4
.
Note in passing that the system proposed to the child is not flawless. It works properly only if the objects provided are those that came with the game itself.
It should be noted that the "cylinders" compartment allows inserting cylinders of the same radius but different lengths, as well as paper clips, a pacifier, a closet key, etc...
Logically, the child who undergoes this group learning can deduce that a closet key and a cylinder are objects of the same kind. This is true, in the sense that these objects share the common property "which fits through that hole".
My daughter, when she was much younger, conducted very interesting experiments on groups using the computer's drive as an input window and sieve. It required a complete disassembly, after being disabled, to reconstruct her approach, which was quite logical. Today, she remains very interested in groups. But these are not the same, and these choices do not please me much.
Back to our child from earlier. A few months later, you will introduce the same child to the Galilean group, by throwing objects at him and encouraging him to catch them. Then, the shape of the objects is no longer important; what matters are their movements. We change groups. For the baby, the object (small enough for him to catch) becomes equivalent to its center of gravity. It is a "point-mass", a "point-particle". The Galilean group manages the dynamics of point-particles.
The classification is then based on the types of movements.
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I can catch this.
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I cannot catch this.
Appreciating the initial conditions, position and velocity vector, at a glance, the baby must recognize the type of movement he is facing, and anticipate.
As he grows up, he may play tennis, always using the Galilean group, which, like the Euclidean group, is a square matrix. As tennis balls move at speeds obviously much lower than the speed of light, he will not need to use the Poincaré group (another square matrix, which manages the relativistic movements of point-particles).
This being said, the point-particles managed by the Galilean group are no longer the points of the Euclidean group, they have attributes.
The specialist in mathematical physics who manipulates the Galilean group no longer speaks of point-particles, but of movements. What he seeks to classify are movements. This leads to an essential aspect of elementary particle physics: one seeks to associate a phenomenological description:
Tell me how your movements are, and I will tell you what you are.
One no longer seeks to know "what a particle is made of", but how it behaves. Thus, neutral particles do not behave in the same way as electrically charged particles. They belong to different species. They have different attributes.
Our baby from earlier, now a high-energy physicist, may examine daily photographs taken in a bubble chamber.
Photographs obtained using a bubble chamber (Schem..)
Left photo: a proton, a neutron and an electron have passed through the field of the chamber. The chamber is subjected to a magnetic field perpendicular to the plane of the figure. The neutron, without electric charge, is unaffected. It goes straight. Charged particles have very different radii of rotation (Larmor radius). Light electrons turn much more sharply in a magnetic field perpendicular to their direction of motion than heavy protons. And both turn in opposite directions.
Set of movements, in a magnetic field, managed by...
But this group does not exist yet. If you find it, you would make happy.
Nevertheless, our man, examining his photos, detects trajectories belonging to different sets of movements.
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This goes straight, it's a neutron.
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This turns to the right slowly, it's a proton.
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This turns sharply to the left, it's an electron.
Right photo: an electron and an anti-electron, born from the same radiation (dotted line), adopt symmetric trajectories, indicating that they have opposite electric charges.
Behavioral classification of species, as sets of movements.
The moment.
This purely geometric object can be considered as a set of attributes. Let's set aside charged particles, we will come back to them later. A "relativistic point-particle" has attributes grouped according to what the mathematician Jean-Marie Souriau, a leader in mathematical physics, called the "moment linked to the Poincaré group".
The attributes of the relativistic point-particle are called:
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Energy E
-
Momentum
p
- Spin
l (related to spin)
- Passage
f
The "moment" is thus: J = { E, p, l, f }
Note in passing:
We have therefore become accustomed, in everything that follows, to denote scalar quantities with thin letters and non-scalar quantities (square matrices, row matrices, column matrices) with bold letters.
Precision: one can then perform all row-column matrix multiplications by handling these thin or bold letters, which is extremely convenient. Let's give an example. The action of the element of the 2D Euclidean group was written:
By introducing:
we will obtain the more compact writing:
g then appears as a matrix, itself composed of submatrices:
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a is a square matrix of format (2,2).
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c is a column vector (the translation vector) of format (2,1).
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0 is a row vector of format (1,2):
In general, the bold 0s are either row vectors or column vectors.
The action is then written:
Given that a r means a x r (but one eventually neglects the matrix multiplication sign).
End of the note, back to the theme of the moment. Let's return to the expression of the moment in the case of the relativistic point-particle. J = { E, p, l, f }
E is a scalar (energy).
p is the vector momentum.
l and f (bold letters) are other vectors (lx, ly, lz) and (fx, fy, fz): the "spin" and the "passage".
Throughout our personal work that we will present in this sub-site Geometrical Physics B (the dynamic groups of physics), the problem will be precisely to make other "attributes" of elementary particles emerge, as components of a richer moment (the charges: electric, baryonic, leptonic, tauonic and the gyromagnetic coefficient).
It was Souriau who, in the 1970s, built the method allowing to reveal the components of the moment of a point-particle, starting from the group that manages it (in relativity, it is the Poincaré group). See the book:
Structure of Dynamical Systems. Dunod 1973
Strongly recommended aspirin.
It is then difficult to go further without implementing a fairly extensive, if not complicated, mathematical toolkit. Perhaps we will do this later, on the site, if there are enthusiasts, in the style "Everything you have always..."