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 (g1 × g2) = det (g1 ) × det (g1 )

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)

( g1 × g2 ) × g3 = g1 × ( g2 × g3 )

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 column-vectors. For an example space-time 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:
(22-bis)

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.