  According to the Schwinger mechanism, a uniform electric field brings about
pair productions in vacuum; the relationship between the production rate and
the electric field is different, depending on the dimension of the system. In
this paper, we make an offer of another model for the pair productions, in
which weak values are incorporated: energy fluctuations trigger the pair
production, and a weak value appears as the velocity of a particle there.
Although our model is only available for the approximation of the pair
production rates, the weak value reveals a new aspect of the pair production.
Especially, within the first order, our estimation approximately agrees with
the exponential decreasing rate of the Landau-Zener tunneling through the mass
energy gap. In other words, such tunneling can be associated with energy
fluctuations via the weak value, when the tunneling gap can be regarded as so
small due to the high electric field.
