Deriving Equation (3) from A. Zee's "QFT in a Nutshell" Chapter I.5

[kexue]

In A. Zee "QFT in a nutshell" in chapter I.5 on page 31 how does equation (3) come about? Just the line above this equation, he gives an equation which needs to be rearranged to give (3). It must be embarrassing simply, but I can't see how. Something with the indices goes on. How does the -g-kk/m^2 gets on the numerator, why and how kk/m^2??

If you don't own the book, http://press.princeton.edu/chapters/s7573.pdf" with page 31 and equation (3).

thank you

 

[hamster143]

Try to multiply both sides of the previous equation by

$$-g_{\mu\rho} + k_{\mu} k_{\rho} / m^2$$

 

[kexue]

thanx hamster143!

Another equation bothering me is (5) in chapter I.10 on page 78. It's the matter action expanded to first order in h. I can't see clearly how eq. (4) and (5) go toegther. What happens to the tensor density -g in (4), it does not appear in (5)? What is the first term on the RHS, the matter action where h=0? Any hints?

thank you again

 

[arkajad]

Use $$-g=1+\eta^{\mu\nu}h_{\mu\nu}+O(h^2)$$. Thus
$$(-g)^{-1/2}=1-\eta^{\mu\nu}h_{\mu\nu}/2+O(h^2)$$.

 

[kexue]

Wow that was fast, thanks a lot arkajad!