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...An
element gp of the Poincaré's 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 precedent 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's 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's 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's 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's 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** }
Index Dynamic Groups Theory