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 understanding 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 of $SO(3)$ , but of a larger group called $Spin(3)$ , one that has two elements corresponding to every element of $SO(3)$ . Such a group exists in any dimension $n$, 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 in $n$ dimensions. In the $n=3$ case it turns out that $Spin(3)\simeq SU(2)$ and in this chapter we will study in detail the relationship of $SO(3)$ and $SU(2)$ This appearance of the unitary group $SU(2)$ is special to geometry in 3 and 4 dimensions, and the theory of quaternions will be used to provide an explanation for this.

中文目录

Chapter contents

• 6.1 The rotation group in three dimensions
• 6.2 Spin groups in three and four dimensions
  • 6.2.1 Quaternions
  • 6.2.2 Rotations and spin groups in four dimensions
  • 6.2.3 Rotations and spin groups in three dimensions
  • 6.2.4 The spin group and $SU(2)$
• 6.3 A summary
• 6.4 For further reading

Previous chapter: 5 Lie Algebras and Lie Algebra Representations

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