Geometrical definition of antimatter.

**Observatoire de Marseille
**

**Abstract **:

...Through a new four components non-connex group, acting on a ten dimensional space, composed by (x,y,z,t) plus six additional dimensions we give a description of particles like photon, proton, neutron, electrons, neutrinos ( e, m and t ) and their anti, through the coadjoint action on the momentum space. Quantum numbers become components of the moments. Matter and antimatter are interpreted as two different movements of mass-points in this

matter movement taking place in the {z
^{i }> 0} half space and antimatter in the
remnant {z^{ }^{i}
< 0} one.

The z-Symmetry :

which there goes with charge conjugation, becomes the definition of matter-antimatter duality.

__1) Introduction.__

...As pointed out by J.M.Souriau in his book [1] the Poincaré group, as a dynamic group for physics, arises a problem about the sign of the mass.

Everything starts from the Lorentz group L, whose element
**L** is axiomaticaly defined by :

(1)

where :

(2)

The Lorentz group acts on space-time :

(3)

through the action :

(4)

The matrix **G **comes from the expression of the Lorentz
metric (with c=1) :

(5)

We know than the Lorentz group is composed by four components :

L_{n} is the neutral componant, which contains
the neutral element **1**, i.e. the peculiar matrix :

(6)

L_{s} , the second component, contains the matrix
:

(7)

which reverses space.

L_{t} , the third component, contains the matrix
:

(8)

which reverses time.

L_{st} , the fourth component, contains the matrix
:

(9)

which reverses both space and time.

From the Lorentz group one builds the Poincaré
group G_{p}, whose element is :

(10)

**C **is a space-time translation :

(11)

...If we use the four components
of the complete Lorentz group L , (10) will be called the complete Poincaré
group. As the Lotentz group, it owns four components :

- Its neutral component :

(12) (4212)

built with the neutral component L_{n} of the
Lorentz group L.

- A second component :

(13)

built with the component L_{s} of the Lorentz
group.