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Choice of the matrix m.
... A group G can be compared to a certain surface. It depends on a certain number of parameters. Let P be this space of the parameters of the group and p a point in this space. The number of these parameters pi is the dimension of the group.
(217)
Shown: the neutral element e (the identity matrix 1).
We can give an increment d p:
(218)
... Then we differentiate the matrix g, which is an element of the group. We obtain a square matrix dg that does not belong to the group. This is called the tangent vector to the group. These tangent vectors form what is known as the Lie algebra of the group (which, incidentally, is not actually an algebra).
We choose to differentiate in the neighborhood of the neutral element:
(219)
and we choose the following anti-action:
(220) AAg( m) = g⁻¹ × dg(g=e) × g
Remark:
Why do we choose the tangent vector to the group at g = 1?
... We could use a more general form, a tangent vector dg at any point of the group. We would obtain the same result, but the calculations would be significantly more cumbersome.
The dimension of the group is n. The matrix g depends on n parameters { pi }.
The element of the Lie algebra dg(g=e) depends on the same number of parameters { d pi }.
The computation of the above anti-action yields the mapping:
(221) { d pi } ⟶ { d p' i }
We introduce the same number of scalars: { J i }
We call this set the moment J of the group. J = { J i }
It is a set of n quantities, n scalars. Sometimes we can represent it as a matrix (e.g., Poincaré's action on its momentum).
{ J i } is the cotangent vector { d pi } to the tangent vector of the group. Duality gives:
(222)
From this conservation of the scalar product, if we know the mapping:
(223) { d pi } ⟶ { d p' i }
we can construct the dual mapping:
(224) { J i } ⟶ { J' i }
This is the essential action we seek, and Souriau calls it the coadjoint action of the group on its momentum space.
The best way to illustrate this concept is to provide an example:
Coadjoint action of the Poincaré group on its momentum space Jp.
Earlier, we presented the generalized Lorentz group. By choosing:
(225)
we obtain the Lorentz group L whose element L satisfies the axiomatic definition:
(226)
The spacetime vector is (227)
With c = 1, we obtain the elementary quadratic form, the Minkowski metric:
(228)
The inverse matrix is (229)
Now introduce a spacetime translation:
(230)
We construct the element gp of the Poincaré group Gp as follows:
(231)
Exercise: Show that it forms a group and compute the inverse matrix:
(232)
The element of the Lie algebra is (233)
and the anti-action:
(234) dgp' = gp⁻¹ × dgp × gp
We observe that
(235) G d L
is an antisymmetric matrix. Let us denote it as:
(236)
whence:
(237)
Let:
(238)
From this, we can construct the anti-action:
(239) dgp' = gp⁻¹ × dgp × gp
which gives us the mapping:
(240)
(240b) (240c)
is the required mapping:
(241)