Need advice with learning renormalization groups

[Alesak]

Hi,
I will be writing my bechelors thesis on the application of renormalization groups and p-adic analysis on stock markets. The problem is that I don't understand it yet. I need advice on the best path how to learn it.

What I know already:
- I've read most of Mac Lanes Algebra, so I know finite-dimensional linear algebra pretty well, some category theory and some group and ring and field etc. theory.
- Basic real analysis, this includes stuff like implicit function theorem, total derivatives and the like.
- Point set topology and some very beginning stuff in algebraic topology
- Smooth manifolds, as in Lee. This does not include more advanced stuff like Riemannian things.
- ODEs but not PDEs
- I'm beginning with Lie theory

What is the optimal path for me next? I guess I will need analytical mechanics, but as I don't know any physics the book from Spivak seems best for me, although the price is bloody.

Also I don't mind learning some physics in the process, like QM and QFT. Unfortunately I don't know any functional or complex analysis. Can you guys comment on this?

Thanks

 

[bhobba]

Well that sounds like a tough ask for a bachelors thesis - I would think it more of Masters level.

Anyway I would start with trying to understand Renormalization:
http://arxiv.org/abs/hep-th/0212049

I think you have enough actual math - its bringing it together that will be hard. The math of this stuff reaches rather dizzying heights.

The following may also be of interest:
http://arxiv.org/pdf/cond-mat/0008300v1.pdf

Thanks
Bill

 

[Alesak]

Thanks a lot, especialy for the first link. I got a whole year to write the thesis, so I hope it will be enough time. Any further comments or advice appreciated.

 

[kloptok]

First of all, I don't know anything about finance or stock markets but I must say it sounds interesting to apply renormalization group techniques to stock markets! Could you perhaps tell us more about your project? It will also be easier to give advice then.

The renormalization group is normally encountered in QFT so I guess it would be a good idea to familiarize yourself a bit with this. If I understand correctly you have not read any QM or analytical mechanics, which of course makes QFT a hard topic. But probably many of your references on the renormalization group will refer to QFT so you will have a hard time following these unless you at least have an inkling of what QFT is about. I am sorry to say that you have a lot of reading to do if you need to study QM, analytical mechanics and QFT in your bachelor's thesis.

 

[Alesak]

First of all, I don't know anything about finance or stock markets but I must say it sounds interesting to apply renormalization group techniques to stock markets! Could you perhaps tell us more about your project? It will also be easier to give advice then.

Well, this seems to be a very practical paper analysing 1987 crash. Other than that, there is a book "Didier Sornette - Why Stock Markets Crash" that treats this topic very nicely. Would make a good read for physicists, it seems.

The renormalization group is normally encountered in QFT so I guess it would be a good idea to familiarize yourself a bit with this. If I understand correctly you have not read any QM or analytical mechanics, which of course makes QFT a hard topic. But probably many of your references on the renormalization group will refer to QFT so you will have a hard time following these unless you at least have an inkling of what QFT is about. I am sorry to say that you have a lot of reading to do if you need to study QM, analytical mechanics and QFT in your bachelor's thesis.

Reading a lot of QM is fine with me, since I find physics very interesting subject in itself. As an economist, unfortunately I don't have access to standard physics education, so what I need is advice from you more experienced gyus what to read and what not.

I already got "The Variational Principles of Mechanics" by Lanczos, but in the first chapter he was talking all the time about manifolds without ever saying it(once, to be fair). So maybe I'll get that book from Spivak, as it is more to the point and shows what's under the hood directly.

I understand I will need some functional analysis - I already got Functional Analysis by Bachman and Narici and will read it through the summer. What other maths do you recommend I read?

Also, how to get into QM with my background? I've seen Quantum Mechanics: Concepts and Applications by Zettili which seems pretty good, but at the beginning they talk about black body radiation as if reader knows it.

 

[kloptok]

I wouldn't worry so much about the maths. Sure, you need functional analysis to truly understand what's going on in QM, but you surely do not need it just to grasp the concepts and do even serious calculations. Most physics undergrads probably don't take any functional analysis at all. Try to find a good book on QM instead, I've heard that Ballentines book is supposed to be good and quite rigorous. Otherwise Sakurai is good (these are both a bit more advanced than your average beginner's QM book but it seems like you have a quite solid math background so it could work). And remember that you are reading QM mostly to be able to understand QFT, not to learn it on its own!

If you mainly want to understand QFT and don't care so much about calculations I would recommend Zee's "QFT in a nutshell", it is quite talkative but deals with a lot of the concepts in QFT. If you want to get into the calculations of QFT you could take a look at the standard introductory texts by Srednicki ("Quantum Field Theory") or Peskin&Schroeder ("Introduction to Quantum Field Theory"). I prefer Srednicki but P&S is really a standard text that you will find anywhere.

I don't know your background but you might be confused by subjects from analytical mechanics used in QFT. I don't think you have to study analytical mechanics in detail but you should know the basics. I don't really have a good reference for this.

You should make it your goal to understand why RG (Renormalization Group) techniques are important in QFT rather than trying to understand QFT thoroughly. And this you furthermore do just to be able to understand how to use it in economy. QFT is a very big subject and if you don't keep your eyes on the ball you will easily get lost.

EDIT: I looked at your link and have a further thought I would like to share. RG techniques are used in high energy physics and statistical physics, and actually the pioneer of the RG was Ken Wilson, a condensed matter/statistical physics guy. The similarities are probably larger between economy and statistical physics, so concentrate on looking at RG and its use in condensed matter/statistical physics. Personally I am more inclined towards particle physics and high energy so this is where my advice (above) mainly comes from. Hopefully you can get some more appropriate advice from someone with more knowledge of RG techniques in statistical physics.

 

[Alesak]

Thanks a lot for recommendations. Zee looks interesing and is not expensive, will be getting that. For QM I've found "Essential Quantum Mechanics" by Gary Bowman, which seems to fit the bill.

Any comments from people with statistical mechanics background would be hugely appreciated.

 

[kloptok]

I found some lecture notes from a Cambridge University course on statistical field theory, available at http://www.damtp.cam.ac.uk/user/rrh/notes/qstat.pdf

There seems to be some stuff in there about renormalization group equations, fixed points etc. which might help. Some literature tips in the beginning also.