a4106
| 6 |
|---|
Components of a group.
We have considered two groups: SO(2) and O(2). The second contains the first.
The first contains the neutral element. We can represent the elements of the group as follows:
(73)
The elements of the first component form a group (a subgroup).
The elements of the second component do not form a group, for many reasons:
- It does not contain the neutral element 1.
- We can choose two matrices in this second component whose product does not belong to this second component. Example:
(74)
The component of the group that contains the neutral element 1 is called the
neutral component of the group.
In the following, we will consider groups with 2, 4, or 8 components.
The Euclidean group.
We can now integrate this extended, enriched group with 2D translation, and obtain:
(75)
and the corresponding action of this Euclidean group:
(76)
Suppose we use our group to manipulate, govern, and study alphabetic letters.
Restrict the set to the letters: A B C D E F G J K L N P Q R S Z
We have several sizes:
(77) A B C D E F G J K L N P Q R S Z A B C D E F G J K L N P Q R S Z
A B C D E F G J K L N P Q R S Z
We know it is impossible to find a group element or a subsequent group action that can transform:
G into G
because their sizes are different. We decide to call their sizes masses, so that G and G are similar to particles, objects, atoms, possessing different masses. Now, this depends on the group acting on this set of objects. If I use:
(78)
Suppose this "universe" is filled with:
(79)
having a certain spectrum of sizes (masses) and angles. If I apply any group actions, I will never find objects belonging to the Russian alphabet:
(80)
This will become possible if I use the enriched group, the Euclidean group:
(80b)
Then my "universe" will become:
(81)
The group has enriched the "zoo" of letters. But in my zoo, one element is invariant under symmetry, i.e.:
(82)
(83)
(84)
(85)
...In general, any symmetry with respect to a straight line in the plane, which is a "2D mirror," does not change the "nature" of this character.
(86)
I will call this character a "photon" and identify the transformation
(87)
with matter-antimatter duality. Then I obtain a global zoo:
(88)
We could link letters of the same shape (nature) but different sizes (representing their energies), using Descartes' group:
(89)
...But we will not build a complete analogical model of elementary particles based on alphabetic characters. Anyway, you begin to see where we are heading. Groups have very simple appearances, but hidden properties. These properties depend on their subgroups, which generate the species.
...The Euclidean group goes hand in hand with a Euclidean world, with a Euclidean zoo. The animals of Euclidean geometry are called sphere, cylinder, prisms, plane, straight line, triangles, etc. They are invariant under certain subgroup actions. Souriau calls the subgroup associated with an object belonging to a species the regularity of that object.
For example, spheres centered at a given point O are invariant under the subgroup of rotations around that point.
-
We can consider that invariance is a property of the species called "spheres centered at a point O".
-
Conversely, we can consider that this property defines the species.
