Understanding Zee's QFT: Simplified Gauge Derivative Calculation on Page 236

[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

 

[dextercioby]

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

 

[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.

 

[Bowles]

muchísimas gracias, Rainbow Child!

Very much appreciated, your answer and your effort writting the latex code.

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!

 

[strangerep]

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.

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.

 

[Bowles]

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.)

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.

 

[strangerep]

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.

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.

 

[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 https://www.amazon.com/dp/0071543821/?tag=pfamazon01-20 book is coming out. Sounds like a book for dummies, so I will take look.

 

[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).

 

[masudr]

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

 

[dextercioby]

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

 

[Bowles]

Masudr, look http://www2.physics.utoronto.ca/~luke/PHY2403/References.html" .