Lie Groups for Pedestrians (Dover Books on Physics) [Paperback]

Harry J. Lipkin (Author), Physics (Author)

Book Description

Publication Date: July 15, 2002 | ISBN-10: 0486421856 | ISBN-13: 978-0486421858

This book shows how the well-known methods of angular momentum algebra can be extended to treat other Lie groups. Chapters cover isospin; the three-dimensional harmonic oscillator; algebras of operators that change the number of particles; permutations, bookkeeping, and Young diagrams; and more. 1966 edition.

Product Details

• Paperback: 192 pages
• Publisher: Dover Publications (July 15, 2002)
• Language: English
• ISBN-10: 0486421856
• ISBN-13: 978-0486421858
• Product Dimensions: 8.6 x 5.3 x 0.4 inches
• Shipping Weight: 6.4 ounces (View shipping rates and policies)
• Average Customer Review: 4.0 out of 5 stars  See all reviews (7 customer reviews)
• Amazon Best Sellers Rank: #595,838 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

52 of 52 people found the following review helpful:

5.0 out of 5 stars I-spin, U-spin, V all spin for I-spin, November 13, 2004

By 

Jill Malter ([e-mail]) - See all my reviews

This review is from: Lie Groups for Pedestrians (Dover Books on Physics) (Paperback)

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.

34 of 34 people found the following review helpful:

3.0 out of 5 stars I think I need "Lie Groups for Toddlers", May 9, 2008

By 

Buddha Buck (Ithaca, NY USA) - See all my reviews
(REAL NAME)   

This review is from: Lie Groups for Pedestrians (Dover Books on Physics) (Paperback)

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.

34 of 34 people found the following review helpful:

5.0 out of 5 stars Top ten classical but nowadays incomplete review of group theory in Physics, March 20, 2007

By 

Rutwig Campoamor Stursberg (France) - See all my reviews
(REAL NAME)   

This review is from: Lie Groups for Pedestrians (Dover Books on Physics) (Paperback)

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].