f4505

Geometrization of matter and antimatter through

coadjoint action of a group on its momentum space. 4 :

The Twin group. Geometrical description of Dirac's antimatter.

Geometrical interpretations of antimatter after Feynmann and so-called CPT-theorem. (p5)

The equation (16) is the action on the Lie algebra element , corresponding to the group

.The coadjoint action is the dual of this action and is based on the invariance of a scalar. Call S this scalar from which one computes the coadjoint action of the group

on its momentum. We compute the coadjoint action of the group

g 3

from the scalar :

(17)

c dJ + S

Then the coadjoint action of the group

g 3

on its momentum is :

(18)

(4529)

The moment of the group

g 3

is :

(19) J = {

c

, momentum of the group

G }

The extension of the group adds a component

c

to the moment, which obeys (20). In particular, if

, i.e :

(20)

(4531)

its coadjoint action is :

(21)

c

'

= l m c

(22)

(4532)

(23)

(4533)

The equations (22) + (23) identifies to the coadjoint action of the Poincaré group when

Lis the neutral component of the Lorentz group.

We know that we can put the momentum

J p

of the Poincaré group

g p

into an antisymmetric matrix :

(24)

(4534)

The its action on this momentum is :

(25)

(4535)

Then we can write :

(26) J = {

c

,

J p }

and :

(27) (4536)

c

'

= l m c

The Dimension of the Poincaré group is ten. The dimension of this extended group is eleven, due to adding the new variable

f

. (

l

= ± 1 ) and (

m

= ± 1 ) are not new dimensions of the group.

This method can be extended as many times as one wants. Consider the following matrix :

(28)

(4537)

The Poincaré group depends owns ten dimensions. The set

( f 1

,

f 2

,

f 3

,

f 4

,

f 5

,

f 5 )

adds si more dimensions. The scalar (

l 1

,

l 2

,

l 3

,

l 4

,

l 5

,

l 5

) are fixed and do not correspond to new dimensions.

The coadjoint action of the group on its momentum

(29) J = {

c 1

,

c 2

,

c 3

,

c 4

,

c 5

,

c 6

,

J p }

is :

(30)

(4538)

c

'

i = l i m c i

with

i = { 1 , 2 , 3 , 4 , 5 , 6 }

References .

[1] J.P.Petit & P.Midy : Geometrization of matter and antimatter 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 antimatter. Geometrical Physics B ,

1

, march 1998.

[2] J.P.Petit & P.Midy : Geometrization of matter and antimatter through coadjoint action of a group on its momentum space. 2 : Geometrical description of Dirac's antimatter. Geometrical Physics B,

2

, march 1998.

[3] J.P.Petit and P.Midy : Geometrization of matter and antimatter through coadjoint action of a group on its momentum space. 3 : Geometrical description of Dirac's antimatter. A first geometrical interpretation of antimatter after Feynmann and so-called CPT-theorem. Geometrical Physics B ,

3

, march 1998.

[4] J.M.Souriau : Structure des Systèmes Dynamiques, Dunod-France Ed. 1972 and Birkhauser Ed. 1997.

[5] J.M.Souriau : Géométrie et relativité. Ed. Hermann-France, 1964.

[6] P.M.Dirac : "A theory of protons and electrons", Dec. 6th 1929, published in proceedings of Royal Society ( London), 1930 : A

126

, pp. 360-365

[7] R.Feynman : "The reason for antiparticles" in "Elementary particles and the laws of physics". Cambridge University Press 1987.

Acknowledgements .

This work was supported by french CNRS and Brevets et Développements Dreyer company, France.

Déposé sous pli cacheté à l'Académie des Sciences de Paris, 1998.

Copyright french Academy of Science, Paris, 1998.