I turns out that these transformations have a convenient mathematical form.

For transformations on a real space we can drop the imaginary number.

[http://en.wikipedia.org/wiki/Euler's_rotation_theorem]

Taylor series

Start with a rotation in 2-d. Imagine that it is a rotation about the z axis by .

Choose the matrix Jz as shown below. We can then test to see if indeed the above matrix can be obtained using the exponential form and the Taylor series.

So for a simple rotation about the z-axis we see that one can define a matrix J that can be used to build a rotation about the z axis.

In general there are rotations about each axis  and you extend the above methods to develop a three component generator with a three component parameter .

That is there is a vector operator  and a set of parameters that can be used to carry out any rotation.  To make sense of the above equation we simply use a Taylor series expansion for the exponential function. For the word operator one substitute the more specific word matrix which is a special case of the general form.  We have found matrix representations for the more general members of the group of rotations which are operators in that they take a state and get a new state.  I can ask you to pick up a book and rotate it. You are operating on the book. You are not a matrix. However mathematically you could represent your actions on the book in the form of a matrix. Thus giving the matrix to a fellow student they would be able to extract which operation you performed ie which rotation you had performed. So operation is a non-specific implementation of a rotation. It includes physical realizations, matrix manipulation, any other implementation.

The interesting fact is that using the operator J which can be used to generate a rotation has separated the rotation into two convenient parts, a  set of parameters and an independent operation.

The structures that we have uncovered can be labeled in several different ways. For our example we could state the size of the matrix needed to perform the rotation. Monopole=1, dipole=3… One can also use the generator J to label the state. In QM this operator is the angular momentum operator. The structures then possess and intrinsic angular momentum. In particle physics we build elementary particles from the irreducible elements of the rotation group and label them by the their angular momentum which is defined as J the generator of the rotations.

The possible irreducible representation increase when one includes imaginary numbers.

J operators for quantum spin states:

 Spin 0 è I or 1 Spin ½è Pauli matrices Spin 1èVector Bosons Spin 3/2èDelta