06009 - Groups, Representations and Physics [Jones].pdf

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Preface to the Second Edition
In this second edition I have taken the opportunity to expand the scope
of the book somewhat by including a new chapter (Chapter 9) on the
Cartan-Weyi-Dynkin approach to Lie algebras. This is a generalization
to more complicated Lie algebras of the method of raising and lowering
operators used to obtain the irreps of SU(2).
It
is a systematic and unified
approach, which allows one to classify all possible simple Lie algebras
and, in principle, to find all their irreducible representations. I hope that
inclusion of this topic, albeit in a necessarily rather condensed form, will
extend the usefulness of the book.
I am grateful to those readers who took the trouble to contact me
pointing out various errors in the previous edition, and I have corrected
those remaining errors of which I am aware.
H F
Jones
London, February /998
Preface to the First Edition
It
was with some trepidation that I decided to write yet another book on
group theory and physics. There are, after all, quite a large number
already. However, in the first place some of the best of these were,
inexplicably, out of print, and secondly I thought I perceived a niche for
a book which, while by no means skimping the physical applications,
exposes the student to the power and elegance of abstract mathematics.
There is, indeed, a fascinating interplay between mathematics and
physics. In some cases the need to understand and formulate a physical
problem provides a stimulus for the development of the relevant
mathematics, as for example in Newton's development of the calculus as
a tool for calculating planetary orbits. In others a mathematical formal-
ism already developed turns out to be tailor-made for physics. Thus the
theory of vector spaces is exactly what one needs for the general
formulation of quantum mechanics, and the representation theory of
groups is precisely the mathematical framework needed when one
considers the action of symmetry transformations on quantum systems.
With this in mind, I have developed the mathematical theory in a
more formal fashion than is strictly necessary for physical applications.
Thus, for example, I have dealt with representations in the coordinate-
free language of linear transformations acting on vector spaces, rather
than simply as matrices.
It
is my experience that students, far from
being put off by this, are fascinated by what may well be their first
exposure to the rigorous axiomatic approach, and ultimately obtain a
deeper understanding of the subject. However, since the book is aimed
primarily at physicists, I have been careful to include many examples
and illustrations of the various mathematical structures and theorems. I
have also included for the reader's convenience a short glossary of the
mathematical symbols used.
xii
Preface
The first four chapters give a systematic development of the theory of
finite groups and their representations. Applications to various physical
systems are given in Chapter 5 and the problems which follow.
Although finite groups are of direct importance in solid state and
molecular physics, the bulk of the applications to physics are concerned
with continuous groups. In dealing with these groups I have adopted an
intermediate level of rigour, not worrying overduly about proofs of
convergence which are the proper concern of mathematicians. The
theory of the rotation groups S0(2) and S0(3) is developed in Chapter
6. following which I am able to extend the range of the physical
examples.
My own interests as a field theorist have influenced the subject matter
and scope of the last three chapters. Chapter H is concerned with the
SU(N) groups, Chapter 9 with the Lorentz and Poincare groups. with
particular emphasis on the concept of helicity, which leads to remark-
able simplifications even in a non-relativistic context. In view of the
widespread interest in gauge theories, due to their remarkable successes
in recent years, I have included a final chapter on gauge groups. This is
necessarily somewhat impressionistic, particularly on the field theory
aspect, but I hope it conveys the essential points.
The book, based loosely on lectures given to physics students at
Imperial College, is aimed primarily at third-year physics undergradu-
ates with a good mathematical background, but it should also be useful
to first-year postgraduates in solid state. atomic or elementary particle
physics. The first four chapters might also be helpful to mathematics
students.
In view of this intended readership, a reasonable acquaintance with
quantum mechanics is assumed. Initially only the wave mechanical
formulation is used, but later on this becomes increasingly restrictive
and I go over to Dirac notation. For readers not familiar with this
formalism, a brief account is given in Appendix A. For similar reasons
Appendix B gives a review of the standard quantum mechanical
treatment of angular momentum, in the Dirac formalism. Appendix C is
a derivation of the invariant integration measure for S0(3 ). This is
rather technical, and can certainly be skipped at a first reading, hut does
make use of a rather elegant construction in spherical geometry.
Appendix D fills in the necessary background for the last section of
Chapter 9 on relativistic scattering, and Appendix E is a crash course in
Lagrangian mechanics for those interested in gauge field theories.
A short bibliography is provided so that the interested reader is able
to pursue any particular topic in greater detail, having, I hope. obtained
Preface
X Ill
a firm grounding in the basics. He or she can test that grounding against
the problems at the end of each chapted. Sketch answers are given at
the end of the book.
H F
Jones
London, January /989
t'Starred' problems are rather difficult, and for the dedicated reader only.

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