Pair Production in a Uniform Electric Field

We study the Klein-Gordon field coupled with an external uniform vector potential. We compute pair production in a finite time t using the semiclassical approximation, and show that, after the interaction of the Klein-Gordon field with the external potential, when ħ → 0 the average number of produced pairs is zero. There is agreement with the classical limit because the classical limit involves no production of pairs. We compared our results with those of Schwinger. Finally we saw that the random variable N(t)=′′ net number of pairs produced at time t′′ is in the semiclassical limit a stochastic Poisson process.