f4505 *Geometrization of matter and antimatter through the coadjoint action of a group on its momentum space. 4: * The Twin group. Geometrical description of Dirac's antimatter. Geometrical interpretations of antimatter after Feynman and the so-called CPT theorem. (p5)
Equation (16) is the action on the element of the Lie algebra, corresponding to the group . The coadjoint action is the dual of this action and is based on the invariance of a scalar. Let S be this scalar from which one calculates the coadjoint action of the group on its momentum. We calculate the coadjoint action of the group g3 from the scalar:
(17) c dJ + S
The coadjoint action of the group g3 on its momentum is then:
(18) (4529)
The momentum of the group g3 is:
(19) J = { c, momentum of the group G }
The extension of the group adds a component c to the momentum, which obeys (20). In particular, if , that is:
(20) (4531)
its coadjoint action is:
(21) c' = l m c
(22) (4532)
(23) (4533)
Equations (22) + (23) correspond 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 Jp of the Poincaré group gp into an antisymmetric matrix:
(24) (4534)
Its action on this momentum is:
(25) (4535)
We can then write:
(26) J = { c, Jp }
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 the addition of the new variable f. (l = ± 1) and (m = ± 1) do not represent new dimensions of the group.
This method can be extended as many times as desired. Consider the following matrix:
(28) (4537)
The Poincaré group has ten dimensions. The set (f1, f2, f3, f4, f5, f5) adds six more dimensions. The scalars (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, Jp }
is:
(30) (4538) c'i = li m ci with i = {1, 2, 3, 4, 5, 6}
References.
[1] J.P. Petit & P. Midy: Geometrization of matter and antimatter through the 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 the 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 the coadjoint action of a group on its momentum space. 3: Geometrical description of Dirac's antimatter. A first geometrical interpretation of antimatter after Feynman and the 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", December 6th 1929, published in the proceedings of the 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 the French CNRS and by the company Brevets et Développements Dreyer, France.
Deposited in a sealed envelope at the Académie des Sciences de Paris, 1998.
Copyright Académie des Sciences de France, Paris, 1998.
