Reddit mentions: The best group theory books

Interesting question! I'm not a physicist, but I can offer some insights into language, math, and how to learn them.

MATH AS A LANGUAGE

The idea that math is a language has some merits. For our purposes, we might define a language as an ordered triple (M,S,G). No, not monosodium glutamate! Here, these represent

• M = set of meanings or ideas;
  
• S = set of symbols and words to represent those meanings; and
  
• G = set of grammatical rules for combining words.
  
  Like natural languages such as English, math
  
  
• has its own set of meanings, its own symbols and words, and its own grammar for combining those symbols;
  
• allows people to communicate about complex abstractions in a standardized format by defining them in terms of simpler abstractions;
  
• evolves and grows in a fairly decentralized way; and
  
• facilitates the sharing of ideas between a vast array of human endeavors.
  
  LANGUAGE OF MATH VS. NATURAL LANGUAGES
  
  Languages such as English and Spanish have different symbols and words, as well as different grammars. However, the meanings and ideas in English and Spanish are largely the same. For example, as humans, we all have a need for water, so English and Spanish both have a word that represents the idea of water.
  
  As a result, if you already know one natural language such as English, then when you learn Spanish, you're mostly learning a new way to describe ideas you've already had.
  
  On the other hand, to learn math, you need to learn not only a new set of symbols and a new grammar, but also an entirely new set of ideas. Whereas we all need water, we don't all have a need to discuss topology.
  
  EXAMPLE
  The term "compact set" isn't a different word for a small collection of objects (something with which you might already be familiar). Instead, it's a term that represents a novel and fundamental idea from the field of topology. Specifically, a compact set is a set for which every open cover has a finite subcover.
  
  That idea builds upon several other ideas (the notion of a topology, of an open set, of an open cover, etc.). The only way to fully understand that idea is to start by learning the more basic ideas of which it is composed, then consider a range of examples and non-examples, then identify the properties common to those examples, then establish theorems relating those properties to other properties, and so on.
  
  Keep in mind that all of this depends on the setting, as well. Let's take the underlying notion of "open set." Depending on the book, this may be defined as
  
  
• a certain kind of subset of the set of real numbers;
  
• a certain kind of subset of R^n ;
  
• a certain kind of subset of a metric space; or
  
• a certain kind of subset of a topological space.
  
  The definitions are all consistent with each other, but they may appear quite different, since they're each tailored for a particular level of generality.
  
  POSSIBLE APPROACH
  The approach advised by /u/docmedic is probably the best: skim through a book or a set of course materials on the subject you want to learn, find out the prerequisites for that subject, find out the prerequisites for the prerequisites, and so on until you find materials that don't assume more than you currently know. Then work your way back up.
  
  That said, you don't necessarily need to learn everything in the prerequisite subjects in order to progress to the next set of topics; you could talk to professors and students to help you figure out which parts are needed and which aren't.
  
  Also, you may not need one hundred percent mastery of the topics you do learn. Math is best learned cyclically, I think. Try to gain as deep an understanding as you feasibly can, move on to a more advanced topic, and eventually go back to the prerequisite topic as you encounter it in more advanced contexts to see what new insights you've picked up.
  
  Lastly, don't be discouraged if a particular book is too dense. One of the comments mentioned Principles of Mathematical Analysis by Rudin as a place to start for analysis. That's an excellent book, but it works within a fairly abstract setting, and it's famously concise.
  
  Starting right away with a more abstract setting is fine if you look at lots of examples and can make sense of the abstraction, but it's probably a good idea to get several books nonetheless. I'd get some that provide lots of motivation, in order to improve your understanding, and others that are concise, so that you can better determine what's important.
  
  SOME "MATH FOR PHYSICISTS" BOOKS
  
  
• Mathematics of Classical and Quantum Physics
  
• Mathematics for Physicists
  
• Group theory in a nutshell for physicists
  
  This is just a sample.
  
  CONCLUSION
  There is not likely to be a "fast-track" to learning the language of math, at least not in the sense that you seem to be hoping for, but there might be ways to make this endeavor more manageable.
  
  If you want any help getting up to speed on advanced undergraduate math (it sounds like that's about your current level), let me know.
  
  Good luck!
  Greg at Higher Math Help

Hello! I'm interested in trying to cultivate a better understanding/interest/mastery of mathematics for myself. For some context:

 




To be frank, Math has always been my least favorite subject. I do love learning, and my primary interests are Animation, Literature, History, Philosophy, Politics, Ecology & Biology. (I'm a Digital Media Major with an Evolutionary Biology minor) Throughout highschool I started off in the "honors" section with Algebra I, Geometry, and Algebra II. (Although, it was a small school, most of the really "excelling" students either doubled up with Geometry early on or qualified to skip Algebra I, meaning that most of the students I was around - as per Honors English, Bio, etc - were taking Math courses a grade ahead of me, taking Algebra II while I took Geometry, Pre-Calc while I took Algebra II, and AP/BC Calc/Calc I while I took Pre-Calc)

By my senior year though, I took a level down, and took Pre-Calculus in the "advanced" level. Not the lowest, that would be "College Prep," (man, Honors, Advanced, and College Prep - those are some really condescending names lol - of course in Junior & Senior year the APs open up, so all the kids who were in Honors went on to APs, and Honors became a bit lower in standard from that point on) but since I had never been doing great in Math I decided to take it a bit easier as I focused on other things.

So my point is, throughout High School I never really grappled with Math outside of necessity for completing courses, I never did all that well (I mean, grade-wise I was fine, Cs, Bs and occasional As) and pretty much forgot much of it after I needed to.

Currently I'm a sophmore in University. For my first year I kinda skirted around taking Math, since I had never done that well & hadn't enjoyed it much, so I wound up taking Statistics second semester of freshman year. I did okay, I got a C+ which is one of my worse grades, but considering my skills in the subject was acceptable. My professor was well-meaning and helpful outside of classes, but she had a very thick accent & I was very distracted for much of that semester.

Now this semester I'm taking Applied Finite Mathematics, and am doing alright. Much of the content so far has been a retread, but that's fine for me since I forgot most of the stuff & the presentation is far better this time, it's sinking in quite a bit easier. So far we've been going over the basics of Set Theory, Probability, Permutations, and some other stuff - kinda slowly tbh.

 




Well that was quite a bit of a preamble, tl;dr I was never all that good at or interested in math. However, I want to foster a healthier engagement with mathematics and so far have found entrance points of interest in discussions on the history and philosophy of mathematics. I think I could come to a better understanding and maybe even appreciation for math if I studied it on my own in some fashion.

So I've been looking into it, and I see that Dover publishes quite a range of affordable, slightly old math textbooks. Now, considering my background, (I am probably quite rusty but somewhat secure in Elementary Algebra, and to be honest I would not trust anything I could vaguely remember from 2 years ago in "Advanced" Pre-Calculus) what would be a good book to try and read/practice with/work through to make math 1) more approachable to me, 2) get a better and more rewarding understanding by attacking the stuff on my own, and/or 3) broaden my knowledge and ability in various math subjects?

Here are some interesting ones I've found via cursory search, I've so far just been looking at Dover's selections but feel free to recommend other stuff, just keep in mind I'd have to keep a rather small budget, especially since this is really on the side (considering my course of study, I really won't have to take any more math courses):
Prelude to Mathematics
A Book of Set Theory - More relevant to my current course & have heard good things about it
Linear Algebra
Number Theory
A Book of Abstract Algebra
Basic Algebra I
Calculus: An Intuitive and Physical Approach
Probability Theory: A Concise Course
A Course on Group Theory
Elementary Functional Analysis

> It's fair to say that isn't what people are talking about here, but I guess that is what my original question was meant to refer to. I've read that when adding SSB to a theory with massless Goldstone bosons, that certain properties of the massless bosons carry over into the massive bosons (for example the "longitudinal polarization" of the massive boson being given by the massless one, though I don't know any of the details of how).

> If that's the right way of characterizing it, then on the surface it seems that there is a conversion of some kind going on (i.e. massless to massive) rather than being two otherwise completely unrelated types of particle. I guess that's what I'm trying to understand; is it a conversion, and if so, in what ways are they related or not related.

It's actually the locality of gauge invariance that causes the "conversion" most directly. What I mean by that is that if you have a quantum field theory with the right kind of potential, if you impose global gauge invariance, then you wind up with, say, one massive boson and one massless Goldstone boson. But if you impose local gauge invariance on the same theory, you instead wind up with two massive bosons. One of them will be the same massive boson from the globally invariant case, in the sense that it's the same field involved. But the other one comes from the gauge field that enters into the covariant derivative. The term in the Lagrangian that represented the Goldstone boson in the globally invariant case gets canceled out by the term resulting from the gauge transformation of the gauge field. (This is kind of hard to explain without math, so I'd suggest you take a look at chapter 14 of Halzen and Martin, in particular sections 14.7-14.8.)

Anyway, the main point is that in one theory, you have 5 degrees of freedom from particle polarizations and one more from a gauge transformation, and in the other theory you have 6 degrees of freedom from polarizations alone. If you want to call that a conversion, then you can. But personally I prefer not to, since to many people "conversion" implies some sort of physical process that you go through to turn one thing into another, and of course that's not what's happening here. (Basically I want to avoid prompting people to ask "why don't we convert photons into massive particles to slow them down?")

> Yeah, that's called a "quasiparticle" or perhaps a "collective excitation" right? I hesitated to ask about them because I understand these are not really fundamental but are more like an approximation or a different way of describing the system -- like describing conduction in terms of electron holes rather than electrons, or coordinates in polar form rather than linear.

I'm not really sure, actually - I don't know that much about the theory of superconductivity, at least not in enough detail to figure out what happens to photons inside a superconductor.

> One thing I may be confused on, and it might just be bad wording so I'm going to ask. You said there are "really four symmetries -- or more precisely, four generators" of SU(2)xU(1). Is it right to say that there is a symmetry for each generator? And what is the difference then between a symmetry and a generator? I was under the impression that SU(2)xU(1) was itself a single symmetry/group, and the generators were related to the charges rather than the quanta. Is that wrong?

Yeah, that was bad wording. Forget the thing I said about four symmetries; that was a relic from something I'd started typing up before it occurred to me that I could probably just talk about generators without confusing you too much. All I meant was that the symmetry has four degrees of freedom, in a sense; they correspond to the four generators of the SU(2)xU(1) group.

Now, you could also construct a theory that had a U(1)xU(1)xU(1)xU(1) symmetry, and that would also have four generators and thus four degrees of freedom to the symmetry. But the commutation relations of the generators would be different. In matrix notation (in the fundamental representation), the generators of SU(2)xU(1) are the three Pauli matrices and the number 1 (or the 2x2 identity matrix, if you prefer), but the generators of U(1)^4 are just the number 1, four times. Generators play a role in group theory similar (in certain respects) to the role that basis vectors play in linear algebra, so having different commutation relations among the generators is kind of like having two spaces with the same number of dimensions but with different metrics.

As for learning more about group theory, I'm no expert on it myself, but I've heard good things about this book by Wu-Ki Tung.

A couple good ones to get started are:

• A Classical Introduction to Modern Number Theory - Ireland & Rosen
  
  
• Primes of the form x^(2)+ny^(2) - David Cox
  
  These will introduce a lot of the main ideas in a way that you can understand with only the basics of abstract algebra. Having background in Galois Theory helps (you can use Dummit & Foote for a solid intro). Cox's book is probably one of the best reads in the subject. Then there are a few that introduce the basic formalism rigorously:
  
  
• First two/three chapters of Algebraic Number Theory - Neukirch
  
  
• Milne's Online Notes (PDF)
  
  
• A Course in Arithmetic - Serre (PDF)
  
  These cover the basics and can be found from many sources via amazon and google. After this, you can look at some more specialized stuff, though the two main directions are Analytic Number Theory and Langlands Program:
  
  
• Analytic Number Theory: Introduction to Analytic Number Theory - Apostal
  
  
  
• Local Fields: Local Fields - Serre
  
  
• Cyclotomic Fields: Introduction to Cyclotomic Fields - Washington
  
  
• Class Field Theory: Algebraic Number Theory - Cassels & Frolich, the final chapters of Neukirch, Artin L-Functions: A Historical Approach - Snyder (PDF)
  
  
• Elliptic Curves: Rational Points on Elliptic Curves - Silverman (undergrad level), The Arithmetic of Elliptic Curves - Silverman
  
  
• Modular Forms: A First Course in Modular Forms - Diamond & Shurman, Introduction to Elliptic Curves and Modular Forms - Koblitz, Modular Functions and Modular Forms - Milne (PDF), Automorphic Forms and Representations - Bump
  
  
• Langlands Program: (The theory that ties absolutely everything together) An Introduction to Langlands Program - Various, An Elementary Introduction to the Langlands Program - Gelbart (PDF)
  
  
• Prerequisites for the Langlands Program - Knapp (PDF) - This has a basic overview of most everything encountered in algebraic number theory along with many more references. Check it out!
  
  This may be looking far into the future, for someone just getting into Algebraic Number Theory, but I figure it's good to have a reference to point people to. I have my own biases and preferences, if anyone else wants to add something, go for it!

Agnostic here. I'm afraid it is not so easy to rule out the presence of brilliance and religion in a scientist or mathematician. Here is a list of living scientists who are christians (it is only a part of a much larger list going back several centuries).

Here are some examples with whose work I am more or less directly familiar.

John Polkinghorne was a student of Paul Dirac, and he has written a couple of books that are very lucid introductions to Quantum Mechanics.

Christopher Isham has written books on

• Quantum Theory,
  
• Modern Differential Geometry for Physicists and
  
• Lectures On Groups And Vector Spaces For Physicists.
  
  I have only actually read the first one on Quantum Theory.
  
  Edward Nelson is a mathematical physicist who has also contributed to quantum theory.
  
  Don Knuth is a mathematician and computer scientist who created the computer typesetting system, TEX, as well as a number of important theoretical developments, (e.g. the Robinson–Schensted–Knuth correspondence, and the Knuth–Bendix completion algorithm)
  
  Maddening, ain't it? But that's just how it is.

https://en.wikipedia.org/wiki/Schur_orthogonality_relations

The concept is that if a representation is reducible, the entire representation can be written as a direct sum of at least two irreducible representations. That particular formula is a consequence of writing the group representation as an h-dimensional vector space where the vectors are the orthogonal, irreducible representations and the vector components are matrix elements.

So in more plain english, that formula comes from orthogonality.

I would also add that this is something you really shouldn't worry about if your goal is any flavor of chemistry. Chemical Applications of Group Theory by Cotton is a book you should have if you're the type of person to ask these kind of questions. It goes way more into the actual math and machinery than what you need, so if something isn't explained in Cotton, it's fair to say that spending time trying to understand it will be a waste of time.

Also, that meaning for n is incorrect. It's usually true, but it's not a given that all vibrations will be either IR or Raman active. Especially when there's high symmetry. Not to mention that this formula is just the reduction formula and only predicts vibrations if you remove the proper 6/5 representations while using the proper coordinates.

Zee just came out with a group theory for physics book this month

http://www.amazon.com/Group-Theory-Nutshell-Physicists-Zee/dp/0691162697/ref=sr_1_1?ie=UTF8&qid=1458956358&sr=8-1&keywords=zee+group+theory

I can't promise that its what you're looking for or that its a stellar book (since it came out this month), but from how well received his GR and QFT book is, I wouldn't expect anything so much less.

This might seem like a strange recommendation at first, but bear with me. Atkins and Freidman, Molecular Quantum Mechanics has a fantastic section on symmetries and group theory. Since chemists deal with molecular shapes, they have really got that application of group theory down. It's a little removed from gauge symmetries, but the book really teaches representation theory and symmetry adapted bases. I wouldn't buy the book. But it's worth an afternoon in the library.

One with fewer pictures, but not as hands-off as Georgi is Hamermesh Group Theory ... . It has the advantage of being dirt cheap. It was the standard about 50 years ago, before Weinberg and Salam and the explosion of group theory applications in HEP.

Group Theory has enormous applications to physics. Check this book out.

J.F. Cornwell, Group theory in physics: an introduction (link)

W. Ludwig, Symmetries in physics: group theory applied to physical problems(link)

M. Tinkham, Group theory and quantum mechanics (link)

W.-K. Tung, Group theory in physics (link)

E.P. Wigner, Group theory and its applications to the quantum mechanics of atomic spectra (link1, link2)

N. Jeevanjee, An Introduction to Tensors and Group Theory for Physicists (link)

G. Costa, Symmetries and Group Theory in Particle Physics: An Introduction to Space-Time and Internal Symmetries (link)

B. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction (link)

R. McWeeny, Symmetry: An Introduction to Group Theory and Its Applications (Dover Books on Physics)(link)

Group Theory in a Nutshell for Physicists by Anthony Zee.

Some sources that come to my mind.

Problems From The Book

Mathematical Reflections problems

Putnam problems

IMC problems

Problems in Group Theory