Solving Schwartz QFT Eqn 5.26 to Get Eqn 5.27

[merrypark3]

Hello.
From Schwartz QFT BOOK,
How could Eqn 5.26 can be Eqn 5.27?

$d \Pi_{LIPS}=(2 \pi) ^{4} \delta^{4}(\Sigma p) \frac{d^{3} p_{3}}{(2 \pi) ^{3}} \frac{1}{2 E_{3}} \frac{d^{3} p_{4}}{(2 \pi)^{3}} \frac{1}{2 E_{4}}$ Eqn(5.26)


$d \Pi_{LIPS}=\frac{1}{16 \pi ^{2}} dΩ ∫ d p_{f} \frac{{p_{f}}^2}{E_{3}} \frac{1}{E_{4}} \delta ( E_{3} + E_{4} - E_{CM})$ Eqn(5.27)

 

[WannabeNewton]

Just integrate over the $\delta$-function and then switch to spherical coordinates in momentum space. Keep in mind $\delta^4 (\Sigma p) = \delta^4 (p^{\mu}_1 + p^{\mu}_2 - p^{\mu}_3 - p^{\mu}_4)$ so separate the $\delta$-function into products over the 3-vectors and the energies.

 

[merrypark3]

Integrate over? In (5.26), there is no integration?

 

[WannabeNewton]

It's implicit.

 

[Avodyne]

Yeah, there really shouldn't be an integral sign in 5.27 if there isn't one in 5.26. Also, p_3 has changed its name to p_f. Also, while 5.27 is Lorentz invariant, he's adopted a specific frame (the CM frame) in 5.27.

 

[merrypark3]

OK. as $\vec{p_{3}}=-\vec{p_{4}}$ , we can insert integration (over $\vec{p_{4}}$ ) in Eqn(5.26) without altering the original. got it.

 

[merrypark3]

thanks. I would ask some more questions about Shwartz QFT textbook. I hope to solve all the exercises of this book within 2 years, though had solved only up to ch.4

 

[WannabeNewton]

I hope to solve all the exercises of this book within 2 years, though had solved only up to ch.4

Cool, well good luck! I'm working through the book as well actually. I'm on ch.7 problems. So it looks like we have the same goals :)

 

[merrypark3]

Good. Good luck! This book is quite well written.

 

[WannabeNewton]

This book is quite well written.

Haha yes, it is the first QFT book I've personally come across that actually feels like a true physics book. It almost feels like cheating having this book in possession when my class's assigned text is (unfortunately) Peskin and Schroeder since the former provides all the intuition that the latter completely lacks, at least in Part I (I haven't even looked Parts II and beyond).

EDIT: actually Aitchison and Hey is a really awesome physics book as well, George Jones told me about it.