- #1

- 83

- 1

I have encountered a little problem. I want to show

that the

**explicit**form of the Feynman propagator for massless scalar fields is given by:

[tex]

\begin{align}

G_F(x) & = - \lim_{\epsilon \to +0} \int \dfrac{\mathrm{d}^{4}k}{(2 \pi)^{4}} \dfrac{1}{k^{2} + i \epsilon} \mathrm{e}^{- i k x}

\\

& = - \lim_{\epsilon \to +0} \dfrac{1}{4 \pi^{2}} \dfrac{1}{x^{2} -i \epsilon}

\end{align}

[/tex]

And I would like to do that

**directly**, i.e., without starting from the massive case and considering the limit [itex] m \to 0 [/itex].

I found a script where one can find a derivation of that result,

http://mo.pa.msu.edu/phy853/lectures/lectures.pdf

on pages 58-59.

There, the integration is split up into the imaginary and real part.

But for the imaginary part, the author finds:

[tex]

\begin{align}

G_F,i(x) & = \dfrac{1}{4 \pi^{2} r} \int_{0}^{\infty} \mathrm{d}k \cos(k x_{0}) \sin(k r)

\\

& = \dfrac{1}{16 \pi^{2} r i} \int_{0}^{\infty} \mathrm{d}k \left[ \mathrm{e}^{ik(x_{0}+r)} - \mathrm{e}^{-ik(x_{0}+r)} + \mathrm{e}^{-ik(x_{0}-r)} - \mathrm{e}^{ik(x_{0}-r)} \right]

\\

& = - \frac{1}{8 \pi^{2} r} \left[ \dfrac{1}{x_{0} + r} - \dfrac{1}{x_{0} - r} \right]

\\

& = - \frac{1}{4 \pi^{2} x^{2}}

\end{align}

[/tex]

Here, I don't understand how the integration is performed. I think if you integrate every exponential term it will

**diverge**at [itex]\infty[/itex], but somehow the author ends up with a

**finite**term.

Does anyone understand what is going on here?

Furthermore, I think we should end up with Cauchy's principal value and not just 1/x^2, right?

Maybe, someone has a more "

*elegant*" way of deriving the massless propagator?