8
(91)
This coadjoint action can be written in matrix form.
The matrix of the Poincaré group is:
(92)
its transpose is:
(93)
Consider the matrix:
(94)
That is, we will put the momentum
(95)
Jp = {
M,
P }
in matrix form and form the product:
(96)
(97)
(98)
which I can identify with the matrix:
(99)
Jp is therefore the momentum of the Poincaré group, written in matrix form. And the coadjoint action is written:
(100)
As an exercise, the reader can, based on the axioms, check that this is indeed an
action
.
The momentum of the Poincaré group can be explicitly written as follows:
(101)
This matrix is antisymmetric (which implies that its main diagonal is made up of zeros).
Mis the matrix:
(102)
Let's specify it:
(103)
It is indeed an antisymmetric matrix, the hypothesis formulated from the beginning, which depends on six parameters:
(104)
( lx , ly , lz , fx , fy , fz )
The last three ( fx , fy , fz ) are the components of a vector, the vector-
passage f
:
(105)
The first three ( lx , ly , lz ) are the independent components of a (3,3) antisymmetric matrix, the
rotation l
:
(106)
Thus:
(107)
The vector
Pis the
four-momentum-energy vector
:
(108)
We can then explicitly write the momentum of the Poincaré group, in its full generality:
(109)
We check that it is indeed an object with ten components (the same number as the dimensions of the group).
(110)
Jp = { E , p
x
, p
y
, p
z
, f
x
, f
y
, f
z
, l
x
, l
y
, l
z
} = { E ,
p
, f , l
}
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