a4117

Choice of the matrix
m .

...A
group Gcan be compared to a certain surface. It depends on certain number
of parameters. Let Pbe this space of the parameters of the group and
pa point of this space. The number of these parameters pi
is the dimension of the group.
(217)

Shown : the neutral element e ( the unity matrix 1 ).
We can give an
increement d p
:
(218)

...Then
we differentiate the matrix g which is the element of the group. We
get a square matrixdg which does
not belong to the group. One calls it the "tangent vector to the group".These
tangent vectors forms what one calls the "Lie-Algebra" of the group ( which
not an algebra, by the way ).
We chose to differentiate
at the vicinity of the neutral element :
(219)

and we choose the following
anti-action :
(220)

AAg(m)
= g-1 x
dg(g=e) x
g

Remark :
Why do we choose the tangent vector to the group at g = 1 ?

...We
could use a more general , a tangent vector dgat any point of the group. We would get the same result, but the calculation
would be much more tedious.

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 will provide the mapping :
(221)

{ d
pi
} -----> { d pp'i
}

We introduce the same
number of scalars :

{ Ji }

We call this set the
momentum J of the group.

J = { Ji
}

It is a set of n quantities,
n scalars. Sometimes we can put it into a matrix ( Poincaré's action
on its momentum ).

{ Ji } is
the cotangent vector { d p
i } to the tangent vector of the group. The duality gives :

(222)

From this conservation of the scalar product, if we know the mapping :

(223)

{ d
p
i } -----> { d p'
i }

we may build the dual
mapping :
(224)

{ Ji } ----->
{ J'i }

This is the essential
action we search, and Souriau calls it coadjoint action of the group on its
momentum space.

The better way to illustrate
this conceot is to give an example :

Coadjoint action of
the Poincaré group on its momentum space Jp .

Above, we presented the
generalized Lorentz group. Chosing :
(225)

we get the Lorentz group
Lwhose element L obeys the axiomatic definition :
(226)

The space-time vector is
(227)

With c = 1 we get the
elementary quadratic form, the Minkowski metric :
(228)

The inverse matrix is

(229)

Introduce now a space-time
translation :
(230)

we build the element
gp of the Poincaré group Gp as follows
:
(231)

Exercise : show it forms a group an compute the inverse matrix :
(232)

The element of the Lie
algebra is
(233)

and the anti-action :

(234)

dgp'
= gp-1 x
dgp
x
gp

We notice that

(235)

G d
L

is an antisymmetrical
matrix. Call it :
(236)

whence :
(237)

Let :
(238)

from this, we can build the anti-action :
(239)

dgp'
= gp-1 x
dgp
x
gp

which will give us the
mapping :
(240)

(240b)

(240c)

is the required mapping
:
(241)

Index Dynamic Groups Theory