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We need:
Dual actions.
We have built, above, an action:
(200)
and an anti-action:
(201)
The first can refer to a column vector m:
(202) m' = g x m
and the second to a row vector n:
(203) n' = n x g-1
m belongs to a certain space M
n belongs to another space N.
Form the scalar:
(204) S = n m Note that:
(205) n' **m' **= n x g-1 x g x m
...We will say that the two considered actions are dual. Similarly the two spaces M and 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 Marxism-Leninism, from the beginning. Think about the communist and the munist.
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 column vector m, n must be a row vector. Then these two vectors 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 group on its "momentum space". We are going to build 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)