THE DYNAMIC GROUPS OF PHYSICS
Third axiom of
group theory:
Any element of the group
must own its inverse , written g^{1} , defined by:
(15)
g × g^{1} = g^{1} × g = 1
In our example:
(16)
i.e: β
=  α or:
(17)
g^{1} ( a ) = g (  a )
Here the calculation
of the inverse matrix is trivial.
What is the condition
for a given square matrix to own its inverse?
To
any square matrix we can associate a scalar called determinant . For
definition see any book devoted to linear calculus . This determinant
is codified: det ( g )
In addition we have a
general theorem:
det (g_{1} × g_{2})
= det (g_{1} ) × det (g_{1} )
The determinant of a diagonal matrix is:
(18)
As a consequence:
det ( 1 ) = 1
for 1 is a diagonal matrix.
From the definition of
the inverse of a matrix:
g × g^{1} = g^{1} × g = 1
Then:
(19)
det ( g × g^{1} ) = det (g) × det (g^{1} ) = 1
If
det (g) = 0 the condition (19) cannot be satisfied. Sets of matrixes
whose peculiar elements own null determinant don't satisfy the third axiom,
and cannot form a group.
By the way:
(20)
Fourth axiom of group theory:
The multiplication must
be associative, i. e: .
(21)
( g_{1} × g_{2} ) × g_{3} = g_{1} × ( g_{2} × g_{3} )
Matrix multiplication
is basically associative.
Dimension of a group:
As
we will see, a group may act on a space whose points are described by columnvectors.
For an example spacetime points (called "events"):
(22)
This
is a four dimensions space. Different groups may act on it. But the dimension
of a group has nothing to do with the dimension of the space it acts on.
The dimension of
a group (of matrixes) is the number
of parameters which
define these square matrix.
We have given an example
of matrixes, defined by a single parameter
α
So that the dimension
of this group is one.
Notice that:
(22bis)
Remark:
All groups of matrixes
are not commutative , although the group we studied owns this property:
(23)
If such group acts on a
column vector, corresponding to a 2d space:
(23 bis)
it corresponds to rotation around a fixed point, in a plane:
(23 ter)
This operation is obviously
commutative.
You will tend to say: "like all rotations groups".
You're
wrong. Consider the rotations around axis passing by a given point O. Combine
two successive rotations, around different axis. This is not commutative.
Exercise: show that, using orthogonal axis system (OX, OY, OZ), combined
rotations around these axis is not a commutative operation.. Take any object.
 Make a rotation +90°
around OX, then a rotation +90° around OZ
Return to initial conditions
and:
 Make a rotation +90°
around OZ, then a rotation +90° around OX
Compare the results.
Group's action
A
group G is composed by square matrixes g . They can be multiplied.
We will say that a group may act on itself .
The group may also act on a space, made of points, described by column vectors.
Example:
(24)
If we write:
(25)
the action of the group
on this space becomes:
(26)
g × r
In
this peculiar case the action on space identifies to the simple matrix multiplication.
But the concept of an action is much more general.
