تعميم المادة والمضادة للمادة من خلال التأثير المترافق

f4203 Geometrization of matter and antimatter through 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. Obvioulsy :

(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, runing backward in time. In his book [1] J.M.Souriau suggests two solution :

• Either one simply decides that negative masses cannot exist.

• 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 orhochron 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 neven clearly undestood. Through his geometric quantification method, J.M.Souriau shows than the spin must be quanticized [1]. Choosing a coordinate system in which the passage becomes zero, and considering only z-motions, the Jp the momentum matrix becomes :

(40)

where E is the energy, p the modulus of the vector impulsion 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 Schödinger equation.

What about antimatter ?

...In a former book [2] J.M. Souriau developped general relativity in five dimensions, adding an extra dimension z to space-time ( x , y , z , t )

...Then, reference [2], Chater 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.