The applications are to physics, mainly quantum theory. Edit: Forgot to comment on the last part of the questions. I think Wigner is a good read. You'll not learn much about general group theory, but you will learn about representation theory of the Poincare group and some general techniques from representation theory like the Mackey machine for induced representations.
Well, in my dictionary "group theory for physicists" reads as "representation theory for physicists" and in that regard Fulton and Harris is as good as they come. You'll learn all the group theory you need which is just a tiny fragment of all group theory along the way.
John Baez's "Gauge fields, knots and gravity" has a very illuminating chapter on lie groups and lie algebras, which is just at the right level of rigor for a physicist. His chapters on differential geometry are also pretty awesome. Morton Hamermesh's Group Theory and Its Application to Physical Problems is a Dover Press book, so quite inexpensive though the price seems to be up a bit since I bought it in the '90s.
I took a course on group theory in physics based on Cornwell and even though I followed all of the proofs, I had no idea how it might help me solve physical problems until I picked up Tinkham's Group Theory and Quantum Mechanics.
His approach and examples might be considered dated not much on Lie groups and a lot on crystallography but if you're just getting acquainted with the field, I think it's the best around. It gets the elementary ideas really cemented. I personally recommend Georgi's book with a particular focus on SU 3. And there is also Ramond's book , which is along the same lines as Georgi's textbook. Also online there are some notes available from Grossman , 't Hooft , and Slansky.
I see almost all the classical recommendations, all except one. Check out the contents table on the Amazon preview. Just filling in some gaps. Generations of practitioners have used these books, so they underlie what you read about in many of your textbooks. Classical Groups for Physicists , by Brian G. Wybourne Wiley. Is addressed to readers who habitually illustrate and attempt understand abstract mathematical notation a rare species. Once one learns how to use it, one may spend a lifetime doing just that.
Dynamical group treatment for solvable systems a veritable classic. Somewhat chaotic, but has lots of geometrical illustrations and examples, and tracks down nontrivial, non-hackneyed physics applications like few others.
Invaluable in appreciating Wigner-Inonu contractions beyond name-dropping. Easy to develop reliance on. A classic, yeomanly, solid, responsible Lie Group resource; heavily relied on by boomers. This actually means it is useful in illuminating their universally shared "you know"s.
Universally shared bare minimum background on SU 3 , again a "live in the background" boomer mainstay resource. If your teacher throws something on the eightfold way you are unsure about, this one is by far the most likely to resolve it. Explicit, albeit somewhat ponderous; but beware of the odd actual stereotypical misconception: do not use unthinkingly. Lie Algebras and Applications Springer by F Iachello, delightfully tabulates Lie algerbas and their standardized features.
Good, usable resource tables, in the Patera-McKay or the Slansky spirit. Cahn Benjamin Well logically organized, it provides proofs and arguments for the mathematically exigeant physicist, at just the right level: no hidebound pedantic drivel here. Parting notes: For informed grad student work, R Slansky's classic Physics Reports 79 sourcesbook review Group theory for Unified Model Building can hardly disappoint.
Michael Stone's Mathematics for Physics is a pearl—boy, would I have loved it, had it been available in my college years. Finally, a worker's book, not a student's, which I am only adding here because I'd be remiss if I did not point out how truly important and accessible it is for theoretical physicists. Truly, as they quote Hadamard,. Sternberg's book is excellent and illuminating but perhaps a bit hard for a beginner. The book deals with representation theory of Lie groups of matrices.
After reading this I also recommend the Sternberg's book for physical applications and the topological point of view of group theory. The books by J. Cornwell are well written and a mix of formalism and examples. There are several different editions but "Group Theory in Physics vols 1 and 2" are excellent choices containing well-chosen examples. I am surprised no one has mentioned Lipkin yet. His "Lie Groups for Pedestrians" uses notation that is not too out of date, since it was written in the early 60s.
He covers the use of group theory in nuclear physics, elementary particle physics, and in symmetry-breaking theories. From there, it is only a small jump to more modern theories.
Georgi's book mentioned above may be even better, but it is awfully pricey: as a Dover Press book, Lipkin's is quite cheap and easily available. It can even be downloaded as a PDF file from 4shared. Or bought as an e-book from Google.
Even the Preview on Google is not bad, being surprisingly close to complete. Lipkin does assume the readers knows quantum mechanics at about the sophomore physics major level, since the quantum-mechanical angular momentum operator is basic to his whole presentation; he also assumes familiarity with Dirac's bra and ket notation.
But I am sure that is not asking too much. And both books are too old to cover use of group theory with QCD or symmetry breaking. But both these books explain the philosophy of the use of groups in QM, which later authors seem to usually assume you already know. Heine also includes a lot more than most about the application of finite and 'point' crystallographic groups. But he does still seem to take a more mathematically abstrat approach than most physicists need: as Lipkin points out, the interests of a physicist and those of a mathematician in group theory really are different: as an example of the difference, Lipkin even mentions the rank of Lie algebras without ever defining it:.
There is a recent textbook which gives a fairly complete and concise presentation of group theory, covering both structure and representations of both finite and continuous Lie groups, with a brief discussion on applications to music finite groups and elementary particles Lie groups. The target level is advanced undergraduate and beginning graduate.
It is freely available at. The author has also co-published texts on contemporary particles and elementary particle theory, some parts of which discuss real life applications of group theory.
For those who only care about Lie groups and representations i. Systematically emphasizes the role of Lie groups, Lie algebras, and their unitary representation theory in the foundations of quantum mechanics. For erratas, reviews and other posts check out Peter Woit's Home Page. There is no good book aimed at physicists. Gelfand, Graev, and Vilenkin, Les Distributions, vol. Representations of finite groups are covered in Boerner, Representations of Groups: With Special Consideration for the Needs of Modern Physics an old classic written for physicists.
None of these books are good, but they are the best I can think of. Strichartz has written about harmonic analysis on the actual Lorentz group, perhaps it is worthwhile, perhaps I will look at it some day A famous mathematician once told me no one had ever understood Weyl, The Classical Groups. I think much of it is covered by Boerner.
Instead of following the books, I've been teaching group theory for physicists by following these papers below. The idea is to study the papers from top to bottom, and use a traditional books e. Tinkham, Hammermesh, Dresselhaus, Joshi to fill the gaps. These only cover point group and space group symmetries for solid state physics.
For the next semester, I may use also this paper:. But it would be nice to complement these with a paper that uses Lie algebras to solve a simple but interesting and illustrative problem undergrad level. Any suggestions? I'll add to the list these two:. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.
Create a free Team What is Teams? Learn more. Comprehensive book on group theory for physicists? Ask Question. Asked 10 years, 10 months ago. Active 6 months ago. Viewed 41k times. Improve this question. Because if you just want to use group theory in physics then in my experience you won't need anything besides representations. See this question of mine over at MO: mathoverflow. Add a comment. Active Oldest Votes.
To quote a review on Amazon albeit the only one : "This book is an excellent introduction to the use of group theory in physics, especially in crystallography, special relativity and particle physics.
Improve this answer. Although it is certainly rich, it is written in a way that only is internalize-able if you've already seen the material. Every section starts from very general and abstract grounds, making no reference whatsoever to the end goal, so every "final result" seems mysterious and confusing. A good introductory text, I think, sufficiently motivates each idea before it is presented, thus giving you the "big picture".
Georgi with B. Hall would be best. The former offers physical motivation, employs physics notations, covers a humongous range of topics relevant to actual physics, but is a little off-hand and sloppy at times. The latter offers rigorous proofs with very elegant and down-to-earth reasoning, still very readable unlike many other math textbooks.
Wu-Ki Tung, Group theory in physics Its approach isn't go from general to specific, but from intuition to generalization. In specific, the book: Boldly uses ' for mappings see def 2.
I've never seen this kind of notation before, and at first I think using this will make more confusion. But turns out it's not Important theorems are named , not just numbered Avoids study all groups in detail Has many advanced example without proof, because they are just illustrations, not a topic for you to study Proofs are deferred after discussing significance A trivial thing: theorems and definitions have different numbering systems.
I highly recommend this book, even though it's quite old 50 years or so. Zee, Group Theory in a Nutshell for Physicists The book is written in xkcd style: funny and lots of footnotes, with quotes and historic stories. Yet, I think you should take a look at the fruitful bits. They do give you new perspectives.
Jakob Schwichtenberg, Physics from Symmetry Its structure: It starts with special relativity, then the symmetry tools Lie group and Lagrange formalism , then the basic equations free and interaction theory , then their specific applications: quantum mechanics, quantum field theory, classical mechanics, electrodynamics and gravity. Still, this book is a gem and has a lot of results on the applications of group theory especially infinite-dimensional representations of noncompact Lie groups to Physics.
Lie groups for physicsts djvu. Despite the condescending-sounding title, it's far from a watered-down version of Lie theory. It is one of many books by Robert Hermann on Lie theory and physics.
Chapter 9 in this book contains the geometric theory of induced representations. Mackey developed the theory of induced representations in the infinite-dimensional setting required by quantum mechanics. I happen to have a copy of this book in my office. Harmonic analysis as the exploitation of symmetry -- a historical survey pdf.
A leisurely, surprisingly accessible survey about many topics in representation theory and physics from one of the masters. I can't seem to find this reference in the library or anywhere online.
I have not seen it since the late s, so perhaps it's superseded by his book see below on Quantum Field Theory. Quantum Theory of Fields, I: Foundations djvu. The second chapter has a good discussion in the usual Weinberg style of the group theoretical underpinnings of quantum field theory on Minkowski spacetime. Group Theory in Physics, Vol. II djvu.
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