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## Main Question or Discussion Point

Hi,

I'm teaching myself QFT. I don't understand some of the calculations described on pages 29 and 30 of Peskin and Schroeder's book on QFT.

Making a digression, the propagator can be written in momentum space as

[tex]\Delta_{F}(p) = \frac{1}{2E_{p}}\left[\frac{1}{p^0 - (E_p - i\epsilon)}-\frac{1}{p^0 + (E_p - i\epsilon)}\right][/tex]

(Feynman prescription)

Plugging this back into the expression for the Green's function [itex]G(x-x') = \int \frac{d^4 p}{(2\pi)^4}e^{-ip\cdot(x-x')}\frac{1}{(p^0)^2 - E_{p}^2}[/itex] we get

[tex]\Delta_{F}(x-x') = \int \frac{d^{3}p}{(2\pi)^3} \frac{1}{2E_{p}}\int \frac{dp^{0}}{2\pi} e^{ip\cdot(x-x')}\left[\frac{1}{p^0 - (E_p - i\epsilon)}-\frac{1}{p^0 + (E_p - i\epsilon)}\right][/tex]

This doesn't give me what I want, but I think I am close to it as there is a -i and two theta functions that enter into the final expression if this is integrated.

Specifically,

1. I don't understand how the extra factor of {-i} in the expression in the quote box above (from Peskin & Schroeder) comes in.

2. What is the motivation for the choice of the contour shown on page 30?

3. What is the significance of [itex]x^0 > y^0[/itex].

Thanks!

I'm teaching myself QFT. I don't understand some of the calculations described on pages 29 and 30 of Peskin and Schroeder's book on QFT.

The next step is where I have a problem...we can write [itex][\phi(x),\phi(y)] = \langle 0|[\phi(x),\phi(y)]|0\rangle[/itex]. This can be rewritten as a 4 D integral as follows, assuming for now that [itex]x^{0} > y^{0}[/itex]:

[tex]\langle 0|[\phi(x),\phi(y)]|0\rangle = \int \frac{d^{3}p}{(2\pi)^3}\left\{\frac{1}{2E_p}e^{-ip\cdot(x-y)}|_{p^0 = E_{p}} + \frac{1}{-2E_p}}e^{-ip\cdot(x-y)}|_{p^0 = -E_{p}}\right\}[/tex]

I think I'm making a mathematical mistake somewhere, but I haven't been able to figure it out yet.[tex]=_{x^0 > y^0} \int \frac{d^{3}p}{(2\pi)^3}\int \frac{dp^0}{2\pi i}\frac{-1}{p^2-m^2}e^{-ip\cdot(x-y)}[/tex]

Making a digression, the propagator can be written in momentum space as

[tex]\Delta_{F}(p) = \frac{1}{2E_{p}}\left[\frac{1}{p^0 - (E_p - i\epsilon)}-\frac{1}{p^0 + (E_p - i\epsilon)}\right][/tex]

(Feynman prescription)

Plugging this back into the expression for the Green's function [itex]G(x-x') = \int \frac{d^4 p}{(2\pi)^4}e^{-ip\cdot(x-x')}\frac{1}{(p^0)^2 - E_{p}^2}[/itex] we get

[tex]\Delta_{F}(x-x') = \int \frac{d^{3}p}{(2\pi)^3} \frac{1}{2E_{p}}\int \frac{dp^{0}}{2\pi} e^{ip\cdot(x-x')}\left[\frac{1}{p^0 - (E_p - i\epsilon)}-\frac{1}{p^0 + (E_p - i\epsilon)}\right][/tex]

This doesn't give me what I want, but I think I am close to it as there is a -i and two theta functions that enter into the final expression if this is integrated.

Specifically,

1. I don't understand how the extra factor of {-i} in the expression in the quote box above (from Peskin & Schroeder) comes in.

2. What is the motivation for the choice of the contour shown on page 30?

3. What is the significance of [itex]x^0 > y^0[/itex].

Thanks!

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