Chapter 6

The Rotation and Spin

Groups in 3 and 4

Dimensions

Among the basic symmetry groups of the physical world is the orthogonal group
SO(3) of rotations about a point in three dimensional space. The observables
one gets from this group are the components of angular momentum, and under-
standing how the state space of a quantum system behaves as a representation
of this group is a crucial part of the analysis of atomic physics examples and
many others. This is a topic one will find in some version or other in every
quantum mechanics textbook, and in chapter 8 we will discuss it in detail.
Remarkably, it is an experimental fact that the quantum systems in nature
are often representations not ofSO(3), but of a larger group calledSpin(3), one
that has two elements corresponding to every element ofSO(3). Such a group
exists in any dimensionn, always as a “doubled” version of the orthogonal group
SO(n), one that is needed to understand some of the more subtle aspects of
geometry inndimensions. In then= 3 case it turns out thatSpin(3)'SU(2)
and in this chapter we will study in detail the relationship ofSO(3) andSU(2).
This appearance of the unitary groupSU(2) is special to geometry in 3 and 4
dimensions, and the theory of quaternions will be used to provide an explanation
for this.

6.1 The rotation group in three dimensions

Rotations inR^2 about the origin are given by elements ofSO(2), with a counter-
clockwise rotation by an angleθgiven by the matrix

R(θ) =

(

cosθ −sinθ

sinθ cosθ

)