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...An element gp of the Poincaré group Gp is defined by a set of parameters {pi}, whose number is, by definition, the dimension of the group. The matrix dg (g=e) is built from the set {dpi}. So that the preceding mapping corresponds to the anti-action:
(242)
AA : {dpi}
--> {dpi'}
To introduce duality and build the dual mapping, we assume the constancy of the scalar:
(243)
The dimension of the Poincaré group is 10, whence we have ten scalars { Ji }, components of the moment Jp.
Jp
= { J1 , J2 , J3 , J4 , J5 , J6 , J7 , J8 , J9 , J10 }
We decide to split this into two objects:
(244)
Jp= { M , P }
The first, M, is a (4,4) antisymmetrical matrix.
P is the four-vector (4,1)
(245)
Then:
(246)
Jp= { M , P } = { M , p , E}
Write the scalar product as:
(247)
Tr means trace. The constancy of the scalar product gives
(248)
with:
(249)
(249b)
(249c)
but GG = 1. Then:
(250)
(250b)
(251)
i.e:
(252)
(252b)
(252c)
(253)
This coadjoint action can be put into a matrix form:
The element of the Poincaré group is:
(254)
Its transposed is:
(255)
Consider the matrix:
(256)
write:
(257)
(258)
(259)
which will be identified to the matrix:
(260)
Then Jp is the momentum of the Poincaré group, put into a matrix and the coadjoint action is:
(261)
Exercice: check this is an action.
The momentum-matrix corresponds to:
(262)
The diagonal of an antisymmetrical matrix M is made of zeros.
(263)
It depends on six parameters.
(264)
( lx , ly , lz , fx , fy , fz )
Let us write the matrix:
(265)
The last three components ( fx , fy , fz) form what Souriau calls the passage vector f:
(266)
The first ( lx , ly , lz) are the independant component of an antisymmetric matrix (3,3), which Souriau calls the spin matrix.
(267)
whence:
(268)
The vector P i the impulsion-energy four-vector:
(269)
Then the momentum of the Poincaré group can be explicited in its more general form:
(270)
It owns ten components:
(271)
Jp
= { E , px ,py , pz , fx ,fy , fz , fx ,fy , fz } = { E**, p , f , l** }
