Customer Reviews

Lie Groups for Pedestrians (Dover Books on Physics)

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This book is still a very useful resource, nearly four decades after it was first published.

And that's the case even if you aren't exactly a pedestrian. This is the Truth about Lie groups!

While this book is very readable as it takes you through isospin, SU(3), commutation rules, symmetry breaking, the three-dimensional harmonic oscillator, and creation and annihilation operators, the most valuable part is the use of Young diagrams to construct multiplets for SU(3), SU(4), SU(6), and SU(12).

That is, suppose you are taking a course on elementary particles. And you are using some standard text such as Halzen and Martin (also a book that has aged very well). Anyway, you get to page 62 or so and that book tells you that the best way to construct the SU(3) multiplets is to use Young tableaux. But that book doesn't tell you how to use them. This one does.

If you are learning about elementary particles, you can go through this book in a day or two. And you'll be glad you did.

This book, by physicist Harry Lipkin, was intended as a quick introduction to Lie Groups to other physicists like himself working in the mid 1960's. At that time, many physicists had a sophisticated mathematical skill-set, but not one that included Lie Groups and Algebras, nor understood to the degree it is understood today how much it helps to think along those lines. Dr. Lipkin wanted to spread the knowledge of Lie Groups to physicists would would benefit from it.

As such, a "pedestrian" would be expected to be familiar with the then-current formulations of quantum theory, including the matrix and operator representations of quantum mechanics, as well as all the calculus necessary to work those theories. This is evident in chapter 1, where on page 2 section 1.1 is titled "Review of Angular Momentum Algebra", and whose first sentence asks us to consider the operators Jx, Jy, and Jz, which have the "well known" commutation rules [Jx,Jy]=iJz (etc).

Pedagogically, this is sound: start with something you know the reader is familiar with, then show by analogy how that applies to the new topic you are introducing, then expand the techniques into new areas, and then presumably turn it back towards the topic the reader knows and show how Lie Groups (in this case) make things easier.

However, it requires knowing the audience -- or, conversely, being the intended audience. I am not; from the benefit of knowing where physics has gone in the intervening 42 years, I know Lie Groups are important, and I know that my understanding of physics is weak, and I was hoping knowing about Lie Groups would help my understanding of physics. Ultimately, this book lost me before page 5.

I suspect this book no longer has an audience. The importance of Lie Groups in physics is now well-recognized and is taught to physicists-in-training. The folks who would understand this book no longer need it. Historically, it might have been important, but it no longer is.

The book of Lipkin has become a classical reference in group theoretical methods in physics, and is one of the most valuable reviews at the time of the establishment of the Gell-Mann-Ne'eman octet model. Divided into seven chapters and various later written appendixes, this work was originally thought as a comprehensive introduction to the unitary symmetry. This has been achieved in an impressive way, as shows the careful development of the topics and successive refinements. The su(3) symmetry is deduced naturally starting from the annihilation-creation operator formalism employed for the nucleon, and introducing the needed tools step by step. The (1966) more relevant groups SU(3), SU(4), SU(6) and SU(12) groups are analyzed in some detail, as well as some low rank symplectic groups and various subgroups intervening in the state labeling problem, such as the Wigner supermultiplet model. The author makes a self-contained presentation of the combinatorial technique of Young diagrams, which is inspired in the milestone work of M. Hammermesh, but presented here with astonishing simplicity to be applied by the reader without requiring a deep theoretical background.

A quite interesting section is devoted to the experimental predictions obtained from the octet model, like the classical example of the negative hyperon, discovered by Barnes et al. following the theoretical model. In all, this book shows the situation of the global internal symmetries in the 60s.

There is however one surprising fact about the book. In spite of the title, the concept of Lie group is nowhere defined adequately through the book. Although it is commonly understood that the group is meant when working with the corresponding Lie algebra, this can mislead some readers. Also the (informal) definition of Lie algebra given in equation (1.15) on page ten is false, or at least incomplete. A set of operators with some bracket (either of bosonic or fermionic type) defines a Lie algebra only if it is closed with respect to this brackets and additionally satisfies the Jacobi identity. None of this is found in the definition given in the book. To "satisfy commutation relations similar to those of angular momentum operators" is definitively not sufficient for higher rank algebras. I agree that this minor detail is irrelevant for the rest of the book, because the used operators obviously define a Lie algebra, but this can also lead to confusion, since apparently any arbitrary collection of operators would have the same property.

Although this book has aged quite well and remains an important reference, it is no more adequate for those who want an actualized overview of the classification of particles. There are obvious reasons for this, as the non-covered topics correspond to concepts or models that were developed later than the publication of the book. One example is the attribute color (around 1973), introduced to explain some remaining difficulties. This absence obviously extends to QCD (Quantum Chromodynamics). Also the unified theories and the model SU(5) of Georgi-Glashow (1974) are not covered, as well as the symmetry broken down from this group to the reductive group SU(3) x SU(2) x U(1), or the resulting proton decay. Such important absences, easily detected by the expert, are not immediate for the beginner. However, there is no doubt that this book is an excellent introduction to the specific problems of group theory applied to particle physics. In any case, in order to have a larger comprehension of the topic, the text must be completed with the reading of more modern or detailed monographs. Good complements to the book of Lipkin containing later developments and theories would be, for example, the work of Ne'eman [Symétries jauge et variétés de groupe, PUM, Montréal, 1979], the book of Georgi [Lie algebras in particle physics, Perseus Books, Reading, 1982] or the encyclopedic work of Cornwell [Group theory in physics, Academic Press, San Diego, 1984, volume 2].

This is the best book I have come across to explain Lie groups/algebras to someone who has at least taken an introductory quantum mechanics course and is comfortable with the angular momentum operators-more particularly the raising/creation and lowering/annihilation operators. This is not for someone who has watched NOVA and wants to learn more. If you have a good background in group theory and quantum mechanics, or even a very basic understanding of them and you want to learn more about the fascinating word of Lie Algebras/groups, this book is for you.
As a companion to this book I would highly recommend Lie Groups, Lie Algebras, and Some of Their Applications by Robert Gilmore. This book gives an excellent exposition of the subject, going into more detail than Lie Groups for Pedestrians, but not so much detail that you need to have taken a course on group theory in order to digest the information. It also has very informative exercises at the end of the chapter, which in my opinion is a plus.
If you're looking for a quick and dirty introduction to string theory and that ilk, this is not your ticket. If you are looking to expand your horizons of QM theory and second quantization-which leads to wonderful results and 'toys' in not only particle physics, but also in spectroscopy of atoms and molecules-this is your book.

This book is a nice, concise review of Lie algebras and groups as they are used in particle physics. The content is a bit dated, but the explanations of building the mathematical operators used in quantum mechanics are quite lucid. I think this would be an excellent companion to a more modern text on particle physics. This book is NOT Lie groups 101. Topics include the different flavors of spin, harmonic oscillators, multiplets of a few SU(n) groups, and tying everything back to Young diagrams.

The title is a bit misleading- if this is the "pedestrian" approach, I would hate to see what the author thinks is the more exotic route. The book jumps in right at basic angular momentum operators, so if it's been a while you might need to break out notes from a QM class. Important concepts are not defined, or are left as a vague mathematical mumbo-jumbo that doesn't at all hint at the underlying physical concept. Many times the author says "it clearly follows that..." or something similar, but I found it a nontrivial exercise to follow. Despite the fact that I had to go to other sources and push my pencil around a bit (probably my own shortcomings than anything in the book), I found this book easy enough to learn from.

If you are a theorist and your job is to model things, and you simply can't figure out why any of the current mathematical or software packages don't fit your data, then this is the book that you need to read. After recommending this book to 4 fellow physicists and engineers, they were able to apply these techniques to their research, and found it of great use. Looking back, I can say it was a great book to have around for EM, and QM, and my fellow colleagues made use of it when writing up their Science publication papers, which we all enjoy reading and citing.

I won't abound on what have already been said by one of the reviewers: "pedestrian" means "non mathematician, very well versed on Quantum Theory". But I liked to stress that this is not a book to learn Lie Group Theory from scratch. Far from that, I closed (and shelved) it at around page two.