Courses 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.

 

[mpresic3]

As an undergraduate, I took an abstract algebra course out of Herstein, Topics in Algebra. This book covers similar material to Gallian's book. Two years later, I took a Quantum Field Theory course. The abstract algebra course did not help me in any way. I learned a bit about Lie groups in QM a semester earlier.

For example, one problem in Herstein, was show any group of order 9 is abelian. You never have to solve problems of this type in QFT. Algebra is interesting and I do not want to discourage learning it, even for a physicist. The problem solving process might be valuable to a physicist, but other courses could stimulate the neurons just as well. I would say take if if you want,but do not expect it to prepare you more.

On the other hand, I took a course in functional analysis from the math department and I used the concepts they introduced me to to understand the formalism in graduate QM I and II.

 

[mathwonk]

you remind me that i happened to mention that herstein covered some linear algebra. although i did not actually recommend it, I regret even mentioning herstein, as in my experience it is one of the least useful and least insightful of math books. it is oriented only to pure algebraists, and i do not recommend it to anyone else. It is a kind of fun problem oriented book for young students but does not convey understanding of much of anything, to me at least. I do recommend Artin as a good algebra book for most people. Maybe the title should have been a giveaway, "Topics", not "Concepts", nor "Insights".

 

[Omega0]

I know that studying QFT requires understanding Lie Groups and infinitesimal generators as they correspond to symmetry transformations

First, my apologies for my answer before. Reading my own answer it sounds not informative nor motivating.
It is just that I was puzzled for a simple reason: I was introduced in the basic terms already in classical mechanics. SO(3). Basically you can compare the magic with solving a linear differential equation. You will know that you formally solve it via having a matrix in the exponent of the e-function. For the linear pendulum in 2 dimensions this is a nice one. You just write down the equations, the matrix and you can solve the system via the exponential of the matrix. This is where I had my first contact to a generator. (I am not a prof in teaching theoretical physics, I just know what I believe to know - let me and Samantha know if I am writing nonsense).

The point is that from my experience you can take mostly everything into the exponent of an e-function. For example a differential operator. *BAM* here we are, now you write down things like an commutator into the exponent. Hey and here we are, classical mechanics, Poisson brackets and the e-function again.

Having said this, the main thing for me to learn was to understand the "representation" of a group. Looking backward it is so simple and I was really bad in understanding it. Now I got it, I think - for my personal usage ;-) You just have a thing which is called " a group" - this means that you have a set of elements and some mathematics between those, the structure. The mathematics is pretty limited. The nice thing: You can express the same mathematical structure by some standard math, in this case linear algebra. It is just matrices and stuff, nothing you haven't seen before.

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.

You don't have to be familiar with group theory but if you don't have a good level in quantum mechanics then you are lost.

people claim that this book doesn't help much with lie groups and most of the group theory needed for QFT

Sorry, if I am wrong and if my comment distracts you but if you want to "understand" QFT then you need to have "understood" Quantum Theory to some extend. (Understood in the Feynman way - you will never ever undertand it)
Group theory won't be something you will have nightmares about and you need to know in detail. The nightmares may come later ;-)

 

[mpresic3]

By now, it is clear that if the poster wants to take abstract algebra,it is good, but you certainly do not need it before taking QFT. I doubt if the textbooks Istakson/Zuber or Mandl or Zee etc mention in their preface that the students need to be so equipped. Can take QFT now and take algebra later if interested. AbstractAlgebra courses in undergrad and grad seem to lead to showing impossibility of a quintic in terms of radicals, impossibility of trisecting a general angle with straightedge and compass, finite groups, and some rings, a bit linear algebra, and so on. These math courses are part of a mathematician's education, and are not geared to be part of a physicists toolbox. I found functional analysis in the math department to be the exception.
Although I could have done without this one too, I thnk it did help me in the formalism of graduate QM

 

[Omega0]

I doubt if the textbooks Istakson/Zuber or Mandl or Zee etc mention in their preface that the students need to be so equipped.

Where you mention Zee: "Einstein Gravity in a Nutshell" (a nutshell yeah sure :-D ), chapter 1.3, page 39 out of around 860, "Rotation: Invariance and Infinitesimal Transformations", foot note: "If you don't know rotations in the plane extremely well, then perhaps you are not ready for this book. A nodding familiarity with matrices and linear algebra is among the prerequisites."

 

[mpresic3]

Sure. I can only speak from my own experience, but my abstract algebra course did not address rotation, invariance and infinite transformations in any lecture. (My quantum courses did discuss these in depth) The abstract algebra course did address sylow theorems, principle ideal domains, ideals, finite groups, fields, and the like.
Maybe, a treatment out of Artin differs from the one I had out of Herstein. I believe Artin may address rotation and maybe infinite transformations etc. Many times the course catalog or syllabus describes the content of the course under consideration. This might be a place to search for better answers.
Clearly matrices and linear algebra should be among the prerequisites for quantum mechanics as well as quantum field theory. You can get a nodding familiarity (Zee's words) without taking abstract algebra. Moreover, taking abstract algebra may not even touch upon these topics of matrices and (elementary) linear algebra . I know from the first week we used matrices as examples of groups, and moved way beyond that. It was assumed we knew these subjects before taking an upper undergraduate course in mathematics.

 

[Omega0]

Sure. I can only speak from my own experience, but my abstract algebra course did not address rotation, invariance and infinite transformations in any lecture. (My quantum courses did discuss these in depth)

What puzzles me is that you hadn't an introduction about this subject in classical mechanics, the first theoretical course studying physics. Hopefully it is clear that I don't mean this snotty in any way - perhaps I had just luck with my prof who was really brilliant. It doesn't even need to go into the geometric aspects of classical mechanics (buzzwords: symplectic structure, symplectic groups, ...), it is just enough to get into infinitesimal canonical transformations, I would say. I thought this is standard in an undergraduate course.

Moreover, taking abstract algebra may not even touch upon these topics of matrices and (elementary) linear algebra . I know from the first week we used matrices as examples of groups, and moved way beyond that.

I never had such a class. Well, I am absolutely not sort of a luminary in QFT but I know that (for me) the group theoretical part in QFT is mostly harmless. So, reading your impression, I absolutely wouldn't recommend the course. Nothing is better than speaking about things having own experience in this field.

 

[mpresic3]

Yes I did see symplectic groups in classical mechanics when I read the chapters 8, 9 and 10 in Goldstein. But these were in second semester graduate mechanics. I took QM I in the first semester of Grad school concurrent with classical mechanics I. My undergraduate course out of Marion, (Thornton was added later), did not mention symplectic matrices, symplectic structure, groups etc.

My graduate professor in theoretical mechanics was extremely good, and he did address all this. He also taught two semesters in classical mechanics and added material in fluid dynamics. (These days most graduate students get one semester of classical mechanics). Wheh he taught us, he had been teaching the subject for over 30 years.

There is currently no such thing as the first course you study in graduate school. You generally end up taking classical mechanics, electromagnetics, and quantum mechanics , and sometimes mathematical physics concurrently.

But all this is beside the point. The question remains is any of this to be taught from the math professor in the abstract algebra course. Best idea is to talk to the Math professor to see if he or she will address the sections that would profit a field theorist.

Nothing wrong with taking algebra, if you like it. I liked it, but it did not help in the field theory course. It was of more value in the elementary particle course. It is much more valuable when I took a course in group theory in quantum mechanics, but this was not field theory, and even this course could have been completed successfully without it. I understand it is really valuable in a many chemistry courses, but I do not think these courses have abstract algebra as a prerequisite.

 

[bhobba]

Well, as a guy trained in math/computer science but who turned more and more to theoretical/mathematical physics over the years, here is my advice. QFT is hard. You do not know how hard until you do it. Before doing any course on QFT, I heartily recommend getting at least a glimpse of QFT. After doing that, you can decide which way to go. The book I recommend is Student Friendly Quantum Field Theory:
https://www.amazon.com/gp/product/0984513957/?tag=pfamazon01-20

Please give it a read, then decide what to do. I had a somewhat unconventional route to QM. I learned QM proper from Von-Neumann - Mathematical Foundations of QM. For me, it was a breeze - just an extension of Hilbert Spaces I learned during my degree. Then I did Dirac - Principles of QM. As a mathematician, it was maddening - not that I could not understand it - in many ways, it was better than Von-Neumann. It was that damn Dirac Delta Function. Mathematically it was exactly as Von-Neumann described in his book - basically, somewhat dubious. So this sent me on a sojourn that took several years learning about exotica such as Distribution Theory (which every mathematician should know IMHO - it makes Fourier Transforms a snap), Rigged Hilbert Spaces, Nuclear Spaces etc. All as a lead up to the critical theorem - the Nuclear Spectral Theorem. I came out the other end fine, and it was interesting. But I would not recommend it to anyone else. If you are interested, learn about it later - but it is better to learn more QM first.

I suspect it is the same here. However, the journey you take is one you choose based on where your 'heart' leads you. I do, however, always recommend to any mathematically inclined person regardless of interest to study DistributionTheory - my favourite book being:
https://www.amazon.com/dp/0521558905/?tag=pfamazon01-20

It will reward your study many times over.

Thanks
Bill