Should I take a group theory course before QFT?

[samantha_allen]

I know that studying QFT requires understanding Lie Groups and infinitesimal generators as they correspond to symmetry transformations. I want to study or take a course (offered by my university) in QFT in the coming academic year and I have the option to take a abstract algebra course offered by our Mathematics department this semester. The abstract algebra course uses "Contemporary Abstract Algebra" by Gallian.

I am not familiar with group theory at all and I am not sure if this course is going to be useful. It does not seem to talk about Lie groups and doesn't have anything similar as far as I could tell. In addition, I have heard people claim that this book doesn't help much with lie groups and most of the group theory needed for QFT is available in the physics texts themselves.

Is it worth taking this course this semester, or can I take it sometime later when I have the time/ feel like it?

 

[PeroK]

I am not familiar with group theory at all and I am not sure if this course is going to be useful. It does not seem to talk about Lie groups and doesn't have anything similar as far as I could tell. In addition, I have heard people claim that this book doesn't help much with lie groups and most of the group theory needed for QFT is available in the physics texts themselves.

Clearly it can't hurt to learn a bit of group theory, but it's a huge subject; and, Lie Groups are a fairly advanced topic. In mathematics, everything is constructed from the bottom up. Whereas, a physicist has to pick and choose the parts of mathematics that are relevant to the physics being studied. In that sense, for QFT you are going to be just dipping into group theory. It's certainly worth understanding what is meant by the Lorentz and Poincare Groups.

Also, a personal opinion is that the mathematics of QFT is wild and woolly. Perhaps the last thing to do is get into the mindset of the rigorous, pristine world of pure mathematics!

I would suggest studying Noether's theorem in detail. She, as you probably know, was a Lie Group theorist. For QFT, I would focus on the main results and their applications to physics - rather than trying to understand the advanced pure mathematics of Lie Groups themselves.

Good luck!

 

[Orodruin]

Group theory is useful for many different aspects of physics as symmetries tend to be very useful. I would suggest anyone looking to do physics at a higher level to take at least an introductory course in group theory.

I devoted a chapter of my book to the basics, including Lie groups and representations.

 

[PeroK]

I devoted a chapter of my book to the basics, including Lie groups and representations.

That's what a physicist needs. By contrast, the book I have as an introduction to abstract algebra, takes six chapters to get to groups; the ninth chapter is the symmetric group $S_n$; chapter 10 is an introduction to rings; and that's 200 pages! Lie Groups are still a long, long way off.

 

[MathematicalPhysicist]

That's what a physicist needs. By contrast, the book I have as an introduction to abstract algebra, takes six chapters to get to groups; the ninth chapter is the symmetric group $S_n$; chapter 10 is an introduction to rings; and that's 200 pages! Lie Groups are still a long, long way off.

Which book is that?

 

[Orodruin]

That's what a physicist needs.

Well, that was the intention. ;)

 

[fresh_42]

I know that studying QFT requires understanding Lie Groups and infinitesimal generators as they correspond to symmetry transformations. I want to study or take a course (offered by my university) in QFT in the coming academic year and I have the option to take a abstract algebra course offered by our Mathematics department this semester. The abstract algebra course uses "Contemporary Abstract Algebra" by Gallian.

I am not familiar with group theory at all and I am not sure if this course is going to be useful. It does not seem to talk about Lie groups and doesn't have anything similar as far as I could tell. In addition, I have heard people claim that this book doesn't help much with lie groups and most of the group theory needed for QFT is available in the physics texts themselves.

Is it worth taking this course this semester, or can I take it sometime later when I have the time/ feel like it?

Please take the following statement with a grain of salt, will say that there might be exceptions to what I say, but they are not of interest here.

Abstract algebra mainly deals with finite groups. Lie groups are infinite and topological groups, many are algebraic groups. But despite the name, they normally do not occur in an introductory course of group theory.

What you will probably need to know from group theory are some technical terms: group, subgroup, normal subgroup, centralizer, center, normalizer, inner and outer automorphisms, orbits; and of course some rudimentary topology.

It would be far better to study explicitly linear algebraic groups, than studying group theory in general.

 

[George Jones]

I want to study or take a course (offered by my university) in QFT in the coming academic year

Do you know which text, if any, will be used for the QFT course?

 

[vela]

Is it worth taking this course this semester, or can I take it sometime later when I have the time/ feel like it?

I'd take it later. As others have said, Lie groups are an advanced topic. I don't recall seeing them when I took abstract algebra as an undergrad nor in the first part of the graduate course.

 

[mathwonk]

Following up on the post by fresh42, abstract algebra courses usually treat finite groups, and lie groups are the primary example of infinite groups. Moreover the basic examples of lie groups are "linear groups" of invertible matrices. Thus a (maybe second) course in linear algebra would be more useful as background for lie groups than the usual first course in abstract algebra.

Indeed the impression i got just now from scanning the table of contents of Gallian's book on amazon, (kindle edition), is that it contains not only nothing at all on linear, or lie groups, but it contains nothing at all on linear algebra. So it would seem to be the least useful abstract algebra book you could possibly choose. The older standard books on abstract algebra more familiar to me, such as Herstein and M. Artin, do cover important material on linear tranformations that would be applicable.

 

[dextercioby]

My advice to @samantha_allen is this: despite of my signature, do not enroll in a pure math class, if your purpose is to learn serious theoretical physics during the regular university years. Use books on particular subjects of mathematics which are specifically designed for students and future PhD students in physics. In this case, group theory in physics. I recommend a marvelous book by Wu Ki Tung from 1984 with a supplement by H. Georgi's for the Lie Algebra part of the Standard Model.

Georgi, H. - Lie Algebras in Particle Physics (AW, 1982).
Tung, Wu-Ki - Group Theory in Physics (WS, 1984, -+++ Minkowski metric for Lorentz/Poincaré).

The whole discussion on literature is here:
https://physics.stackexchange.com/questions/6108/comprehensive-book-on-group-theory-for-physicists

 

[mathwonk]

If you do consult a pure math book on intro to abstract algebra, i recommend the one by mike artin, Algebra. He emphasizes in his "note to the teacher" that he begins his book with matrix operations rather than permutations because "matrix groups are more important". Indeed the first 8 chapters or so of his book discuss mainly matrix groups. Only the chapter called "More group theory" deals with specifically finite groups, and their numerical formulas such as sylow's theorem.

A lie group is a differentiable manifold that is also a group, so in my opinion, as a novice, learning about manifolds for this subject is more important than learning group theory. A good pure mathematics book that introduces manifolds and then lie groups and algebras, is the one by Frank Warner.

Lie groups and lie algebras go hand in hand by the way, since a lie algebra is a vector space, hence a linear approximation to the lie group, precisely in the sense that the tangent space to a manifold approximates the manifold. Indeed the lie algebra of a lie group can be defined as the tangent space of that lie group at the identity, plus some bracket product structure. A common definition of the lie algebra is as the vector space of left - invariant vector fields on the lie group. But since the group structure allows you to uniquely translate any point to any other point, each tangent vector at the origin can be translated everywhere and yields a left invariant vector field. The realization as a vector field let's you define the bracket product as a commutator of the operation of a vector field as a differential operator on the smooth functions.

One simple example is the circle group of unit length complex numbers. The lie algebra is apparently the real line, which is a translate of the tangent line to the circle at the unit element. The fact that the exponential map sending t to e^(it), maps the real line onto the unit circle, has an importamnt generalization to all lie groups. Namely there is always an "exponential map" from the lie algebra to the lie group, at least locally. Take what I say with large grains of salt, as I have never studied this topic. One way to define this exponential map for matrix groups uses the fact that you can plug a matrix into a convergent power series and it will still converge, and the limit will be an invertible matrix.

A more general definition of the exponential map uses the existence theory for differential equations. As I recall from browsing in J. Frank Aadams' book, Lectures on lie groups it goes something like this. Namely starting from a vector at the identity, use the group translation structure to extend to a vector field near the identity. Then solve this "differential equation" to find a curve passing through the identity and having all velocity vectors agreeing with that vector field. Then run along that curve for time t=1, and stop there, and that point is the exponential map evaluated at that original vector.

 

[fresh_42]

Here is an overview of $SU(2)$:
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/#top

You will see that, as @mathwonk mentioned, it is far closer to differential geometry than it is to group theory.

 

[mathwonk]

The first three sections of chapter 8 of Artin, Algebra, also give a nice elementary treatment of SU(2) and its orthogonal representation, i.e. a map from SU(2) to SO(3). he assumes a minimum of background.

 

[sysprog]

The mapping of the SU(3) non-Abelian Homology Group to the Quarks paved the way to the finding of the Top Quark.

 

[sysprog]

I think that this remark (from @fresh_42) is well-deserving of (favorable) especial isolation:

It would be far better to study explicitly linear algebraic groups, than studying group theory in general.

I think that you're already well aware that you can't really do Physics without doing Mathematics; however, I also think that what @fresh_42 said here could be for you an especially good labor-saving guideline . . .

 

[mathwonk]

While I agree with this advice from fresh_42, some books on linear groups (for example the notes by O'Meara) do assume the reader knows the basic concepts of group theory, such as the subgroup generated by a subset, the center of a group, simplicity, and the commutator of a group. For this reason, a book like Mike Artin's algebra, which is both a beginner's introduction to groups, but also focuses on matrix groups, may be a good choice.

Not being a physicist, this next remark is perhaps wrong, but I would guess that you want a more differential geometric approach and not an algebraic geometric approach. Modern algebraic geometry is a very abstract approach to geometry that mimics the concepts from differential geometry in a purely abstract algebraic way. For example calculus is replaced by commutative algebra in a way that makes many familiar concepts seem completely foreign at first. For example, whereas in differential geometry the lie algebra of a lie group is defined as the tangent space to the manifold underlying the group at its identity element (plus an extra bracket structure), in algebraic group theory the lie algebra is defined as the space of left invariant derivations on the abstract group algebra associated to the group. For this reason, although you may want to study linear groups, I would avoid books with titles like "Linear algebraic groups", e.g. those by Humphreys or Borel, which will often be couched entirely in the language of abstract algebraic geometry.

 

[physicsworks]

My advice to @samantha_allen is this: despite of my signature, do not enroll in a pure math class, if your purpose is to learn serious theoretical physics during the regular university years. Use books on particular subjects of mathematics which are specifically designed for students and future PhD students in physics.

Great advice ineed. I would add to the suggested list A. Zee's book "Group theory in a nutshell for physicists".

 

[fresh_42]

A bit of a problem with our advice here is, that there are indeed some basic concepts that are necessary to know: direct and semi-direct products (important!), isomorphism theorems (important!) and the stabilizer-orbit theorem should have been heard of, too. I think they all can be learned relatively easy on their own, without digging through all cyclic groups, field extensions, or Galois groups. But they are as important as a basic understanding of (set) topology is.

 

[sysprog]

A bit of a problem with our advice here is, that there are indeed some basic concepts that are necessary to know: direct and semi-direct products (important!), isomorphism theorems (important!) and the stabilizer-orbit theorem should have been heard of, too. I think they all can be learned relatively easy on their own, without digging through all cyclic groups, field extensions, or Galois groups. But they are as important as a basic understanding of (set) topology is.

How about that man Galois? $-$ group theory ring theory field theory $-$ all fully elucidated in writing by a guy not old enough to legally be allowed to drink a beer in a Chicago tavern $-$ but he still got into a duel over a woman $-$ and the duel was with a serious sergeant-level military professional $-$ grounds for suspicion . . .

 

[samantha_allen]

I see. As many of you said that an introductory course in group theory is insufficient to understand lie groups and does not deal with the same type of groups, I think I can afford to postpone it to some other time. Instead, I might just read a little bit of group theory aimed towards physics students like @dextercioby suggested. Thank you for guys for the clarification.

 

[fresh_42]

How about that man Galois? $-$ group theory ring theory field theory $-$ all fully elucidated in writing by a guy not old enough to legally be allowed to drink a beer in a Chicago tavern $-$ but he still got into a duel over a woman $-$ and the duel was with a serious sergeant-level military professional $-$ grounds for suspicion . . .

I have a book about the history of mathematics of the 18th and 19th centuries written by Jean Dieudonné. He has included an annex with short biographies of dozens of mathematicians. You would be surprised how many of them died young, swimming or climbing! And, yes, there are other tragedies like Galois or Hausdorff.

I once saw Galois's original paper. I had learned Galois theory before, but that didn't help me at all to understand a word.

 

[Vanadium 50]

A contrarian view:

(1) Should I take X? Sure. It never hurts to learn.

(2) But I can't take everything. Right. So it's really a question of opportunity cost. So will you use this a lot? Or should your spend your time doing something else.

(3) I am a HEP experimenter, and use group theory approximately never. And I know it better than most: I can tell you in SU(3) two octets make a 27, a 10, a 10bar, a singlet and two octets. Every half decade I re-learn Young Tableaux. Is this useful, or will it just make you a better person?

 

[Bandersnatch]

Is this useful, or will it just make you a better person?

I thought the goal was to make one more fun at parties.

 

[Vanadium 50]

I thought the goal was to make one more fun at parties.

I couldn't possibly be more fun at parties.

 

[sysprog]

A contrarian view:

(1) Should I take X? Sure. It never hurts to learn.

(2) But I can't take everything. Right. So it's really a question of opportunity cost. So will you use this a lot? Or should your spend your time doing something else.

(3) I am a HEP experimenter, and use group theory approximately never. And I know it better than most: I can tell you in SU(3) two octets make a 27, a 10, a 10bar, a singlet and two octets. Every half decade I re-learn Young Tableaux. Is this useful, or will it just make you a better person?

I think that knowing how and why an understanding of the SU(3) non-commutative Group helped us to get to the finding of the Top Quark is not in itself a bad thing.

 

[Vanadium 50]

I wasss there at the top quark discovery. How exactly did SU(3) help with the discovery?

 

[sysprog]

I wasss there at the top quark discovery. How exactly did SU(3) help with the discovery?

Well, we (human beings) understood that the SU{3} non-commutative homology group included that member, and that the (non-Abelian -- Abel figured that stuff out) SU(3) group mapped nicely to the Quarks, so we started looking for it -- it took years for us to find it, but knowing about SU(3) helped us to know that it (the top quark) was somewhere to be found.

 

[Vanadium 50]

Well, first, I would argue that SU(2) is more important than SU(3) in this case. I'd also argue that the idea of a top quark had been around since the 1970's. Because I like to argue and would keep pointing out that neither CDF not D0 was designed around finding the top quark because at the time of writing the proposals it was felt that the top would be discovered long before they were operational.

So I disagree that SU(3) was needed by any of the experimenters.

 

[sysprog]

Well, first, I would argue that SU(2) is more important than SU(3) in this case. I'd also argue that the idea of a top quark had been around since the 1970's. Because I like to argue and would keep pointing out that neither CDF not D0 was designed around finding the top quark because at the time of writing the proposals it was felt that the top would be discovered long before they were operational.

So I disagree that SU(3) was needed by any of the experimenters.

I don't disagree with your appreciation of the super-unary group SU(2), and I don't disagree that its membership could have adequately engendered ideations regarding the top quark; however, I think that SU(3) was among groups the main precursor for QCD (quantum-chromo-dynamics -- I'm pretty sure that you already know that acronym, but I'm also pretty sure that not everyone does). I acquired most of the 'teensy' bit of knowledge that I have about the matter from this book (image):

 

[Vanadium 50]

Well, the idea of quarks was motivated by SU(3) flavor, but QCD is SU(3) color. In any event, experimenters really didn't need to know this to discover the top quark.

 

[fresh_42]

But you need to know global Lie groups if you want to enter the GUT race. Every physicist has his / her personal group!

 

[CrysPhys]

Every physicist has his / her personal group!

Physicists are enamored of personal groups because, when they were in high school, they were ostracized by the cool kids.

 

[mathwonk]

Just a know - nothing mathematician's comment related to whether SU(2) or SO(3) is more important. Actually they seem to contain the same information, so theoretically if you know one you also know the other! I.e. "we all know" (if I have this right) there is a surjective homomorphism SU(2)-->SO(3) with kernel {I,-I}, making SU(2) a 2 to 1 topological covering space of SO(3). At first blush then, SU(2) seems more informative since one can recover SO(3) from SU(2) by modding out the center {I,-I}, i.e. SO(3) ≈ SU(2)/{I,-I}. But one can also recover SU(2) as the "universal cover" of SO(3). I.e. since the fundamental group of SO(3) ≈ RP^3, is isomorphic to the group {I,-I}, one can construct SU(2) topologically as the universal (double) cover of SO(3), and then using unique path lifting it seems one can also recover the group operation on SU(2). I.e. choose one of the 2 elements lying over the identity of SO(3) as the identity of SU(2). Then given elements A,B of the double cover, draw any path going from the chosen identity to B, and map the path down to SO(3). Down there multiply that path by the image of the element A. Then lift that product path back up to the unique lifted path starting at A, and it should wind up at the product element AB. This construction is independent of the choice of the initial path from the chosen identity to B, since the universal cover is simply connected, i.e. any 2 such paths are homotopic, hence so are all the paths constructed from it.

oops: now i see the comparison was between SU(2) and SU(3), not SO(3). but i still like my construction of SU(2) from SO(3). It isn't too shocking that the 3 sphere has a group structure, since it is probably the unit quaternions, but I am surprized that P^3 has one. But it is of course obvious, since we just mod out the unit quaternions by {1,-1}. gosh, it seems hard to me to think of an object simultaneously in geometry and group theory. I am only used to abelian examples, like S^1xS^1x...xS^1. where I can see group translation as rotation.

 

[Omega0]

I would be puzzled if you take QFT before standard QM... and then perhaps relativistic QM (but for classical solid state physics for example relatvistic QM is not necessary).
I took QFT naturally after standard QM and long, long before group theory was already introduced - at least in an "all the things you need to know now" level.
Or are you speaking about some Feynman lecture part III level?! If so you will not need any group theory.