Should I take a group theory course before QFT?

[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