THE DYNAMIC GROUPS OF PHYSICS
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 fathers beautiful differential equations.
With differential equations
we build things, which then are used to explain or predict what we call physical
facts.
Historically men began to study and to codify
facts, observations, performing measurements. Then they imagined conservations
laws, and "physical laws". At the begining of the century they began to think
that physical laws could have something to do with geometry.
At the same period, Felix
Klein asked:
What
is a geometry?
Notice he said "a geometry" and not "geometry" (Erlangen's program)
Klein,
Lie, Cartan and others showed that something lied under geometrical appearence.
Geometry was not the last 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: logics. But logics is a flat whose last occupier was Kurt
GÅ“del, a dangerous pyromaniac. With his wellknown theorem he put fire to
the furniture,which was completely destroyed. Since this tragedy, the room
is unoccupied.
That's
for I put a question mark there.
Groups
What's
a group? In the following we limit the investigation to dynamic groups of
physics: a set of square matrixes (n,n) obeing defined axioms. Theses matrixes g, elements
of a group G, act each on another through classical (linecolumn) matricial
multiplication . Among these square matrices we find unity matrixes.
(1bis)
A
group obeys the axioms defined by the Norvegian mathematician Sophus Lie.
These axioms apply to objects which are much more general than matrixes sets.
But we will limit our glance on this peculiar world and use the matrixmultiplication:
×
First axiom of
group theory
The product of two elements g_{1} and g_{2} of
a group G:
(2)
g_{3} = g_{1} x g_{2
}
obeys:
(3)
Let us give an example
of a group of matrix, which depends on a single parameter a . The element
is:
(4)
The product of two elements
gives:
(5)
or:
(6)
g( α )
× g( β )
= g( γ ) = g( α + β ) = g ( γ )
We can write the matrixproduct:
(7)
which is similar to g_{1} and g_{2 }, i.e:
(8)
Example to the contrary: Consider the following set of matrixes which depend
on a single parameter α
(9)
The product of two elements
gives:
(10)
which is basically different from (5).
Second axiom of group theory
In the set of elements
we must find a peculiar one, called neutral element e , which, combined
to any other element, obeys:
(11)
g x e = e x g = g
In groups whose elements
are square matrixes this neutral element e is always the unity matrix 1 .
(12)
g x 1 = 1 x g = g
Notice we use lightfaced
types to describe scalars and bold types for other objects: square or line
or colomn matrixes.
Let us return to the
first example of group:
(13)
Remark that:
(14)
