العمل والرد الفعلي المزدوج

a4116

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 x m

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 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 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