Preface
This book, third in a series, is based on lcctures on applied mathematics
given at the Applied Mathematics Laboratory of the David Taylor Model
Basin. It treats the n-dimensional unitary and rotation groups and their
finite-dimensional representations. The element of volume of these groups
is derived and the concept of integration over their parametric spaces is
treated in detail. A modification of Euler's parametrization of the 3-dimensional
rotation group is given, and this modification is extended to the
n-dimensional rotation group where n is any integer greater than 3. A
similar parametrization of the n-dimensional unitary group is furnished
and the close connection between the 2-dimensional unimodular unitary
group and the 3-dimensional rotation group is clearly explained. This
connection givC8 rise to the spin representations of the 3-dimensional rotation
group which are, precisely speaking, representations of the 2-dimensional
unimodular unitary group rather than of the 3-dimensional rotation
group. These spin representations of the 3-dimensional rotation group,
and also the spin representations of the proper, time-sense preserving
Lorentz group, are treated in considerable detail.
As in the preceding two volumes, care bas been taken to make the
lectures self-contained by giving detailed proofs of the basic theorems.
It is hoped that the material will prove useful to workers in theoretical
physics in view of the increasing importance of rotation groups of higher
dimension than 3.
Washington, D. C.
May 14, 1961
FRANCIS D. MURNAGHAN