هندسة الجسيمات والجسيمات المضادة في فضاء مزدوج الطبقات

a4130

What is the solution ?

...If, as suggested by J.M.Souriau, God, in his infinite wisdom, did not create negative mass and energy particles and prevent physicist to use antichron elements, the theory cannot deal with PT and CPT symmetries.

We present a alternative solution in :

J.P.Petit and P.Midy : "Geometrization of matter and anti-matter through coadjoint action of a group on its momentum space. 4 : The Twin group. Geometrical description of Dirac's anti-matter. Geometrical interpretations of anti-matter after Feynmann and so-called CPT-theorem". Geometrical Physics B, 4, 1998.

...In order to prevent collisions between positive an negative energy particles, we split the evolution space into two folds, which forms the quotient of the group by its orthochron sub-group. We get a twin geometry.

We introduce a fold indix f = ± 1

f = +1 corresponds to the fold F

f = - 1 corresponds to the fold F*.

The Twin-group is :
(400)

...It is still a eight components group. We see that ( m = - 1 ) elements, which correspond to PT-symmetry, go with a fold commutation : f -----> - f

...The momentum space is still composed by four sectors, but negative energy sectors corresponds to particle's movements in the F* fold.
(401)

The subsequent symetries are :
(402) We can now define the new "playing field". (403)

The playing field : a two folds ( F** and F*) space, associated to a two sectors momentum space** ( E > 0** and E < **0 ).

(404)

Movements of ordinary matter. Action of orthochron elements of the group, with l = 1. Charges unchanged.

**Coadjoint action of a ( **l = -1 ; m = 1 ) element of the group on the momentum associated to the movement of normal matter : the new movement corresponds to Dirac's anti-matter.

...On the figure, the line M1 figures the movement of normal, orthochron matter. We figure straight lines because our group does not take account of force field, like gravitational or electromagnetic field. It only runs the behaviour of lonely particles, charged mass-points.

We chose an element in the grey area, corresponding to a ( l = -1 ; m = 1 ) matrix. The ( l = - 1 ) value changes the signs of all the z i. They become negative. The new path is in the second sector, corresponding to anti-matter. As l m = - 1 the charges are reversed. But as time is not reversed, the energy and the mass of the particle remains positive.

*This is a geometric description of ( orthochron ) anti-matter after Dirac.
*
...Two more sectors has to be explored. On the third we examine the impact of ( l = - 1 ; m = - 1 ) element on the momentum and movement.

( l = - 1 ) reverses the {z i}. According to our geometric definition this new movement corresponds to anti-matter, for it takes place in the second sector of space { z 1,z 2, z 3, z 4, z 5, z 6, x, y , z , t }.

( m = - 1 ) gives a PT-symmetry, reverses the signs of ( x, y , z , t )

But ( l m = + 1 ) keeps the charges unchanged.

This is "PT-symmetric anti-matter", so that it is a geometric description of anti-matter after Feynmann.

The movement takes place in the second space sector, in the fold F*.
(406)

( l = -1 ; m = -1 ) **elements transform movement of normal matter into movement of anti-matter **(z-Symmetry) of PT-symmetrical object, runing bacward in time. Geometric description of Feynmann's vision of anti-matter. Does not identify vompletely with Dirac's one : negative mass and negative energy.

The last elements correspond to the sector ( l= 1 ; m = -1 )

( l = 1 ) --- > the movement is still in the matter's sector :

no z-Symmetry.

( m = -1 ) goes with a PT-symmetry. The particule runs backward in time.

( l = -1 ) : C-Symmetry. The charges are reversed.

...This is CPT-symmetrical matter, so that it corresponds to a geometrical interpretation of the so-called "CPT theorem", which asserts that the CPT-symmetric of a particle should be identical to that particle. That's not true. This movement corresponds to an antichron movement. The particle goes backward in time, si that (coadjoint action) its mass and energy become* negative* .

...The movement of a particle which is the CPT-symmetrical of a normal particle takes place in the fold F*.
(407)

( l = 1 ; m = - 1 ) case. Corresponds to CPT-symmetry. But the coadjoint action gives negative mass and energy. The CPT-symmetric of a particle of matter is a particule of matter, but with negative mass.

Index Dynamic Groups Theory