groups and physics coadjoint action momentum

groups and physics coadjoint action momentum

(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 to say, we will put the moment

(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 moment of the Poincaré group, written in matrix form. And the coadjoint action is written as:

(100)

As an exercise, the reader can, relying on the axioms, verify that it is indeed an action.

The moment of the Poincaré group can be made explicit as follows:

(101)

This matrix is antisymmetric (which implies that its main diagonal is composed of zeros). M is the matrix:

(102)

Let us make it explicit:

(103)

It is indeed an antisymmetric matrix, a hypothesis made 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-displacement f:

(105)

The first three ( lx , ly , lz) are the independent components of an antisymmetric (3,3) matrix, the spin l:

(106)

Thus:

(107)

The vector P is the four-vector momentum-energy:

(108)

We can then make explicit the moment of the Poincaré group in its full generality:

(109)

We verify that it is indeed an object with ten components (a number equal to that of the dimensions of the group).

(110) Jp = { E , px , py , pz , fx , fy , fz , lx , ly , lz } = { E , **p , f , l **}