Changes

<math>=\frac{1}{(2\pi)^4}\int d^3ke^{-i\mathbf{k}(x-x')}(2\pi i  \frac{e^{ick(t-t')}-e^{-ick(t-t')}}{2k})\Theta</math><br><br>

<math>=\frac{1}{(2\pi)^4}\int d^3ke^{-i\mathbf{k}(x-x')}(2\pi i  \frac{e^{ick(t-t')}-e^{-ick(t-t')}}{2k})\Theta</math><br><br>

<math>=\frac{-2\pi}{(2\pi)^4}\int \frac{k^2dk}{k} \sin\left({ck(t-t')}\right) 2\pi\int_{-i}^i dze^{-ik|\mathbf{x}-\mathbf{x'}|z}\Theta</math><br><br>

<math>=\frac{-2\pi}{(2\pi)^4}\int \frac{k^2dk}{k} \sin\left({ck(t-t')}\right) 2\pi\int_{-i}^i dze^{-ik|\mathbf{x}-\mathbf{x'}|z}\Theta</math><br><br>

<math>=\frac{-1}{(2\pi)^2}\left(\frac{1}{2i|\mathbf{x}-\mathbf{x'|}}\right)2\int dk sin(ck(t-t')) sin(k|\mathbf{x}-\mathbf{x'})\Theta</math><br><br>

<math>=\frac{-1}{(2\pi)^2}\left(\frac{1}{2i|\mathbf{x}-\mathbf{x'|}}\right)2\int dk sin(ck(t-t')) sin(k|\mathbf{x}-\mathbf{x'}|)\Theta</math><br><br>

But the term <math>2\delta(|\mathbf{x}-\mathbf{x}'|+c(t-t'))\rightarrow 0 \quad\forall\quad t>t'</math><br><br>