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The project.
...Our starting point will be a group Dynamic G, i.e., a family of square matrices 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, the latter being the number of independent parameters defining 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 particles are supposed to travel. Building such a space will take us into a strange country, one that will look like some schizophrenic land. But if you follow this path, you will be closer to physical reality than you have ever been before.
...When we have a space to play with and an action to act upon, we will classify the moment-movements into species and identify these species with elementary particles.
...Above, we stated that the product 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 the oriented Euclidean group and the complete Euclidean group, we need to write an action:
(168)
But these actions, as well as the corresponding actions of dynamic groups on space, such as:
(169)
yield... nothing. They merely carry objects in space, or space-time, or more refined spaces (5-dimensional space, 10-dimensional space).
We must search for something "hidden under the group": its momentum space (all matrix groups possess 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 the coadjoint action of a group on its momentum space and demonstrated it in the early seventies. This will be developed below.
...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 the coadjoint action of a dynamic group on its momentum space? The answer seems to be no.
...Imagine you are a student of Aristotle. Suddenly you have an intuition and you invent a new word to name it:
inertia.
Aristotle arrives. He has been warned by other students that you have invented something new, and he asks:
- Could you explain to us what inertia means?
You will be unable to do so using Aristotle's words. You will have encountered a paradigmatic change.
Jump to the Middle Ages. Try to explain a chemical reaction in terms of the vocabulary of the four elements. Impossible too...
The coadjoint action of a group on its momentum space is a paradigmatic change. It is a new insight into physics.
In fact, physicists handle group actions all the time when they speak about "invariance" and "conservation laws."
Such a conventional physicist will ask:
- Can you explain to me what is the meaning of the coadjoint action of a group on its momentum space, in simple words, if possible?
We answer:
-
Why do you use conservation laws in physics?
-
Oh... because there are conserved quantities: energy, mass, electric charge...
-
Why are they conserved?
-
But, it is a basic principle! ....
-
My dear friend, consider the coadjoint action of a group on its momentum space as a basic principle.
-
What do you mean?
-
Any physics is based on a group structure. If you identify the group, you may build its coadjoint action and the corresponding momentum space. Then the components of the momentum 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. Afterwards, physics will be simply... different.
Actions.
What is an action?
Something linked to a group, which obeys the following axioms:
(170)
Of course, for matrix groups, the composition operation is:
x
(line-column matrix multiplication)
For matrix groups we can write:
(171)
Consider the column-vector:
(172)
where x, for example, represents vectors
(173)
Does
(174)
fit the action's axioms? Let g and g' be two elements of the group G.
(175)
(175b)
We must have:
(176)
Ag ( Ag'(x)) = Ag"(x)
i.e.:
(177)
under the associativity property:
(178)
g" = g x g'
is a group action.
...Notice we put the element g of the group G on the left. What happens if we put it on the right? Then it must be combined with a row-matrix y.
(179)
Ag(y) = y x g
Is it an action?
