Ebook: Groups, Representations and Physics

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 formalism
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 coordinatefree
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.