a4114 | JP-Petit.info


	14

The project.

...Our
starting point will be a group Dynamic G , i.e. a family of square matrixes
g .

...Dynamic
: because time is involved .

...This
group has a certain dimension n .It may act on a space X, which has
its own dimension ( which has nothing to do with the dimension of the group,
this last being the number of independent parameters which define each matrix
g of the set, forming the group G ).

...Now
we need an action , to define a space on which the group acts, its
momentum space. This space is not space-time in which the particles are supposed
to travel. Building such space will take us in a strange country, which will
looks like some schizophrenic land. But if you follow this way you will be
closer to physical reality that you have never been before.

...When
we will have a space to play with, and an action ton act on, we will classify
the moments-movements into species and identify such species to elementary
particles.

...Above
we said that the produce of a group by a vector, corresponding to SO(2) and
O(2), as well as SO(3) and O(3) corresponded to an action :

g x
r

i.e :
(166b)

Notice we could write it in the equivalent way :
(167)

For oriented Euclid's
group and complete Euclid's group we need to write an action :
(168)

But these actions, as
well as the corresponding actions of dynamics groups on space, like :

(169)

give ... nothing. They
just carry objects in space, or space-time, or more refined spaces ( 5 dimensional
space, ten dimensional space ).

We have to search something
"hidden under the group" : itsmomentum space ( all groups of matrixes own
one ) and its

coadjoint
action on its momentum space.

which corresponds to
real physics.

What is physics ?

...Good
question. The French mathematician Jean-Marie Souriau invented the concept
of coadjoint action of a group on its momentum space and showed, in the early
seventies. In the following this will be developped.

...Of
course, the physicist, at the end of the calculation, will ask

Why ?

...In
other terms, it works, but can we give a physical meaning to this concept
of coadjoint action of a dynamic group on its space momentum. The answer seems
to be no.

...Imagine
you are a student of Aristotle. Suddently you have a intuition and you invent
a new word to name it :

inertia.

...Aristotle
arrives. He has been warned by other students that you had invented something
new, and he asks ;

Could you explain to
us what means inertia ?

You will be unable to
do it, using Aristotle's words. You will have encouneted a paradigmatic change.

Jump to the middle-age. Try to explain a chemical reaction in terms of four
elements vocabulary. Impossible too...

The coadjoint action
of a group on its space momentum is a paradigmatic change. It is a new insight
on physics.

In fact, physicists handle
group actions all the time when they speak about "invariance", conservation
laws".

Such conventional physicist will ask :

Can you explain to
me what is the meaning of the caodjoint action of a group on its momentume
space, in simple worlds, if possible ?

We answer :

Why do you use conservation
laws, in physics ?
Aoh... because there
are conservative quantities : energy, mass, electric charge...
Why are they conserved
?
But, it is a basic
principle ! ....
My dear friend, consider
the caodjoint action of a group on its momentum space as a basic principe.
What do you mean ?
Any physics is based
on a group structure. If you identify the group, you may build its caodjoint
action and the corresponding momentum space. Then the components of the moment
are the corresponding physical quantities.
........

Warning. If you are a physicist (even a theoretical physicist...) and if you
read what follows you will undergo a paradigmatic mutation. After, physics
will be simply... different.

Actions.

What is an action ?
Something linked
to a group, which obeys the following axioms :
(170)