Find the Perfect Group Theory Book for Physicists

[Svend]

I have failed a course on group theory for physicists in my university, and i need a good book to learn group theory from because anthony zee's book is simply too hard to read. His book is verbose, glosses over many concepts, and is not very rigorous. Then the exercises in the book are very rigorous and hard, its very hard.

I like rigorous and terse books which goes directly to the point of the subject, instead of being chatty and i would really appreciate if anyone could recommend certain books on the topics.

The topics we need to cover on the course are:

• Groups, discrete and continuous
• Representation Theory
• Tensors, covering and manifold
• Groups SU(2), SU(3), SU(4), SU(N), SO(3) and SO(N)
• Lie Algebra
• Roots, weights and classification of Lie Algebras (Killing cartan, dynkin diagrams and etc...

Or the equivalent of part 1 to 6 of Anthony Zee's book.

Any help here is appreciated!

 

[vanhees71]

I'm not sure, whether this covers all subjects:

M. Hamermesh, Group theory and its applications to physical problems, Dover (1989)

is a classic. It's far far better than Zee!

 

[mathwonk]

I am not a physicist and do not know all those topics, but I am a mathematician, and have taught manifolds, covering spaces, and discrete groups for years. In my opinion one reason you are having a hard time is the ridiculously large amount of material you are trying to cram in. So you should not feel bad that it did not all take at once.

If I may suggest some reasonable sources for some of those topics, first of all I suggest the first 9 chapters of the book Algebra, by Michael Artin. It is an undergraduate introduction to algebra for MIT students, hence no nonsense and high level, but very well explained. Moreover he takes the attitude, appropriate for you, that infinite groups are more important than finite ones, so begins with the basic such examples, namely matrices.

Later he discusses the various special structures one can put on a space where matrices act, namely metrics defined by bilinear forms, and then, uniquely at this elementary level, he discusses an intoduction to representation theory, including a few important examples of matrix groups, like SU(2) and SO(3), I believe.

Another good book, more oriented to topology and manifolds, is Foundations of differentiable manifolds and Lie groups, by Frank Warner. Here you will find a treatment of tensors, missing from Artin's book, as well as manifolds and Lie algebras.

Another succinct and well written but challenging book is Lectures on Lie Groups, by J. Frank Adams. Here there are Stiefel diagrams, and exponential maps, assuming you know basic existence theorems for diffrential equations, but still no Dynkin diagrams. he gives a nice geometric version of the exponential map, a map from the lie algebra to the lie group, and it is nice to know also that this map, for matrix groups, is actually given by the marvellous exponential series! I.e. if M is any matrix, then e^M is an invertible matrix.

[digression: The only compact abelian Lie groups I believe are the tori, a quotient of euclidean space by a discrete lattice, e.g. an elliptic curve E = C/(Z+iZ), where C is the complex numbers, and Z is the integers. Here no doubt the exponential map is just the universal covering (quotient) map C-->E. No other compact real surface can have a continuous group structure I believe, by computing the Euler characteristic. Just as the circle has a group structure inherited as the length one complex numbers, so presumably does the 3-sphere inherit one from being viewed as length one quaternions. I thought then the Cayley numbers ("octonions") would give one on S^7, but they are not associative, so all this structure gives us is a non zero vector field, apparently a necessary but not sufficient condition for a group structure. So most spheres are not themselves topological groups, and yet they give rise to them by looking at their groups of rotations, the matrix groups SO(n).]

Edit: I found this linked discussion on stackexchange rather enlightening and fun (read the comments too).
https://math.stackexchange.com/ques...h-spheres-can-be-lie-groups?noredirect=1&lq=1

Mike Spivak's Differential Geometry vol. 1, is another comprehensive source for manifolds, tensors, and has a short introdution to Lie groups in chapter 9, which he obswrves uses all the previous material in the book, which is quite a long and detailed treatment of smooth manifold theory.

Many people also like books by William Fulton and Joe Harris. who have a book on representation theory which I have not seen.

None of these yet classify lie algebras, for which I do not know what to recommend, but I recall James Humphreys was well regarded for his books.

the basic idea is to understand an abstract group in terms of its action as a group of motions on some geometric space. To compare two groups you have to understand homomorphisms of groups, and continuous or differentiable maps for continuous groups. Finite groups are thus represented by mapping them into permutation groups S(n). Continuous groups are studied by mapping them into matrix groups, subgroups of GL(n), such as those that preserve some notion of length, like SO(n), or SU(n).

As always in calculus, differentiable maps are studied by approximating them by linear objects, and hence a manifold is studied using its tangent spaces. the basic fact here is that the tangent spaces of a smooth group are all essentially the same since translation in the group defines isomorphisms between any two of them, so it suffices to understand the tangent space at the origin. Now fundamentally, that tangent space has a structure of algebra, and the "Lie algebra" asociated to a lie group is nothing but the tangent space at the origin of that lie group, equipped with its lie bracket multiplication. Viewing tangent vectors as differential operators, this lie bracket structure can be defined using composition of differential operators, as a "commutator". (I mention as a warning that there apparently exist books on lie algebras which do not even mention this basic connection between lie algebras and lie groups.)

Finally, one tries to understand all lie groups by understanding all lie algebras, and learning to what extent a lie group is actually determined by its lie algebra.

This is a lot of stuff, good luck, and hang in there. It is rather beautiful.

 

[vanhees71]

True. For me as a physicist to learn about group theory and representations was a revelation. For me it's the only way to know, how to get the quantum (field) theory in a systematic way using the hints we have from classical mechanics (and field theory)!

 

[haushofer]

Brian Hall's book is also good, though a bit too math-oriented for me.

 

[weirdoguy]

It's far far better than Zee!

Which by itself is not that hard to achieve

 

[Svend]

I am not a physicist and do not know all those topics, but I am a mathematician, and have taught manifolds, covering spaces, and discrete groups for years. In my opinion one reason you are having a hard time is the ridiculously large amount of material you are trying to cram in. So you should not feel bad that it did not all take at once.

If I may suggest some reasonable sources for some of those topics, first of all I suggest the first 9 chapters of the book Algebra, by Michael Artin. It is an undergraduate introduction to algebra for MIT students, hence no nonsense and high level, but very well explained. Moreover he takes the attitude, appropriate for you, that infinite groups are more important than finite ones, so begins with the basic such examples, namely matrices.

Later he discusses the various special structures one can put on a space where matrices act, namely metrics defined by bilinear forms, and then, uniquely at this elementary level, he discusses an intoduction to representation theory, including a few important examples of matrix groups, like SU(2) and SO(3), I believe.

Another good book, more oriented to topology and manifolds, is Foundations of differentiable manifolds and Lie groups, by Frank Warner. Here you will find a treatment of tensors, missing from Artin's book, as well as manifolds and Lie algebras.

Another succinct and well written but challenging book is Lectures on Lie Groups, by J. Frank Adams. Here there are Stiefel diagrams, and exponential maps, assuming you know basic existence theorems for diffrential equations, but still no Dynkin diagrams. he gives a nice geometric version of the exponential map, a map from the lie algebra to the lie group, and it is nice to know also that this map, for matrix groups, is actually given by the marvellous exponential series! I.e. if M is any matrix, then e^M is an invertible matrix.

[digression: The only compact abelian Lie groups I believe are the tori, a quotient of euclidean space by a discrete lattice, e.g. an elliptic curve E = C/(Z+iZ), where C is the complex numbers, and Z is the integers. Here no doubt the exponential map is just the universal covering (quotient) map C-->E. No other compact real surface can have a continuous group structure I believe, by computing the Euler characteristic. Just as the circle has a group structure inherited as the length one complex numbers, so presumably does the 3-sphere inherit one from being viewed as length one quaternions. I thought then the Cayley numbers ("octonions") would give one on S^7, but they are not associative, so all this structure gives us is a non zero vector field, apparently a necessary but not sufficient condition for a group structure. So most spheres are not themselves topological groups, and yet they give rise to them by looking at their groups of rotations, the matrix groups SO(n).]

Edit: I found this linked discussion on stackexchange rather enlightening and fun (read the comments too).
https://math.stackexchange.com/ques...h-spheres-can-be-lie-groups?noredirect=1&lq=1

Mike Spivak's Differential Geometry vol. 1, is another comprehensive source for manifolds, tensors, and has a short introdution to Lie groups in chapter 9, which he obswrves uses all the previous material in the book, which is quite a long and detailed treatment of smooth manifold theory.

Many people also like books by William Fulton and Joe Harris. who have a book on representation theory which I have not seen.

None of these yet classify lie algebras, for which I do not know what to recommend, but I recall James Humphreys was well regarded for his books.

the basic idea is to understand an abstract group in terms of its action as a group of motions on some geometric space. To compare two groups you have to understand homomorphisms of groups, and continuous or differentiable maps for continuous groups. Finite groups are thus represented by mapping them into permutation groups S(n). Continuous groups are studied by mapping them into matrix groups, subgroups of GL(n), such as those that preserve some notion of length, like SO(n), or SU(n).

As always in calculus, differentiable maps are studied by approximating them by linear objects, and hence a manifold is studied using its tangent spaces. the basic fact here is that the tangent spaces of a smooth group are all essentially the same since translation in the group defines isomorphisms between any two of them, so it suffices to understand the tangent space at the origin. Now fundamentally, that tangent space has a structure of algebra, and the "Lie algebra" asociated to a lie group is nothing but the tangent space at the origin of that lie group, equipped with its lie bracket multiplication. Viewing tangent vectors as differential operators, this lie bracket structure can be defined using composition of differential operators, as a "commutator". (I mention as a warning that there apparently exist books on lie algebras which do not even mention this basic connection between lie algebras and lie groups.)

Finally, one tries to understand all lie groups by understanding all lie algebras, and learning to what extent a lie group is actually determined by its lie algebra.

This is a lot of stuff, good luck, and hang in there. It is rather beautiful.

Wow, thank you for having taken the time for this reply! You have outlined a pathway for me to take to learn group theory in a deep and meaningful way, will definitely keep these in mind as i go self studying on my own. But for the context of passing the exam i don't think i will have the time to read all of these books thoroughly to learn the material in time. Would you suggest buying each of these books and going through the relevant chapters?.

I know that mathematicians and physicists think differently in regards to math. But i am convinced that a deep understanding of mathematics is neccesary in order to thrive on theoretical physics, (which in the end is just a lot of applied pure maths). So someday i will have to learn all of these math subjects from the axiomatic approach.

 

[Svend]

I'm not sure, whether this covers all subjects:

M. Hamermesh, Group theory and its applications to physical problems, Dover (1989)

is a classic. It's far far better than Zee!

Fynny, that's actually the only other book the course lecturer could refer to, so i guess i will have to get this book then! However reading the contents page, i think it doesn't cover all i need, but the first 5 chapters definitely covers the beginning very well, which is what i found hard on zee's book. So i will order it and at least read the first 3 chapters!

The stuff i think is missing, is about tensors, and the SU(2), SU(3), SU(4), SU(N), SO(3) and SO(N), i think, but i am not sure. Do you know if the book covers this too?

 

[vanhees71]

Yes, he does. He starts with finite groups but also covers the Lie groups important for physics. What I find a bit too short is the coverage of the pseudo-orthogonal groups (including the Lorentz group). For this I recommend

R. U. Sexl and H. K. Urbantke, Relativity, Groups, Particles,
Springer, Wien (2001).

Of course also in Weinberg's QFT books you find a lot about this topic (vol. 1 on the Poincare group, vol. 2 on semisimple compact Lie groups as needed for gauge theory).

 

[dextercioby]

Try Wu Ki Tung'1984 book on group theory for physicists published by World Scientific.