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Prologue.
...Physics
is like a cake:
(1)
- First floor: observations,
experiments. - Second floor: differential equations.
- Third floor: geometry.
- Fourth floor: group theory.
Groups rule geometry,
which begets beautiful differential equations.
With differential equations
we build things, which are then used to explain or predict what we call physical
facts.
...Historically, men began to study and codify
facts, observations, and measurements. Then they imagined conservation
laws, and "physical laws." At the beginning of the century, they began to think
that physical laws could have something to do with geometry.
At the same time, Felix
Klein asked:
What
is a geometry?
Note that he said "a geometry" and not "geometry" (Erlangen Program).
...Klein,
Lie, Cartan, and others showed that something lay beneath geometrical appearances.
Geometry was not the top floor, the nec plus ultra of knowledge in physics.
From a group structure, one can build a geometry.
In the following, we will
try to show the link between groups, geometry, and physics.
By the way, upon groups, what?
...I
would tend to say: logic. But logic is a floor whose last occupant was Kurt
Gödel, a dangerous pyromaniac. With his well-known theorem, he set fire to
the furniture, which was completely destroyed. Since this tragedy, the room
has remained unoccupied.
...That is why I put a question mark there.
Groups.
...What is
a group? In the following, we limit the investigation to dynamic groups in physics: a set of square matrices (n,n) obeying defined axioms. These matrices
g, elements
of a group G, act on each other through classical (row-column) matrix multiplication. Among these square matrices, we find identity matrices.
(1-bis)
...A
group obeys the axioms defined by the Norwegian mathematician Sophus Lie.
These axioms apply to objects that are much more general than sets of matrices.
But we will limit our focus to this particular world and use matrix multiplication:
x
1 - First axiom of
group theory:
The product of two elements
g1 and g2 of
a group G:
(2)
g3
= g1 x g2
obeys:
(3)
Let us give an example
of a group of matrices, which depends on a single parameter a. The element
is:
(4)
The product of two elements
gives:
(5)
or:
(6)
g(a)
x
g(b)
= g( g ) = g( a
- b ) = g ( g )
We can write the matrix product
as:
(7)
which is similar to g1
and g2, i.e.:
(8)
Counter-example: Consider the following set of matrices which depend
on a single parameter a.
(9)
The product of two elements
gives:
(10)
which is fundamentally different from (5).
2 - Second axiom
of group theory:
In the set of elements,
we must find a particular one, called the neutral element e, which, combined
with any other element, obeys:
(11)
g x e = e x g = g
In groups whose elements
are square matrices, this neutral element e is always the identity matrix
1.
(12)
g x 1
= 1 x g = g
Note that we use lightfaced type to describe scalars and bold type for other objects: square, row, or column matrices.
Let us return to the
first example of a group:
(13)
Note that:
(14)
