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 Eis 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.
