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 g₃ from the scalar :

(17)

c dJ + S

Then the coadjoint action of the group g₃ on its momentum is :

(18) (4529)

The moment of the group g₃ 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 L is 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

( f¹ ,f² , f³ , f⁴ , f⁵ , f⁵ )

adds si more dimensions. The scalar ( l₁ ,l₂ , l₃ , l₄ , l₅ , l₅ ) are fixed and do not correspond to new dimensions.

The coadjoint action of the group on its momentum

(29)

J = { c¹ , c² , c³ , c⁴ , c⁵ , c⁶ , J_p }

is :

(30) (4538)

c'ⁱ = l_i m cⁱ with i = { 1 , 2 , 3 , 4 , 5 , 6 }

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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.