Zee's QFT in a nutshell

1. Jan 20, 2008

Bowles

In Zee's QFT in a nutshell on page 236 between equations (1) and (2), the authors goes to polar coordinates and gets a new gauge derivative.Sure it 's simple, but I can't see how he gets it.

thanks for any help

2. Jan 20, 2008

dextercioby

Make a printscreen and post it here to see what it's about.

3. Jan 20, 2008

Rainbow Child

Equation (1) is

$$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\left(\mathcal{D}\phi\right)^\dagger\mathcal{D}\phi+\mu^2\phi^\dagger\phi-\lambda\let(\phi^\dagger\phi\right)^2\quad (1)$$

Polar coordinates means $$\phi=\rho e^{i\theta},\phi^\dagger=\rho e^{-i\theta}$$, thus

$$\mathcal{D}_\mu \phi=\left(\partial_\mu-i e A_\mu\right)\rho e^{i\theta}=\partial_\mu(\rho e^{i\theta})-i e A_\mu (\rho e^{i\theta})=(\partial_\mu \rho)e^{i\theta}+\rho i e^{i \theta}\partial_\mu \theta-i e A_\mu \rho e^{i\theta}=\left(\partial_\mu \rho+i\rho (\partial_\mu \theta-e A_\mu \right)e^{i\theta}$$

and

$$\left(\mathcal{D}_\mu \phi\right)^\dagger=}=\left(\partial_\mu \rho-i\rho (\partial_\mu \theta-e A_\mu \right)e^{-i\theta}$$

Plug the above equations into (1) and you will arrive at equation (2) of Zee's book.

4. Jan 20, 2008

Bowles

muchísimas gracias, Rainbow Child!

Unfortunately harmless looking equations like these above and all the little tricks you need to know to manipulate them are never explained proper in almost all QFT texts. Stellar example in this regard certainly Peskin&Schroeder, but also Zee lacks here.

Edit:granted, it's only silly product rule above, but why not pointing that out to the reader? Giving such small hints here and there would so much ease the pain reading QFT books!

Last edited: Jan 20, 2008
5. Jan 20, 2008

strangerep

Well, Zee is trying to cover a very large area with a broadbrush (as he explains very
clearly near the start of the book).

But what examples did you have in mind from Peskin&Schroeder? (I learned a lot by
self-studying P&S. Sure, it was hard at times, but there were only maybe one or two cases
where I had to beg others for help. Prof Peskin's online errata list was of course essential.)

Cheers.

6. Jan 21, 2008

Bowles

I know opinions differ on P&S, or on just any QFT texts in general. For me, P&S did not work. As I said before, what bothers me most with QFT texts is that they do not provide careful enough explanations of the little tricks and techniques to get from equation X to equation Y. Given that QFT is a big mix of group theory, complex integrals, Dirac delta functions, tensors, pertubation techniques and physical intuition, it is hard for the beginner to rearrange those equations.

Just look at P&S first chapter, every true beginner is just lost here! It is just a listing of equations and sentences like 'after some calculating it can be seen'. But this calculating includes residue calculus, branch cuts, dirac deltas, operators, surface integrals etc.

7. Jan 21, 2008

strangerep

Hmmm, ok, I see what you mean. Every textbook is difficult if one is not yet proficient
in its prerequisites.

Have you tried Greiner's series of theoretical physics texts? He makes a large effort
not to commit the kind of sins you mention.

8. Jan 22, 2008

Bowles

You right, Greiner is great at that, have not checked him lately, though. Of course not so beautifully streamlined like P&S, or so original, compact and 'big picture' like Zee.

Also heard a Quantum Field Theory Demystified book is coming out. Sounds like a book for dummies, so I will take look.

9. Jan 22, 2008

Haelfix

Thats sort of the point of taking a class in it, where the proffessor actually does it step by step. Yes you can selfstudy, but going through each and every equation takes a tremendous amount of time and effort. Its worth doing it once for one book (eg most people do it with P&S or say with Coleman's lecture notes), but after awhile the algebra becomes rather tedious when you know the main results and have the physical intution behind you.

For instance, I do not remember the details of many of the calculations I once did, just the main results and perhaps the general scheme of how you derive the answer (eg I have to use an ellpitic integral for this problem, or here I must use dimensional regularization).

10. Jan 27, 2008

masudr

I couldn't find Coleman's lecture notes. Can you help me with that?

11. Jan 27, 2008

dextercioby

He might refer to Coleman's known book: "Aspects of symmetry".

12. Jan 27, 2008

Bowles

Masudr, look here.