Peskin and Schroeder derivation of Klein-Gordon propagator

[chern]

In page 30 of book "An introduction to quantum field theory" by Peskin and Schroeder in the derivation of Klein-Gordon propagator, why p^0=-E_p in the second step in equation (2.54). and why change "ip(x-y)" to "-ip(x-y)"? I thought a lot time, but get no idea. Thank you for your giving me an explanation.

 

[strangerep]

In page 30 of book "An introduction to quantum field theory" by Peskin and Schroeder in the derivation of Klein-Gordon propagator, why p^0=-E_p in the second step in equation (2.54). and why change "ip(x-y)" to "-ip(x-y)"?

It took me a long time to figure that out too, when I first studied P+S.

First, look at this 1D integral:
$$
\int_{-\infty}^{+\infty} dp \; e^{-ipx} ~.
$$ If you perform a change of dummy variable $p \to p' = -p$, what do you get?

So in the 2nd step of (2.54), they're just converting the $e^{ip\cdot(x-y)}$ of the 2nd term in the last line on the previous page 29, so that both exponentials are the same, i.e., $e^{-ip\cdot(x-y)}$. (The latter explains "why p^0=-E_p").

 

[chern]

Thank you!