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The project.
...Our starting point will be a dynamic group 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—this last being the number of independent parameters defining each matrix g in the set forming the group G).
...Now we need an action, to define a space on which the group acts—their momentum space (or phase space). This space is not spacetime, where particles are supposed to move. Constructing such a space will lead us into a strange land, resembling a schizophrenic terrain. But if you follow this path, you will be closer to physical reality than you have ever been before.
...Once we have a space to work with and an action to act upon, we can classify momentum-movements into species, and identify these species with elementary particles.
...Earlier, we noted that the product of a group with a vector—corresponding to SO(2) and O(2), as well as SO(3) and O(3)—constitutes an action: g × r
that is:
(166b)
Note that we can write this equivalently:
(167)
For the oriented Euclidean group and the full Euclidean group, we must define an action:
(168)
But these actions, as well as the corresponding actions of dynamic groups on space, such as:
(169)
produce... nothing. They merely move objects in space, or spacetime, or more refined spaces (five-dimensional space, ten-dimensional space).
We must seek something "hidden beneath 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 1970s. This point will be developed further.
...Of course, once calculations are complete, the physicist will ask:
Why?
...In other words, it works—but can we give a physical meaning to the 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 invent a new word to name it:
inertia.
...Aristotle arrives. He’s been informed by other students that you’ve invented something new, and he asks:
—Could you explain what inertia means?
You cannot do so using Aristotle’s vocabulary. You have encountered a paradigm shift.
...Jump to the Middle Ages. Try to explain a chemical reaction using the vocabulary of the four elements. That’s also impossible.
The coadjoint action of a group on its momentum space constitutes a paradigm shift. It is a new approach to physics.
In fact, physicists constantly handle group actions whenever they speak of "invariance" or "conservation laws."
A conventional physicist would then ask:
—Could you explain, in simple terms if possible, what the coadjoint action of a group on its momentum space means?
We reply:
—Why do you use conservation laws in physics?
—Well... because conserved quantities exist: energy, mass, electric charge...
—Why are they conserved?
—But that’s a fundamental principle!
—My dear friend, consider the coadjoint action of a group on its momentum space as a fundamental principle.
—What do you mean?
—All physics rests on a group structure. If you identify the group, you can construct its coadjoint action and the corresponding momentum space. Then the components of momentum become the corresponding physical quantities.
—.........
Warning. If you are a physicist (even a theoretical one) and you read what follows, you will undergo a paradigmatic transformation. Afterward, physics will simply be... different.
Actions.
What is an action?
Something linked to a group, obeying the following axioms:
(170)
Of course, for matrix groups, the composition operation is:
×
(matrix multiplication, row by column)
For matrix groups, we can write:
(171)
Consider the column vector:
(172)
where x, for example, represents vectors (173)
Does (174)
satisfy the axioms of an action? Let g and g' be two elements of the group G.
(175)
(175b)
We must have:
(176) Ag(Ag'(x)) = Ag''(x)
that is:
(177)
due to the associativity property:
(178) g'' = g × g'
Thus, it is indeed a group action.
...Notice we placed the group element g on the left. What happens if we place it on the right? Then it must be combined with a row matrix y.
(179) Ag(y) = y × g
Is this an action?