativistic Schrödinger equation.
What about antimatter ?
...In a former book [2] J.M. Souriau developed general relativity in five dimensions, adding an extra dimension z to space-time ( x , y , z , t )
...Then, reference [2], Chapter VII , page 413, he identifies the inversion of the fifth dimension ( z ---> - z ) to the charge conjugation ( or charge inversion, or C-symmetry ) transforming matter into anti-matter.
