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Conclusion of this first part devoted to geometry.

A word about Geometrical Physics B and group theory.

( Elements of group theory, applied to physics, are given at the begining
of the sub-site Goemetrical Physics B , "Dynamic Groups of Physics" ) .

We have introduced new geometric concepts.

• Twin geometries, inspired intially by Andréi Sakharov's idea : There
  is not only a single universe, but two, which A. Sakharov called, in 1967
  "twin universes".

• These two universes don't live in distant places, but lie "at the same
  place". We gave the (primitive) didactic image of checkers, with two games,
  one being played on black squares and the other on white ones.

• This is a didactic image of a more refined geometric structure, in which
  the universe, as a whole, is composed by two distinct (but interacting) folds.
  These folds are 4d hypersurfaces, which can be considered as "the two-folds
  cover of a skeleton-manifold".

• As in general relativity, we assume that particles follow geodesic of each
  hypersurface. One of these is supposed to be our space-time. The other is
  supposed to be twin space-time.

A priori three kinds of particles are supposed to cruise along geodesics
in each fold, which are, schematically :

• matter

• anti-matter.

• photons.

So that, in the second fold, the second universe, that we can call ghost fold,
or ghost universe, we would find :

• ghost matter

• ghost anti-matter

• ghost photons.

( all that is explained in details in " Geometrical Physics B : Dynamic
Groups ins Physics" ).

-The two folds are distinct and their geodesic lines are distinct too. So
that a photon, travelling on a geodesic of our fold F, cannot jump and follow
a "ghost geodesic", which belongs to the ghost universe, the ghost fold F*.
As a conclusion, light emetted by matter (or anti-matter) in our fold, cannot
reach the other universe and be received by some ghost particle. If some living
creatures exist in the fold F*, they cannot see our stars, our galaxies, anything
that lies in our fold, on pure geometrical grounds.

• Conversely, a ghost photon, emitted by a particle of ghost matter (or ghost
  anti-matter) in the ghost fold F* ( or ghost universe ), travels on a geodesics
  of this fold and cannot jump to the other fold, ours. So that it cannot be
  received, captured by any massive particle located in our universe. As a conclusion,
  the structures of the twin universe, or shadow universe, or ghost universe,
  whatever the name we choose, are basically invisible to us. If there
  are structures of any kind in this second universe, we cannot observe it,
  by optical means, for the same reason : on pure geometrical grounds.

• This idea is close to the superstring advanced idea. Many researchers,
  from the superstring community, are now conviced that two worlds exist, which
  communicate only through gravitational field.
  Witten ( Field medal winner ), Duff, Green Schwarz, the Nobel Price winner
  Abdus Salam.....
  see a recent paper of Michael Duff, in Scientific American, entitled "the
  new superstring theory", translated in french (Pour la Science Journal, april
  1998 ).
  Duff imagines matter "on a wall" and some invisible matter "on another wall,
  parallel to the first.
  The idea of two universe, two entities, unable to see each other and communicating
  only through gravitational force, was initially due to Green, Swharz and the
  Nobel Price winner Abdus Salam.

The general idea is to extend the number of dimensions. In classical physics,
this number is four : (x , y , z , t), corresponding to space-time. Modern
theoretical physics tends to extend that number, in general to ten.

Then all is based on group theories and symmetries . A symmetry is
not only the familiar symmetries of the 2d or 3d space, like

• Symmetry with respect to a point.

• Symmetry with respect to a plane

• Symmetry with respect to a straight line.

• or rotational symmetry ( periodic objects, crystals ).

An object which remains unchanged through a translation owns this "kind of
symmetry".

There are also, for example, symmetries with respect to time. Consider the
movement of a test particle, located at a distance r from a mass-point M.

G being the constant of gravity, the movement obeys, in
newtonian dynamics, the following differential equation :
(142)

which owns a peculiar solution :
(143)

this last being time-reversible . We get
a symmetry with respect to time, a T-symmetry.

Particles own a set of peculiar symmetries. This forms
a set of strong constraints, if one wants to build the group which runs the
"things".
At the present time the superstring men are facing a wall.
Their tool box offers too many possibilities, so that they don't talk about
theory, but about theories. Many uses to say : "among the million of possible
theories..."

With my collegue Pierre Midy we have approached the problem
on a different angle, using a tool called "the coadjoint action of a group
on its momentum space". See the book of Jean-Marie Souriau :
"Structure of Dynamical Systems",Birkhauser Ed. 1997".

(See also Geometrical Physics B, Dynamic groups ins
Physics ) .

With such tool it has been possible to geometrize elementary
particles such as proton, neutron, electron, photon, neutrinos, and their
antis. But we do not deal with deeper structure ( quarks). See our paper :

J.P.Petit and P.Midy : Geometrization of matter and
anti-matter 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 anti-matter. Geometrical Physics
B, 1, 1998.

This paper contains a geometrical definition of anti-matter
.

Briefly, to classical space-time : { x , y , z , t } we
add six more dimensions, additional dimensions :

{ z
1, z 2,z
3,z 4,z
5,z 6
}

We can link these scalar to a vector z
. Similarly we can define the space-time vector :
(144)

We can consider that a particle "live" in a ten dimensional
space :
(145)

(146)

Or, simply :

z ---> - z

which means :

z 1 -----> - z
1

z 2 -----> - z
2

z 3 -----> - z
3

z 4 -----> - z
4

z 5 -----> - z
5

z 6 -----> - z
6

All the additional dimensions are reversed.

The introduction of additional dimensions modifies the
Dynamic Group . See the book of Souriau, Birkhauser Ed. 1997.

In non quantum relativistic physics the dynamic group
is the Poincaré's group. One extends to the quantum world, introducing
a fifth dimension z (Souriau,
1964). Furthermore the Kostant-Kirilov-Souriau method makes it possible to
build the Klein-Gordon equation from the "Central extension of the Poincaré
group", the new dynamic group.

We deal with a generalized extended Poincaré's
group ("Petit's group"), whose coadjoint action on its momentum gives the
six classical quantum numbers :

q : electric charge

cB : baryonic number

cL : leptonic number

cm
: muonic number

ct
: tauonic number

v :
gyromagnetic constant.

Then a particle is defined by the set :

{ q , cB , cL
, cm , ct
, v ,
E , px , py, pz, s }

E is its energy

{ px , py, pz} is its
impulsion vector

s is its spin.

For example, an electron corresponds to :

q : electric charge = - 1

cB : baryonic number
= 0

cL : leptonic number
= 1

cm
: muonic number = 0

ct
: tauonic number = 0

v : gyromagnetic constant.
= ve

s = 1/2

and anti proton to :

q : electric charge = - 1

cB : baryonic number
= -1

cL : leptonic number
= 1

cm
: muonic number = 0

ct
: tauonic number = 0

v : gyromagnetic constant.
= - ve

s = 1/2

a photon to :

q : electric charge = 0

cB : baryonic number
= 0

cL : leptonic number
= 0

cm
: muonic number = 0

ct
: tauonic number = 0

v : gyromagnetic constant.
= 0

s = 1

In Dirac's anti-matter all the charges are reversed (
including the electric charge ).

As all the charges of photons are zero it explains why
the photons is its own antiparticle, because :

• 0 = + 0

Then this method gives a first geometrization of (usual)
elementary particles. The description is limited to the components of the
nuclei.