| 16 |
We need :
Dual actions.
We have built, above,
an action :
(200)
an anti-action :
(201)
The first can refer to
any column vector m :
(202)
m' = g xm
and the second to any
line vector n :
(203)
n' = n
x g-1
m belong to a
certain space M
n belongs to another
space N.
Form the scalar :
(204)
S = n m
Notice that :
(205)
n' m' =
n x g-1
x g xm
...We
will say that the two considered actions are dual. Similarly the two spaces
Mand N, to which belong m and n are dual spaces :
N = M* or M = N*
Usually, we say that
if m is a vector, n is its covector.
The prefix co is typical
of the duality. As noticed by Souriau, the duality exists in politics and
he adds :
- The duality was present
in the Marxism Leninism, from the begining. Think about the munist and the
communist.
Take another point of view. Suppose we have one action, and we want to build
its dual.
Schematically :
(206)
...In
order to form a scalar product with the colum-vector m , n
must be a line-vector. Then these two vector must be defined by the same
number of scalars :
(207)
then we search the dual
action :
(208)
n' = Ag(n)
so that the scalar product
:
(209)
be invariant. We
must have :
(210)
n' m'
= n m
We have :
(211)
m' = g
x m
(212)
Ag(n)
x g
x m
= n m
whose solution
is :
(213)
Ag(n)
= n x g-1
Towards building the
essential action, or coadjoint action of a group on its momentum ( after
Souriau ).
We search an action of
the goup ont its "momentum space". We are going to built it as the dual of
an anti-action :
(214)
AAg(m)
= g-1 x
m x g
...In
the preceding section m was a vector. But in (214) it is a matrix.
We will take a matrix, which depends on a certain number of parameters :
{ m1 , m2 , . . . .
, mn }
We must imagine a dual
set of scalars :
{ n1 , n2 , . . . .
, nn }
so that :
(215)
Schematically :
(216)
Index Dynamic Groups Theory