Question 5.10: The Proof of (5.148) Prove (5.148) by replacing (q + ξP)µ by......
The Proof of (5.148)
Prove (5.148)
\int \mathrm{d}^4 x \exp (\mathrm{i} x \cdot(q+\xi P)) \delta\left(x^2\right) \varepsilon\left(x^0\right)=\mathrm{i} 4 \pi^2 \delta\left[(q+\xi P)^2\right] \varepsilon\left(q_0+\xi P_0\right) (5.148)
by replacing (q+\xi P)_{\mu} by k_{\mu} and evaluating the Fourier transformation of (5.148), i.e., by proving that
\frac{\mathrm{i}}{4 \pi^{2}} \int\limits \exp (-\mathrm{i} k \cdot x) \delta\left(k^{2}\right) \varepsilon\left(k_{0}\right) \mathrm{d}^{4} k=\delta\left(x^{2}\right) \varepsilon\left(x_{0}\right). (1)
Performing the k_{0} integration turns the left-hand side into
\begin{aligned} I & =\frac{\mathrm{i}}{4 \pi^{2}} \int\limits \exp \left(-\mathrm{i}\left(k_{0} x_{0}-{k} \cdot {x}\right)\right) \delta\left(k_{0}^{2}-{k}^{2}\right) \varepsilon\left(k_{0}\right) \mathrm{d} k_{0} \mathrm{~d}^{3} k \\ & =\frac{\mathrm{i}}{4 \pi^{2}} \int\limits \exp \left(-\mathrm{i}\left(k_{0} x_{0}-{k} \cdot {x}\right)\right) \delta\left(\left(k_{0}-k\right)\left(k_{0}+k\right)\right) \varepsilon\left(k_{0}\right) \mathrm{d} k_{0} \mathrm{~d}^{3} k \end{aligned}
\begin{aligned} = & \frac{\mathrm{i}}{4 \pi^{2}} \int \frac{1}{2 k}\left[\exp \left(-\mathrm{i} k\left(x^{0}-r \cos \theta\right)\right)\right. \\ & \left.-\exp \left(+\mathrm{i} k\left(x^{0}+r \cos \theta\right)\right)\right] \mathrm{d}^{3} k & (2) \end{aligned}
with
\begin{aligned} & k:=\sqrt{{k}^{2}}, \\ & r=\sqrt{{x}^{2}}, \\ & {k} \cdot {x}=k r \cos \theta . & (3) \end{aligned}
We evaluate the angular integration
\begin{aligned} I= & \frac{\mathrm{i}}{2 \pi} \int\limits_{0}^{\infty} \mathrm{d} k \frac{k^{2}}{2 k}\left[\frac{1}{\mathrm{i} k r}\left(\exp \left(-\mathrm{i} k\left(x^{0}-r\right)\right)-\exp \left(-\mathrm{i} k\left(x^{0}+r\right)\right)\right)\right. \\ & \left.-\frac{1}{\mathrm{i} k r}\left(\exp \left(\mathrm{i} k\left(x^{0}+r\right)\right)-\exp \left(\mathrm{i} k\left(x^{0}-r\right)\right)\right)\right] \\ = & \frac{1}{4 \pi r} \int\limits_{-\infty}^{\infty} \mathrm{d} k\left[\exp \left(-\mathrm{i} k\left(x^{0}-r\right)\right)-\exp \left(\mathrm{i} k\left(x^{0}+r\right)\right)\right] & (4) \end{aligned}
and obtain a difference of \delta functions, which yields the postulated result:
\begin{aligned} I & =\frac{2 \pi}{4 \pi r}\left[\delta\left(x^{0}-r\right)-\delta\left(x^{0}+r\right)\right] \\ & =\left.\delta\left(x^{2}\right)\right|_{x^{0}=r}-\left.\delta\left(x^{2}\right)\right|_{x^{0}=-r} \\ & =\delta\left(x^{2}\right) \varepsilon\left(x_{0}\right) . & (5) \end{aligned}