التعريف الهندسي للمادة المضادة

a4127

A geometrical definition of anti-matter.

...As mentioned by Souriau in 1964 in "Géometry and Relativité", Editions Hermann, chapter VII "La Relativité à Cinq Dimensions" ( the five dimensional relativity ), page 413, "the inversion of the fifth dimension corresponds to the charge conjugation".

...It is true if the anti-matter corresponds to Dirac's definition. Let us give an a priori geometric definition of anti-matter. We can figure space with dimensions :
(368)

This can be figured schematically as follows, with fibered space-time :
(369)

...We decide that matter's movements correspond to positive z i 's values and anti-matter's movements to negative ones, which corresponds to :
(370)

It is easy to modify the group in order to integrate this in it.
(371)

This becomes a four-components group ( l = ± 1 ) x 2 ( the extended orthochron group owns two connex components).

The component ( l = +1 ) is a sub-group.

...Clearly, the ( l = - 1 ) elements change the signs of the additional variables. We decide that it corresponds to matter anti-matter duality, on pure geometric grounds.

Let :
(380)

Then we can write, in a more compact way :
(381)

**l **= 1 corresponds to the orthochron sub-group.
(382)

Introduce what we will call a : " l-commuter " :
(383)

It belongs to the second component. But any element of this second component can be written :
(384) go = glc x go

being an element of the orthochron component of the group.

Schematically :
(385)

Left : the movement space, with two half-spaces, corresponding to

(z i > 0) movements ( matter )

and
(z i > 0) movements ( anti-matter )

Between the two the : z i = 0 movements ( photons ).

...Right, the four components group. All are orthochron. All movements correspond to positive energy ( below, momentum space ).

Call the ( l = - 1 ) elements "anti-elements".

We have figured the l-commuter anti-element.

...Normal orthochron elements transform a momentum corresponding to a positive energy movement J1+ into another positive energy movement J2+.

...But anti-elements transform positive energy matter's movement into positive energy anti-matter's movement ( J1+ -----> J3+ ) in momentum space. The figurative point is in the quarter which corresponds to anti-matter.

The corresponding paths are figured in the evolution space
(385b)

The calculation of the coadjoint action of the group
(386)

on its momentum gives :
(387)

see :
J.P.Petit and P.Midy : "Geometrization of matter and anti-matter through coadjoint action of a group on its momentum space. 2 : Geometrical description of Dirac's anti-matter". Geometrical Physics B, 2 , 1998.

Index Dynamic Groups Theory