Geometrical description of Dirac's antimatter

...vWe extend the precedent group to a four-components orthochron set. This operation gives a geometrical interpretation of antimatter after Dirac.

** **

__1) Introduction :__

...In a former paper [1] we have presented a description of elementary particles ins a ten-dimensional space, i.e. space-time (x,y,z,t) plus six additional dimensions :

(1)

We presented a 16-dimensions group, an extension of the
Poincaré orthochron subgroup, acting on :

- its 16-dimensions momentum space

- its 10-dimensional movement space.

The six additional components of the momentum have been identified to the charges of the particles :

(2)

so that the momentum becomes :

(3)

(4)

after J.M.Souriau [1].

We have figured the link between the species of moments and the species of movement, suggesting that :

- The movement of matter corresponds to { z^{
i }> 0 } sector.

- The movement of antimatter corresponds to { z^{
i }< 0 } sector.

- The movement of photons corresponds to { z^{
i }= 0 } plane.

All that must be now justified.

__2) Introducing a four components group. Geometrization
of Dirac's antimatter.__

...The precedent 16-dimensional
group had two components, correspondong to the two orthochron components of
the Lorentz group, L_{n} ( neutral component ) and L_{s} ,
with :

(5)

Our group was an extension of the orthochron Poincaré sub-group :

(6)

and we wrote it :

(7)

The corresponding coadjoint action was :

(8)

with :

(9)

...In such a group no element transforms the movement of a matter mass-point into the movement of an antimatter mass-point, and vice versa. According to the chosen definition of antimatter, through a :

(10)

some element should reverse the additional dimensions. With :

(11)

we can write the precedent group into a more compact form
:

(12)

It contains the neutral element :

(13)

The matrix that reverses the additional dimensions is be the following orthochron commuter :

(14)

We can duplicate the precedent group through the operation
:

(15)

It is equivalent to write the new four component group, whose element is :

(16)

The corresponding coadjoint action is :

(17)

We see that ( l = - 1 ) reverses the charges. In that case the inversion of the additional dimensions :

(18)

goes with a :

(19)

which corresponds to Dirac's description of antimatter
[4], so that the present paper represents a geometrization of antimatter after
Dirac.