Solve Peskin's QFT Eqn 2.54 with Hints

[ananya J]

QFT Peskin p.30 eqn 2.54

Homework Statement

i am perplexed with eqn 2.54 peskins introductory qft. just can't make out how to arrive at it from the previous step. i think that there are dirac delta funtions involved but simply can't make it out. can somebody help? provide some hint? thanks in advance for ur time

3. The Attempt at a Solution
$$\int\ \frac{d^3p} {(2\pi)^3}\ \{ \frac {1}{2E_p}\ e^{-ip.(x-y)}\left|_{p^0 = E_p}\ +\ \frac {1}{-2E_p}\ e^{-ip.(x-y)}\left|_{p^0 = -E_p}\ \}= \int\ \frac{d^3p} {(2\pi)^3}\ \int\ \frac{dp^0} {2p^0}\ e^{-ip.(x-y)}\ \{ \delta (p_0-E_p) +\delta (p_0+E_p)\ \}$$ $$= \int\ \frac{d^3p} {(2\pi)^3}\ \int\ dp^0\ e^{-ip.(x-y)}\ \delta(p^2-m^2)$$

dont know if iam on the right track.pls correct me if am wrong.

 

[Thaakisfox]

It would be good, if you would write down the equation you want to prove, since I don't have the mentioned book...

 

[daschaich]

What P&S are doing in Eqn. 2.54 is re-writing a three-dimensional integral as a four-dimensional integral:
$$\int\frac{d^3p}{(2\pi)^3}\frac{1}{2E_p}[\exp(-ip\cdot(x - y)) - \exp(ip\cdot (x - y))] = \int\frac{d^3p}{(2\pi)^3}\int\frac{dp^0}{2\pi i}\ \frac{-1}{p^2 - m^2}\exp(-ip\cdot(x - y)),$$
where $$x^0 > y^0$$.

What I would do to understand this is start from the latter form and perform the $$p^0$$ integral. Break up the denominator into
$$p^2 - m^2 = (p^0)^2 - \textbf p^2 - m^2,$$
which has poles at
$$p^0 = \pm \sqrt{\textbf p^2 + m^2} = \pm E_p.$$
Contour integration should produce the first expression in the original post [which P&S give as an intermediate step] without too much trouble.

 

[ananya J]

i jumped into conclusions before reading the text further. sorry. anyways thanks so much for ur time & help.