You are here

Rotations, Quaternions, and Double Groups

Simon L. Altmann
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
, on

From the author’s introduction:

… quaternions appear to exude an air of nineteenth-century decay, as a rather unsuccessful species in the struggle-for-life of mathematical ideas. Mathematicians, admittedly, still keep a warm place in their hearts for the remarkable algebraic properties of quaternions but, alas, such enthusiasm means little to the harder-headed physical scientist.

The author writes as a mathematical physicist with, presumably, one foot in both the hard- and soft-headed camps. However, his sympathies are clear:

Anyone who has ever used other parametrizations of the rotation group will, within hours of taking up the quaternion parametrization, lament his or her misspent youth…

He explains why this is, of course, and does a rather good job. Although he pays a great deal of attention to the geometry and topology of rotations, the author’s primary interest derives from angular momentum operators in quantum mechanics and the rotation operators that arise thereby.

The book begins with “All You Need to Know about Symmetries, Matrices and Groups.” This chapter is intended to provide a common basis of language and notation and a background in algebra for a variety of potential readers. An immediate clue that this is not purely an algebra text is the first section that treats symmetry operators on configuration space.

There is considerable attention given in the text, as one might expect, to the groups SU(2) and SO(3), their relationship and representations. A discussion of irreducible representations for SO(3) leads to spherical harmonics, and from there to Pauli matrices. Spinor representations of the rotation group — representations that correspond to half-integral values of the angular momentum — are treated extensively. Another key element is the study of projective representations motivated by the two-to-one mapping of SU(2) onto SO(3).

Double groups are an ad hoc invention of the physicist Hans Bethe who was attempting to distinguish 2π rotations from null rotations for finite rotation groups. Bethe essentially doubled the number of elements in the finite rotation group and defined a compatible group operation to make the double group well-defined. In practice, use of the double group can be quite awkward. Anything that can be done with the double group can be done more effectively with the original group and projective representations corresponding to the spinor factor system. So the double group is a kind of mathematical museum piece.

This book is probably most appropriate for graduate students studying mathematical physics, especially those who are trying to understand the subtleties of rotations and angular momentum.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

0. Notation. Conventions. How to Use This Book

1. Introduction
2. All You Need to Know about Symmetries, Matrices, and Groups
3. A Primer on Rotations and Rotation Matrices
4. Rotations and Angular Momentum
5. Tensor Bases: Introduction to Spinors
6. The Bilinear Transformation
7. Rotations and SU(2). The Stereographic Projection
8. Projective Representations
9. The Geometry of Rotations
10. The Topology of Rotations
11. The Spinor Representations
12. The Algebra of Rotations: Quaternions
13. Double Groups
14. The Irreducible Representations of SO(3)
15. Examples and Applications
16. Solutions to Problems