1.1 Guide to Further Reading

We have tried to limit our prerequisites to the bare minimum. This includes basic facts about Lie groups, Lie algebras, and their representations--especially finite-dimensional unitary representations of compact Lie groups. We will not need the structure theory for simple Lie groups. We do, however, assume a little familiarity with the classical Lie groups ${\rm {GL}}(n)$, ${\rm {SL}}(n)$, $\O (n)$, ${\rm SO}(n)$, ${\rm U}(n)$, and ${\rm SU}(n)$, as well as their Lie algebras.

There are countless books on Lie groups, Lie algebras and their representations, but the text by Hall [14] has everything we need, and more. Sternberg's introduction to group theory and physics [34] includes an excellent account of applications to particle physics. To see the subject more through the eyes of a physicist, try the books by Lipkin [20] or Tinkham [38]. Georgi's text [11] shows how the subject looks to one of the inventors of grand unified theories.

Starting in Section 3.1 we assume familiarity with exterior algebras, and in Section 3.2 we also use Clifford algebras. For what we need, Chevalley's book [6] is more than sufficient.

For the interested reader, there are many introductions to particle physics where one can learn the dynamics after getting a taste of the algebra here. It might be good to start by reading Griffiths' introductory book [13] together with Sudbery's text specially designed for mathematicians [36]. The book by Huang [17] delves as deep as one can go into the Standard Model without a heavy dose of quantum field theory, and the book by Lee [19] is full of practical wisdom. For more information on grand unified theories, see the textbooks by Ross [31] and Mohapatra [21].

Particle physics relies heavily on quantum field theory. There are many books on this subject, none of which make it easy. Prerequisites include a good understanding of classical mechanics, classical field theory and quantum mechanics. Many physicists consider the books by Brown [5] and Ryder [32] to be the most approachable. The text by Peskin and Schroeder [30] offers a lot of physical insight, and we have also found Zee's book [40] very useful in this respect. Srednicki's text [35] is clear about many details that other books gloss over--and it costs nothing! Of course, these books are geared toward physicists: mathematicians may find the lack of rigor frustrating. Ticciati [37] provides a nice introduction for mathematicians, but anyone serious about this subject should quickly accept the fact that quantum field theory has not been made rigorous: this is a project for the century to come.

Particle physics also relies heavily on geometry, especially gauge theory. This subject is easier to develop in a rigorous way, so there are plenty of texts that describe the applications to physics, but which a mathematician can easily understand. Naber's books are a great place to start [22,23], and one of us has also written an elementary introduction [3]. Isham's text is elegant and concise [18], and many people swear by Nakahara [24]. The quantum field theory texts mentioned above also discuss gauge theory, but in language less familiar to mathematicians.

Finally, few things are more enjoyable than the history of nuclear and particle physics--a romantic tale full of heroic figures and tightly linked to the dark drama of World War II, the Manhattan Project, and the ensuing Cold War. Crease and Mann [8] give a very readable introduction. To dig deeper, try the book by Segrè [33], or the still more detailed treatments by Pais [25] and Hoddeson et al [16].