How do I simplify the integral in P&S Ch. 6 to obtain Eq. 6.70?

[JohnPhys]

Hey All,

A fellow grad student and I have been working our way through Peskin & Schroeder's QFT book (not for a course), & we seem to have hit a snag in chapter 6.

Has anyone worked through the details of how to complete the angular part of the integral on page 201 that leads to Eq. 6.70? Can anyone point me in the right direction?

Particularly, I'm wondering how to go from
$$\int_0^1 d\xi \int \frac{d\Omega_k}{4\pi}\frac{1}{\left[\xi \hat{k} \cdot p^{\prime} + (1 - \xi) \hat{k} \cdot p \right]^2}$$

to

$$\int_0^1 d\xi \frac{1}{\left[\xi p^{\prime} + (1- \xi) p\right]^2}$$

It *looks* like they just assumed that the angle between k and p as well as k and p' is the same, but I can't think of a way to justify that.

What am I missing?

Any help would be greatly appreciated.

EDIT: I've noticed this has gotten a decent amount of views, with no replies! Should this be in a different forum? Do I need to supply more info? Just let me know. Thanks!

--John

 

[JohnPhys]

Argh

Never mind, it's obvious.

 

[Ratzinger]

After you got me interested, can you tell me how it's done?

By the way I wish there were a permanent Peskin and Schroeder thread where all these 'obvious' calculations and steps between the equations they show would be discussed.
Just for fun I bought that book some time ago and I try once in awhile to go through some pages. I wish it had an appendix 'how to do integrals in QFT' or something like that.

 

[nrqed]

After you got me interested, can you tell me how it's done?

it's ''obvious'' because you may write the denominator as ${\hat k} \cdot (\xi p' +(1-\xi) p )$, all this squared. Then the result follows from the equation just above (after the sentence ''The last two terms are easy to evaluate'').

By the way I wish there were a permanent Peskin and Schroeder thread where all these 'obvious' calculations and steps between the equations they show would be discussed.
Just for fun I bought that book some time ago and I try once in awhile to go through some pages. I wish it had an appendix 'how to do integrals in QFT' or something like that.

A thread would be a good idea...Maybe a ''chapter 1 thread'', a chapter 2 thread and so on.

 

[JohnPhys]

nrqed hit the nail on the head. It's amazing how much clearer things look after a nights sleep!

A P&S section would be quite helpul. As I mentioned, a fellow grad student and I are working our way through that and various other texts.
We've been through most of Chapters 1-5,9, and all of Chapter 6 in pretty good detail (filling in all of the "just a little algebra", "after the smoke clears", & "obviously" steps), but we still have some questions on some of them.

 

[nrqed]

nrqed hit the nail on the head. It's amazing how much clearer things look after a nights sleep!

A P&S section would be quite helpul. As I mentioned, a fellow grad student and I are working our way through that and various other texts.
We've been through most of Chapters 1-5,9, and all of Chapter 6 in pretty good detail (filling in all of the "just a little algebra", "after the smoke clears", & "obviously" steps), but we still have some questions on some of them.

What other books are you looking at?

Iguess that we call always start threads either here or in the quantum physics board (if the questions are more about interpretation than about doing a specific calculation).

I always look for people to discuss QFT with so feel free to post questions/comments/observations (even if they are of the type: I think I understand this concept..here's my take on it...).

I have also three books of solved problems in QFT and Particle Physics which would be nice to go through as a group sometimes.

Regards

Patrick

 

[JohnPhys]

Well, it's not really a "QFT" book, but we've gone through Griffith's particle physics book in a few places (chapters 4,6,7, & 10, the QED and Weak Int.). Along the way, whenever we get stumped by something in P&S, we try looking at Stone (not so helpful), Ryder (too formal for my tastes, but has a good derivation of the Dirac equation in Ch. 3), Itzykson & Zuber (typos and other issues with that one), Kaku, and Weinberg.

This independent study session came about because a friend and I took a class on QFT (1 sem., WAY too little time), and we realized we didn't really understand how to do anything in QED! (too much time talking about renormailzing the charge and the mass as opposed to actually doing a cross-section). So, we just started hacking our way through the QED stuff in Griffiths and Ch.5 of P&S, and expanded the theory and topics where we felt necessary. We ended up really liking P&S on the whole, but there's still a lot of that fragmentation that occurs.

Favorite Quote from P&S: "To make the above more nearly equal..."

What solved problem books in QFT are out there?

 

[nrqed]

Well, it's not really a "QFT" book, but we've gone through Griffith's particle physics book in a few places (chapters 4,6,7, & 10, the QED and Weak Int.). Along the way, whenever we get stumped by something in P&S, we try looking at Stone (not so helpful), Ryder (too formal for my tastes, but has a good derivation of the Dirac equation in Ch. 3), Itzykson & Zuber (typos and other issues with that one), Kaku, and Weinberg.

This independent study session came about because a friend and I took a class on QFT (1 sem., WAY too little time), and we realized we didn't really understand how to do anything in QED! (too much time talking about renormailzing the charge and the mass as opposed to actually doing a cross-section). So, we just started hacking our way through the QED stuff in Griffiths and Ch.5 of P&S, and expanded the theory and topics where we felt necessary. We ended up really liking P&S on the whole, but there's still a lot of that fragmentation that occurs.

Favorite Quote from P&S: "To make the above more nearly equal..."

What solved problem books in QFT are out there?

1 week !

Are you still working through cross section and decay rate calculations and so on?

I like Ryder for pegagogy and Weinberg for rigor but don't like much Kaku.

Halzen and Martin is good too at the intro level. You would probably *love* Greiner; he has books on QED, QCD the weak interaction, QM, E&M, Nuclear Physics, Stat Mech, etc etc. His books show a lot of *very* explicit calculations. He shows all the steps clearly and very explicitly.

Problem books:

Problem Book in Quantum Feild Theory by Radovanovic, Springer Verlag

Selected Problems in Theoretical Physics with Solutions, Di Giacomo et al, World Scientific (here "theoretical physics" means particle physics)

gauge Theory of Elementary Particle Physics: Problems and Solutions by Cheng and Li (as a companion book to their Particle Physics book): Oxford Press

 

[JohnPhys]

Hehe, I think I may have given the wrong impression. The course we took was in the fall, so we've spent an entire semester going through the material I listed.

I'll have to check out Grenier. With Ryder, I just feel like I can't actually *do* anything after going through sections on it. Too removed from actual applications.

So yeah, if you have Q's about P&S Ch. 6 (even problem 6.1!), I could probably help.

 

[nrqed]

Hehe, I think I may have given the wrong impression. The course we took was in the fall, so we've spent an entire semester going through the material I listed.

I'll have to check out Grenier. With Ryder, I just feel like I can't actually *do* anything after going through sections on it. Too removed from actual applications.

So yeah, if you have Q's about P&S Ch. 6 (even problem 6.1!), I could probably help.

You could have a look at the thread "Virtual Particles" on the quantum physics board (look at the last few posts) and offer your opinion, for starters.

And if you have questions on calculatiosn of cross sections or decay rates or on renormalization, I could maybe help.

Regards