[QUE/QFT-15001] QFT-PROBLEM

Prove that the probability per unit volume per unit time that the external potential \[ \vec{A}= (0,0,a\cos\omega t), A_0=0\] creates an electron-positron pair in the vacuum is given by \[ R = \frac{2}{3} \frac{e^2}{4\pi} \Big(\frac{|a|^2}{2} \Big) \omega^2 \Big( 1+ \frac{2m^2}{\omega^2}\Big) \sqrt{1- \frac{4m^2}{\omega^2}} \]