What are the Mathematical Prerequisites for QFT?

[MathematicalPhysicist]

Can someone please inform me what are the formal mathematical tools used in QFT?

I plan to learn the maths beforehand or in parallel with QFT in the summer, and I don't like how physicists treat maths so that's it.

 

[Fredrik]

You don't need any new math for an introductory QFT course (assuming that you have already studied the usual stuff before that). On the other hand, for mathematically rigorous QFT, you'd need a lot of functional analysis. The theory of distributions is an important part of that. (Take a look at Streater & Wightman and you'll see).

To really understand gauge theories you'd also need differential geometry (including connections on fiber bundles) and stuff about representations of non-abelian Lie groups and the corresponding Lie algebras.

If you're interested in the mathematical stuff, I'd recommend that you start with chapter 2 of Weinberg (the only way to really understand what a "particle" is), and in parallel study the stuff about Lie groups and representations that you need to understand it.

 

[MathematicalPhysicist]

I plan in the summer or when I would have spare time (as if (-:) reading:
Maggiore for its solved problems...
From there I think I will keep going through Weinberg.
Now for Lie Groups and algerbas, I have Brian Hall's textbook, the problem is that he keeps referencing to other textbooks for other proofs and treatments of the theory, which is a problem because I can't buy all of textbooks or lend all of them, can someone advise a few textbooks which cover all or the principal theory of Lie?

 

[Haelfix]

I strongly recommend against learning the mathematical side of QFT until you understand the more heurestic physicist treatment.

Most of the mathematics required for that can be learned on the side (eg group theory, complex analysis, and some differential geometry) and is usually presented hand in hand... Not too difficult to grasp (although the notation is somewhat idiosyncratic)

Once you get into the mathematical treatment, you basically have to relearn everything from scratch and it gets into distribution theory and some sophisticated algebra (c* algebras, moyal products, and so forth) as well as more advanced concepts in differential geometry (like jet bundles) and functional analysis (a priori estimates, Schwarz spaces etc etc).

Yes, you will be frustrated for awhile with the physics treatment, but its imo essential in order to develop some semblance of an intuition for what's going on. Everyone's felt the same way.

 

[MathematicalPhysicist]

I just feel I don't know what's going on there.
I can do the rudimentary tasks in each textbook, but I want to understand the mathematical methods I am using, not just use them.
And ofcourse understand the physics.

 

[vitruvianman]

Can someone please inform me what are the formal mathematical tools used in QFT?

I plan to learn the maths beforehand or in parallel with QFT in the summer, and I don't like how physicists treat maths so that's it.

You can have a look at Prof. Sidney Coleman's lectures.

http://www.physics.harvard.edu/about/Phys253.html

Here are his legendary lectures about QFT

 

[Haelfix]

I'm going to say something that might offend a few of my colleagues who happen to work on the mathematical side of QFT, but in general the way things work out is the following:

The more rigorous you get when dealing with this subject, the further from physical reality you go. Worse, the more rigorous you get, the more time you will spend trying to understand it (and believe me, it is not simple).

Just about every physicist I know has had the exact same problem you are experiencing sometime in grad school. In my case, I wasted about a month of my life pouring through the mathematical side with identically zero results and I managed only to confuse myself and to ask increasingly abstract questions that seem to have no end, and further stretched my mathematics acumen beyond my existing grad level competency. Rigor mortis if you will.

Learning the standard way (see Coleman lectures above) is about two orders of magnitude easier, further reaching and possibly indicative of more mathematical structure than simply fiddling around with modifying Wightman or Haag-Ruelle axioms in order to create a single four dimensional field theory that actually reproduces something that we know is true since the 50s.

The bottomline is that the mathematics themselves are unknown, extremely sophisticated/hard as well as having plausibility problems, indeed many of the socalled texts and papers on rigorous qft are conceivably in the wrong direction altogether. I'm not saying that this is a dead end, one day the mathematics will be ironed out (it has too), but we are objectively far from that point. What we do know, is that the physicist treatment is at least approximately correct, and as such is pretty much required material for everyone.

I recommend Weinberg volume 1 for a very logical presentation of the material (be warned it is harder than most qft texts precisely b/c of this fact). If that still does not satisfy you, well seek out Streeter-Wightman, but do keep in mind that material is sort of a siren call and merely a first pedestrian step into a tunnel that is quite dark.

Your best bet is to decide what exactly is confusing you about the physics treatment, and then ask questions.

 

[meopemuk]

I recommend Weinberg volume 1 for a very logical presentation of the material (be warned it is harder than most qft texts precisely b/c of this fact).

I totally agree. This is the best textbook ever written about QFT. Unfortunately, Weinberg omits many important subjects (which is probably the reason why people find it difficult to read). For example, I couldn't find any mention of the Fock space, which is kind of odd.

My suggestion is to complement Weinberg's book with reading journal articles cited in the text. Usually these are classical references, which fill most gaps.

Eugene.

 

[MathematicalPhysicist]

Well, anyway I will continue in maths graduate studies and not physics.

Still I don't understand from you why not learn the maths hand in hand with the physics, besides the fact that it will take more than the summer which awaits me.

 

[meopemuk]

Still I don't understand from you why not learn the maths hand in hand with the physics, besides the fact that it will take more than the summer which awaits me.

It depends on your goal. If you want to learn mathematics of QFT as a set of abstract definitions, theorems, proofs, etc. that's one thing. There are many books on "axiomatic" or "algebraic" QFT. However, as Haelfix said, there is no complete and universally accepted axiomatic framework. It is often not clear how (if) these mathematical constructions are relevant to observable physics.

On the other hand, if you want to understand the physical meaning of QFT and how it helps to solve real life problems, then Weinberg's book is the best choice. He also gives a healthy dose of relevant and rigorous mathematics, but not more than is necessary to illustrate physical ideas and to provide computational tools.

Eugene.

 

[Fredrik]

You could probably spend a year studying functional analysis and introductory stuff about rigorous QFT, and still not have any use for what you've learned in a traditional QFT course. These subjects are that different.

You mentioned Brian Hall's book. I've studied parts of that too. I felt that Chris Isham's book "Modern differential geometry for physicists" taught me most of the things I wanted to know that Hall didn't cover. It could also be a good idea to get a book on differential geometry that covers some Lie group and Lie algebra stuff. "Introduction to smooth manifolds" by John Lee is an excellent book, with a chapter and several smaller sections about Lie groups and Lie algebras. Unfortunately I haven't read those chapters, but if they're anything like the parts I've read, you won't be disappointed.

And I still recommend that you start with Weinberg chapter 2, and study the math you need for that in parallel.

The books by Srednicki and Zee are very popular too, so you might want to check them out too. I haven't read them myself.

 

[MathematicalPhysicist]

The problem is that I don't want to be given formulas and having believing them to be true on faith, which is the way I so far have been learning QM2 course, the lecturer just gives us formulas and the steps towards arriving at them he briskly at a fast pace describing them without any meaning, all this course seems to be plotting equations.
After I am finished with this course I will have learned nothing, and will remember nothing from it.

 

[martin_blckrs]

I subscribe to every single word Haelfix wrote. Exactly the same experience in my case - I've been doing a lot of math during my studies, including Functional Analysis, Differential Geometry,... and I was very keen to understand the mathematical structure of QFT. I spent a lot of time trying to understand QFT on a rigorous level and the outcome was that the further I went, the less I understood. You should really first understand the physical concepts behind it, in order to later deal with the mathematical subtleties involved. It's very hard (close to impossible) to do it the other way round.

So my advice would be: start with some easy to read book, like Peskin&Schroeder or Zee and then proceed to whatever my be interesting to you.

 

[DarMM]

If that still does not satisfy you, well seek out Streeter-Wightman, but do keep in mind that material is sort of a siren call and merely a first pedestrian step into a tunnel that is quite dark.

Very good way of putting it.

Personally I found that I did come out the other end of the tunnel and eventually understood it all, however it was probably intellectually the hardest thing I've ever done. The only upshot is that in the end you understand that renormalization isn't a pile of nonsense and actually is mathematically rigorous.

There might be an analogy with website design here, QFT is like a website that provides factually correct information but isn't W3C complient. Most don't care, it is perfect at what you want it for. Others want it to be technically perfect as well.

I'd agree with others here, you're better at trying to understand it from Peskin and Schroder, e.t.c. first. It can only ease the transition to the hard stuff later, if you still want to do that.

I just thought I'd say one thing

The bottomline is that the mathematics themselves are unknown, extremely sophisticated/hard as well as having plausibility problems, indeed many of the socalled texts and papers on rigorous qft are conceivably in the wrong direction altogether. I'm not saying that this is a dead end, one day the mathematics will be ironed out (it has too), but we are objectively far from that point.

I would rather say that the mathematics may not be natural. There's no problem with any of the papers and texts being incorrect, they achieve their aim in the end. The only real problem is that they're perhaps not the right approach. That is, even though they are rigorous formulations of QFTs, maybe there is a better mathematical language that expresses QFT rigorously and concisely, rather than rigorously with nine billion pages of measure theory as is currently the case.

 

[vitruvianman]

mathphysicist u can also look at scholarpedia the gauge theories page, written by gerard 't hooft